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\begin{document}

\title[Direct Sums and Other Objects with Matroids]
{Direct Sums and Tutte Functions of Graphs and Other Objects
with Matroids}

\author{Seth Chaiken}
\address{Computer Science Department\\
The University at Albany (SUNY)\\
Albany, NY 12222, U.S.A.}
\email{\tt sdc@cs.albany.edu}

\begin{abstract}

The polynomials expressing the conditions that
a parametrized $P$-ported Tutte function is well-defined
for matroids
each have one factor $I(Q)$, where $Q$ is a $P$-quotient.
Since the elements are
labelled, the methodology applies
to objects such as graphs with ports
for which similar theorems can be proven.
We abstract graphs to objects that have
ported Tutte functions
because they have matroids, but might
have different Tutte function values
on two objects with the same matroid.
Two new theorems are given
and are used to generalize known conditions
for graphs without port edges to graphs with port edges.
In some cases, the condition becomes
$I(Q)r+I(Q')s=0$, where $r,s$ are
polynomials in $x,y,X,Y$.
The abstraction is then used to
characterize ported Tutte functions
of an object or a combination of two
objects, whose matroid or oriented
matroid is a direct matroid or
oriented matroid sum.
This extends with ports some
known strong Tutte
function and multiplicative
Tutte function results.
\end{abstract}

\subjclass[2000]{Primary 05B35; Secondary 05C99, 05C15, 57M25, 94C05}

\keywords{Tutte function, Tutte polynomial}

\thanks{Version of \today.}

\maketitle
\pagestyle{headings}

%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}

When $P=\emptyset$, the facts about separator-strong Tutte functions
of matroid direct sums easily follow from the formula
$T(M^1\oplus M^2)T(\emptyset)=T(M^1)T(M^2)$.  For example,
$T$ is strong if and only if $T(\emptyset)=T(\emptyset)^2$.
The theory of separator-strong Tutte functions of graphs covered
in \cite{Ellis-Monaghan-Traldi}  follows from the fact that any
minor closed family of graphs (see below) $\mathcal{G}$ is partitioned into
subfamilies $\mathcal{G}_k$, each with just one indecomposible,
$E_k$, the edgeless graph with $k$ unlabelled vertices,
if $\mathcal{G}_k\neq\emptyset$.
Tutte function formulas for disjoint and one-point graph unions,
and the conditions for strongness (defined $T(G^1)T(G^2)=T(G)$ if
matroids $M(G^1)\oplus M(G^2)=M(G)$) are then
derived \cite{Ellis-Monaghan-Traldi}
in terms of the values $\alpha_k=I(E_k)$.  Life is simple because
matroid $M(E_k)=\emptyset$ for all $k$.

The corresponding facts become more complex when the definitions
are naturally extended to $P$-ported matroids and graphs
or to vertex labelled graphs.  As with
matroids, a $P$-ported graph $G$ has some of the edges in $P$ and the rest,
$E(G)$, satisfy $E(G)\cap P=\emptyset$.  Deletion, contraction,
$P$-minors, $P$-families and $P$-quotients (i.e., indecomposibles)
are also defined as they are for $P$-ported matroids or oriented matroids.
As in \cite{Ellis-Monaghan-Traldi},
deletion of an isthmus (i.e., coloop in the matroid) and
contraction of a loop is forbidden within Tutte equations.

The main difficulty is illustrated by the following example.
Let
$G$ be the circle graph of the five edges
ordered $(e,p,f,q,r)$ and take $P=\{p,q,r\}$.
So, $e$ and $f$ are a series pair connected to $P$, but
the $P$-quotient graphs $Q_1=G/e\setminus f$ and
$Q_2=G/f\setminus e$ are different graphs, even though they have
the same matroid.  $Q_1$ is the path $qrp$ and $Q_2$ is the path $pqr$.
Function $T$ might satisfy \eqref{TA}
and \eqref{TSSM} even if $T(Q_1)=I(Q_1)\neq I(Q_2)=T(Q_2)$.
So, if this $G\in\mathcal{G}$, a necessary condition
for $T$ to be a $P$-ported
(separator-strong, as always) Tutte function would be
\[
I(Q_1)(x_ey_f - y_fX_e) = I(Q_2)(x_fy_e - y_eX_f).
\]
This equation does not have the form of
those in the ZBR theorem for graphs \cite{Ellis-Monaghan-Traldi}
because the latter's equations, like the equations in
Theorem \ref{BigTheorem}, each has a single factor
$I(Q)$ depending on one indecomposible.

The example relies on the elements of $P$ being labelled.
This leads us to formulate a extension of Tutte function theory
for vertex labelled graphs.
When the outcome, Theorem \ref{ZBRWellBehaved} is applied to graphs as in
\cite{Ellis-Monaghan-Traldi}, we get Corollary \ref{PZBRGraphCor2}
which demonstrates that the above example illustrates the
\emph{only} situation where the $P$-ported ZBR equations
of Theorem \ref{BigTheorem} must be modified.

Figure \ref{c4p2TwoFigure} illustrates
the same phenonemon as the first, in a smaller graph, when
the objects in $\mathcal{G}$ are graphs whose vertices are
labelled by disjoint sets.  Again, two different graphs have the
same oriented matroid.

\begin{figure}
\input{c4p2Two.pdf_t}
\caption{\label{c4p2TwoFigure}
Two Tutte computation trees for the same graph with set labelled
vertices.  The indecomposibles at the bottom of the first tree
are named $Q_1$, $Q_2$ and $Q_3$ from left to right.  For non-separators,
contraction
edges are slanted left and  deletion edges are slanted right.  An edge
representing a separator is drawn vertically.}
\end{figure}

The expressions from the two Tutte computation trees in Figure
\ref{c4p2TwoFigure}
are
\[
I(Q_1)x_ex_f+I(Q_2)x_ey_f+I(Q_3)y_eX_f
\]
and
\[
I(Q_1)x_ex_f+I(Q_3)x_fY_e+I(Q_2)y_fX_e=
I(Q_1)x_ex_f+I(Q_2)y_fX_e+I(Q_3)x_fY_e,
\]
which are equal if and only if
\begin{equation}
\label{BadZBRExampleEq}
I(Q_2)(y_fX_e - x_ey_f)=
I(Q_3)(y_eX_f - x_fY_e).
\end{equation}
$Q_2$ and $Q_3$ are isomorphic as edge-labelled graphs
but are different when the vertex
labels are present.

There are two  complications introduced
into $P$-families of matroids when $P\neq\emptyset$.
First, one matroid
might have more than one $P$-quotient, i.e., indecomposible.
The most simple
example is a dyad matroid composed of one port and one
non-port element; and its $P$-minors.
Therefore, minimal $P$-minor closed families might have
more than one indecomposible.  Some will share
matroids or oriented matroids and others will not.
The second, which also occurs with the
minor closed families of graphs \cite{Ellis-Monaghan-Traldi}
in the original $P=\emptyset$ form, is that
the family is partitioned into disjoint $P$-minor closed
subfamilies.  Each subclass has its own indecomposibles,
$E_k$ in the case of graphs.  Again,
indecomposibles in different subclasses
share matroids as do
the $E_k$ all of whose matroids are $\emptyset$.
When $P\neq 0$, the indecomposibles of different
subclasses might or might not share matroids
or oriented matroids.

The ZBR theorem for graphs in \cite{Ellis-Monaghan-Traldi}
has conditions analogous to those in Corollary \ref{ZBRmatroids},
except the factor $\alpha$ is replaced by $\alpha_k=I(E_k)$
depending on the subfamily.

$P$-ported matroids or oriented matroids can be combined
by matroid direct sum $\oplus$.  Graphs can be combined
by disjoint union $\amalg$ or by a one-point union;
then each such combination $G$ of $G^1$ and $G^2$, if defined,
satisfies $M(G)=M(G^1)\oplus M(G^2)$.
Zaslavsky's definition\cite{MR93a:05047} for
matroids is immediately extended:

\begin{definition}
A \textbf{strong} $P$-ported Tutte function $T$ on a $P$-family
$\mathcal{C}$ of matroids
or oriented matroids satisfies $T(M^1)T(M^2)=T(M^1\oplus M^2)$
when $M^1, M^2$ and $M^1\oplus M^2$ are all in $\mathcal{C}$.
\end{definition}

Note that such a strong Tutte function is a separator-strong
Tutte function with $X_e=T(U^e_1)$ and
$Y_e=T(U^e_0)$ for all $e\in E(\mathcal{N})$.

We will give extensions of definitions of strong Tutte functions
and of multiplicative Tutte functions of graphs below when
we define $P$-families of objects with matroids or oriented
matroids.

\section{Objects with Matroids or Oriented Matroids}

It is useful to think that a $P$-ported Tutte computation tree may have
objects $N$ for its node labels such as graphs.
Each object $N$ has an associated
a $P$-ported matroid or oriented matroid $M(N)$.
Elements are defined $S(N)=S(M(N))$,
each $p\in S(N)\cap P$ is called a port, and
$E(N)=S(N)\setminus P$.  Loops, coloops and non-separators of $N$
are characterized by their status in $M(N)$.  So we
say $N$ is an \emph{object with a matroid or an oriented matroid}.
Often, but not always, $N$ will be some matroid or oriented
matroid representation.

Contraction $N/e$ and deletion $N\setminus e$
of object $N$ are defined when $e\in E(N)$, and
$e$ is not a coloop in $M(N)$ and
$e$ is not a loop in $M(N)$, respectively.
Under those conditions, $M(N/e)=M(N)/e$ and
$M(N\setminus e)=M(N)\setminus e$ (as matroids or
oriented matroids).
Thus $P$-minors are defined,
and an indecomposible or $P$-quotient
is a $P$-minor $Q$ for which $S(Q)=S(M(Q))\subseteq P$.

\begin{definition}
\label{OMOMdef}
An $P$-ported object $N$ with a matroid or oriented matroid
is described above together with $M(N)$, $E(N)$, $S(N)$,
$P$-minors, etc.
A \textbf{$P$-family of objects $\mathcal{N}$}
is a $P$-minor closed class of
objects with matroid or oriented matroids.
\end{definition}

Tutte computation trees are defined for such $N$.
The matroid $M(N)$ of course
constrains the structure of these trees.
It is possible
(as when the edgeless graphs $G_k$ have different vertex sets but all
$M(G_k)=\emptyset$)
for different objects,
even different indecomposibles, to have the same
matroid or oriented matroid.  It also natually occurs
that
$N/e\setminus f$ $\neq$
$N/f\setminus e$ (as objects) even though
$M(N)/e\setminus f$ $=$
$M(N)/f\setminus e$.  The latter equation when
$e,f$ are in parallel or in series (see Proposition
\ref{SameMinorProp}) is critical to the matroid ZBR theorems.
It is also conceivable that $N/e/f\neq N/f/e$ or
$N\setminus e\setminus f\neq N\setminus f\setminus e$.

Since every $P$-minor $N'$ of $N$ has matroid
or oriented matroid $M(N')$ the same as the corresponding
minor of $M(N)$, we observe:

\begin{lem}
\label{ObjectTreeValueLemma}
The Tutte computation trees for $M(N)$ are in a one-to-one
correspondance with the
Tutte computation trees for $N$
where corresponding trees are isomorphic.  In each
isomorphism,
corresponding
branches have the same labels ``$e$-contracted'' or
``$e$-deleted'' with $e\in E(N)=E(M(N))$, and a node
labelled $N_i$ in the tree for $N$ corresponds to
a node labelled $M(N_i)$ in the tree for $M(N)$.

Each computation tree value is given by
the activities expansion \eqref{PAE} reinterpreted for
objects.
\end{lem}

We can still talk about Tutte decompositions and a
Tutte computation tree for $N$ even  without
a Tutte function.  If we are given values $I(Q)$
for the indecomposibles, each Tutte computation tree
for $N$ yields a value in the $R$-module generated by
the $I(Q)$.
The Tutte decompositions, and the universal
Tutte polynomial (if it exists!)
of each $N\in \mathcal{N}$
are determined by $M(N)$ and the indecomposibles, i.e., $P$-quotients
$Q$ in $N$, which of course satisfy $Q \in \mathcal{N}$.
This generalizes Zaslavsky's discussion\cite{MR93a:05047}.

\begin{definition}[Separator-strong $P$-ported Tutte function on objects]

Function
$T$ on $\mathcal{N}$
is a $P$-ported separator-strong Tutte function
on $\mathcal{N}$ into
the ring $R$
containing parameters $x_e, y_e, X_e, Y_e$,
or an $R$-module containing the initial values
$T(Q)=I(Q)$ for indecomposibles,
when for all $N\in\mathcal{N}$,
\eqref{TA} and \eqref{TSSM} are
satisfied for each
$e\in E(N)$.
\end{definition}

Therefore:

\begin{prop}
$T$ is a $P$-ported separator-strong Tutte function
on $\mathcal{N}$ if and only if for each $N\in\mathcal{N}$,
all Tutte computation trees for $T(N)$ yield polynomial expressions
that are equal in the range ring or $R$-module.
\end{prop}

We develop our first ZBR-type theorem for $P$-ported objects
with matroids or oriented matroids.  It is the
generalization of the ZBR theorem for graphs as given
by Ellis-Monaghan and Traldi\cite{Ellis-Monaghan-Traldi}.
It depends on a lemma similar to one of theirs.

\begin{lem}
\label{DisjSubclassLem}
Suppose $P$-family $\mathcal{N}$ is partitioned into
disjoint $P$-minor closed subfamilies $\{\mathcal{N}_{\pi}\}$.
Then $T$ is a Tutte function on $\mathcal{N}$ if and
only if $T$ restricted to $\mathcal{N}_{\pi}$ is
a Tutte function for each $\mathcal{N}_{\pi}$.
\end{lem}

\begin{thm}
\label{ZBRWildFamily}
Suppose $P$-family $\mathcal{N}$ is partitioned into
disjoint $P$-minor closed subfamilies $\{\mathcal{N}_{\pi}\}$,
and each initial value $I(Q)$ depends only on
the matroid or oriented matroid $M(Q)$ and on the
$\pi$ for which $Q\in\mathcal{N}_{\pi}$,

Then $T$ is a Tutte function with given parameters $(x,y,X,Y)$
and initial values $I(Q)$ if and only if it satisfies
the equations of Theorem \ref{BigTheorem}, interpreted
for families of objects with matroids or oriented matroids.
\end{thm}

\begin{proof}
As in \cite{Ellis-Monaghan-Traldi}, lemma
\ref{DisjSubclassLem} lets us prove the
theorem for each $\pi$ separately.

By Lemma \ref{ObjectTreeValueLemma}, $T$ is a
$P$-ported Tutte function of family
of objects $\mathcal{N}_{\pi}$ if and only
if function $T'(M(N))=T(N)$ on the $P$-family
$\mathcal{C}^{\pi}=\{M(N)\mid N\in\mathcal{N}_{\pi}\}$ is a $P$-ported Tutte
function, since by hypothesis
$I'(M(Q))=I(Q)=I(M(Q))$ for corresponding
indecomposibles $Q\in\mathcal{N}_{\pi}$ and
$M(Q)\in\mathcal{C}^{\pi}$.
The conclusion follows
from Theorem \ref{BigTheorem} applied to $\mathcal{C}^{\pi}$.
\end{proof}

Ellis-Monaghan and Traldi's ZBR theorem for graphs
refers to one initial value $\mathcal{\alpha}_k=I(E_k)$
for each non-empty subclass of graphs, with unlabelled vertices, that
have $k$ graph components.  One natural ported generalization
is to partition the $P$-ported graphs $G$
according to (1) how many
graph components $k$, (2) $P'=P\cap S(G)$ and (3)
$\nu:P'\rightarrow\{1,\ldots,k\}$, where $\nu(p)$ is which
component contains edge $p$.  Theorem
\ref{ZBRWildFamily} tells us:

\begin{cor}
\label{PZBRGraphCor1}

Let a $P$-minor closed collection $\mathcal{G}$ of graphs with unlabelled
vertices be partitioned into $\mathcal{G}_{k,P',\nu}$.  Suppose initial
values $I(G)=I_{k,P',\nu}(M(G))$ are given that depend only on the
part and the matroid or oriented matroid of $G\in\mathcal{G}_{k,P',\nu}$.
Then there is $T$,
a $P$-ported separator-strong parametrized Tutte function
of graphs $\mathcal{G}$ satisfying $T(Q)=I(Q)=I_{k,P',\nu}(M(Q))$ whenever
$P$-quotient $Q\in\mathcal{G}_{k,P',\nu}$ if and only if the identities
of Theorem \ref{BigTheorem}, interpreted for graphs, are satisfied with
the given $I(Q)$.
\end{cor}

The next ZBR-type theorem  addresses the problem
illustrated by equation \eqref{BadZBRExampleEq}. It requires that
the $P$-family satisfy the following:

\begin{definition}
Object $N\in\mathcal{N}$ is \textbf{well-behaved} when
for every independent set $C\subseteq E(N)$ and
coindependent set $D\subseteq E(N)$ for which
$C\cap D=\emptyset$, each of the
$|C\dunion D|!$ orders
of contracting $C$ and deleting $D$ produces the
same $P$-minor (which is an object) of $N$.

Specifically, let
$C=\{c_1,\ldots,c_j\}$,
$D=\{d_{j+1},\ldots,d_k\}$
and $R_i(N')=N'/c_i$ if $1\le i \le j$
and $N'\setminus d_i$ if $j+1\le i \le k$.
The condition is
$R_1\circ\cdots\circ R_k(N) = R_{\sigma_1}\circ\cdots\circ R_{\sigma_k}(N)$
for every permutation $\sigma$ of $\{1,\ldots,k\}$.

$\mathcal{N}$ is \textbf{well-behaved} when
each $N\in\mathcal{N}$ is well-behaved.

\end{definition}

By definition \ref{OMOMdef}
all the minors are defined and
$M(R_1\circ\cdots\circ R_k(N)) = M(R_{\sigma_1}\circ\cdots\circ R_{\sigma_k}(N))$
independently of whether $N$ is well-behaved or not.
The point is that the objects themselves are the same.
We give two examples.

\begin{definition}[Graphs with set-labelled vertices]
\label{GSLVDefinition}
The elements of such a graph $S(G)=E(G)\dunion(P\cap S(G))$ are edges.
The vertices are labelled with non-empty
finite sets so the two sets labelling distinct vertices in one
graph are disjoint.  Only non-loop edges $e\not\in P$
can be contracted; when an edge is contracted, its two endpoints
are replaced by one vertex whose label is the union of the
labels of the two endpoints.  Only non-isthmus edges $e\not\in P$
can be deleted; deletion doesn't change labels.  The graph
has its graphic matroid if it is undirected and its
oriented graphic matroid if it is directed.
\end{definition}

A graph with set-labelled vertices is well-behaved because
the minor obtained  by contracting forest $C$
and deleting $D$ is determined by merging all the vertex labels
of each graph component of $C$ and removing edges $C\cup D$.
The deletions do not affect the vertex labels.  Hence
the vertex labels are not affected by the order
of the operations.

\begin{definition}[Graphs with set-labelled components]
\label{GSLCDefinition}
The elements of such a graph $S(G)=E(G)\dunion(P\cap S(G))$ are edges.
The path-connected components are labelled by
non-empty finite sets so two components in the same
graph always have disjoint labels.  In other words,
the set labels of the components are a partition $\pi_V$.
Only non-loop edges $e\not\in P$
can be contracted and only non-isthmus edges
$e\not\in P$
can be deleted.
The component labels are unchanged by these minor operations.
Definition \ref{GSLVDefinition} specifies the
matroids or oriented matroids.
\end{definition}

A non-well-behaved $P$-family $\mathcal{C}!$
can be constructed from
any $P$-family of matroids $\mathcal{C}$ with
some $M\in\mathcal{C}$ with $|E(M)|\geq 2$.
Each member of $\mathcal{C}!$ is formed from
some $M\in\mathcal{C}$ together with some
history of deletions and contractions that
can be applied to $M$.  Let $c_e$ and $d_e$ be symbols
for contracting and deleting $e\in E(\mathcal{C})$
respectively; a history $h$ is a string of
such symbols.
Let $M|h$ be the $P$-minor obtained by
performing history $h$ on $M$, assuming each step
is defined.  The objects of $\mathcal{C}!$
are all pairs $(M,h)$ for which
$P$-minor $M|h\in\mathcal{C}$ is defined.
The matroid of $(M,h)$ is $M|h$, which
determines the element set, loops and coloops.
If $e\in E(M|h)$ is not
a loop, then define $(M,h)/e=(M,hc_e)$.
Similarly, if $e\in E(M|h)$ is not
a coloop, $(M,h)\setminus e=(M,hd_e)$.

The point of this example is that even if
the $P$-family is not well-behaved and so the
indecomposibles do carry information about their
history, Theorem \ref{ZBRWildFamily} tells
us that the Tutte function
is still well defined if the initial values depend only
on the matroid, or the oriented matroid, of the
indecomposible. (See Questions, \ref{Qsubsection}.)

The examples forced us to recognize that
for $N$ an object with a matroid or oriented matroid $M(N)$
with
${e,f}\in E(N)$ in series or in parallel,  it might
happen that $N/e\setminus f\neq N/f\setminus e$ even though,
by Proposition \ref{SameMinorProp},
$M(N)/e\setminus f= M(N)/f\setminus e$.  Note that
Proposition \ref{SameMinorProp} is about
$\cdot/e\setminus f$ and
$\cdot/f\setminus e$ which are not commutations
of the same two operations.

\begin{thm}[ZBR Theorem for well-behaved
$P$-families of objects with matroids or
oriented matroids]
\label{ZBRWellBehaved}

Supposed $\mathcal{N}$ is well-behaved.
The following two statements are equivalent.
\begin{enumerate}
\item $T$ from $\mathcal{N}$ to $R$ or an $R$-module is a $P$-ported
separator-strong parametrized
$P$-ported Tutte function with $R$-parameters $(x, y, X, Y)$ whose values
$T(Q)$ on $P$-quotients $Q\in\mathcal{N}$ are the initial
values $I(Q)$.
\item
For every $N\in\mathcal{N}$:
\begin{enumerate}
\item
If $M(N)=U^{ef}_1\oplus M(Q)=U^{ef}_1\oplus M(Q')$ with
$P$-quotients $Q=N/e\setminus f$ and $Q'=N/f\setminus e$,
\[
I(Q)(x_e Y_f - y_f X_e ) =
I(Q')(x_f Y_e - y_e X_f).
\]
\item
If $M(N)=U^{efg}_2\oplus M(Q)=U^{efg}_2\oplus M(Q')$ with
$P$-quotients $Q=N/e\setminus f/g$
and $Q'=N/f\setminus e/g$,
\[
I(Q)X_g(x_e y_f - y_f X_e ) =
I(Q')X_g(x_f y_e - y_e X_f ).
\]
\item
If $M(N)=U^{efg}_1\oplus M(Q)=U^{efg}_1\oplus M(Q')$ with
$P$-quotients $Q=N/e\setminus f\setminus g$
and $Q'=N/f\setminus e\setminus g$,
\[
I(Q)Y_g(x_e Y_f - y_f x_e) =
I(Q')Y_g(x_f Y_e - y_e x_f).
\]
\item
If $\{e,f\}=E(M(N))$ is a parallel pair connected to $P$,
\[
I(Q)(x_e Y_f - y_f x_e) =
I(Q')(x_f Y_e - y_e x_f)
\]
where $P$-quotients $Q=N/e\setminus f$
and $Q'=N/f\setminus e$.
\item
If $\{e,f\}=E(M(N))$ is a series pair connected to $P$,
\[
I(Q)(x_e y_f - y_f X_e) =
I(Q')(x_f y_e - y_e X_f)
\]
where $P$-quotients
$Q=N/e\setminus f$
and $Q'=N/f\setminus e$.
\end{enumerate}
\end{enumerate}
\end{thm}

\begin{proof}
All the steps of the proof given for
Theorem \ref{BigTheorem} can be adapted.
The hypothesis the $\mathcal{N}$ is well-behaved
means that $N'/e/f = N'/f/e$, $N'/e\setminus f = N'\setminus f/e$, etc.,
(equalities of objects) are true respectively whenever a step of the
form
$M'/e/f = M'/f/e$, $M'/e\setminus f = M'\setminus f/e$, etc.,
respectively occurs for matroids or oriented matroids.

The calculations done in all cases of Theorem \ref{BigTheorem}
of where
Proposition \ref{SameMinorProp} is applied to write
$Q = M'/e\setminus f = M'/f\setminus e$ when $e,f$ are in
series or parallel are replaced by
$Q = N'/e\setminus f$ and $Q' = N'/f\setminus e$.
Matroids or oriented matroids are replaced
by objects with matroids or oriented matroids in
the calculations.
Notes to guide the reader appear in the proof.
\end{proof}

\begin{cor}
\label{IVEqualSerParCor}
A $P$-family of objects with matroids satisfies a ZBR-type theorem
with the identities given in Theorem \ref{BigTheorem} if,
in
addition $\mathcal{N}$ being well-behaved,
the initial values $I$ satisfy
$I(N/e\setminus f)=I(N/f\setminus e)$ when $\{e,f\}=E(N)$ is a series
or parallel pair,
$I(N/e\setminus f/g)=I(N/f\setminus e/g)$ when $\{e,f,g\}=E(N)$ is a
triangle and
$I(N/e\setminus f\setminus g)=I(N/f\setminus e\setminus g)$
when $\{e,f,g\}=E(N)$
is a triad.
\end{cor}

\begin{cor}
A $P$-family of objects with matroids satisfies a ZBR-type theorem
with the identities given in Theorem \ref{BigTheorem} if, in
addition to $\mathcal{N}$ being well-behaved,
the object $P$-quotients
$N/e\setminus f=N/f\setminus e$ when $\{e,f\}=E(N)$ is a series
or parallel pair,
$N/e\setminus f/g=N/f\setminus e/g$ when $\{e,f,g\}=E(N)$ is a
triangle and
$N/e\setminus f\setminus g=N/f\setminus e\setminus g$ when $\{e,f,g\}=E(N)$
is a triad.
\end{cor}

\begin{proof}
Clearly, if $N/e\setminus f=N/f\setminus e$ then
$I(N/e\setminus f)=I(N/f\setminus e)$, etc.
\end{proof}

We conclude with second ported generalization of
Ellis-Monaghan and Traldi's ZBR theorem for graphs,
besides Corollary \ref{PZBRGraphCor1}.

\begin{cor}
\label{PZBRGraphCor2}
Let $\mathcal{G}$ be a ported $P$-family of graphs with
unlabelled vertices, as in Corollary \ref{PZBRGraphCor1}.
Then there is $T$, $P$-ported separator-strong parametrized Tutte function
of graphs $\mathcal{G}$ satisfying $T(Q)=I(Q)$
for all $P$-quotients $Q\in\mathcal{G}$ if and only if
for every $G\in\mathcal{G}$, each case applies:
\begin{enumerate}
\item[a-d]
Revise the corresponding case of Theorem \ref{ZBRWellBehaved}
by writing $Q'=Q$, i.e., replace $Q'$ by $Q$.
\item[e]
If $E(G)=\{e,f\}$ is a series pair connected to $P$,
\[
I(Q)(x_e y_f - y_f X_e) =
I(Q')(x_f y_e - y_e X_f)
\]
where $P$-quotients
$Q=G/e\setminus f$
and $Q'=G/f\setminus e$.  (This is case (e) of Theorem \ref{ZBRWellBehaved}
\emph{verbatim}.)
\end{enumerate}
\end{cor}

\begin{proof}
$\mathcal{G}$ is a well-behaved $P$-family of ported objects
with matroids or oriented matroids, so Theorem \ref{ZBRWellBehaved}
applies.
In all cases of Theorem \ref{ZBRWellBehaved} but the
last, the two object minors are the same graph
because the contracted edges are path-connected,
so the equations of Theorem \ref{ZBRWellBehaved} are simplified.
\end{proof}

Both Corollaries \ref{PZBRGraphCor1} and \ref{PZBRGraphCor2} reduce
to the ZBR theorem for graphs when $P=\emptyset$.  The first
uses the property that $\mathcal{G}$ is partitioned into
minor-closed subclasses with indecomposibles $E_k$
for which the
initial values depend only on the matroid or oriented matroid to generalize
the original ZBR equations.  As with matroids, we find again that
the Tutte functions can distinguish different orientations of the
the same undirected graphs.
The second relies on the
commutivity of the graph minor operations and generalizes
the fact that different initial values may be assigned to
different indecomposibles, but then the
conditions sufficient for the initial values to
extend to a Tutte function must be stronger.

\section{Direct and Other Sums}

\label{DirectSec}

It is a common situation that $\{N^1,N^2,N\}\subseteq\mathcal{N}$
and their matroids or oriented matroids
satisfy $M(N^1)\oplus M(N^2)=M(N)$.
Tutte computation trees help.  The proposition below applies
even to non-well-behaved $N$ when the symbols
$/B^j_i|P_j$ refer to sequences of deletions and contractions.

\begin{definition}
If $\mathcal{T}_1$ and $\mathcal{T}_2$ are Tutte computation trees then
$\mathcal{T}_1\cdot \mathcal{T}_2$ is the tree obtained by appending
a separate copy of $\mathcal{T}_2$ at each leaf of $\mathcal{T}_1$.
The root is the root of the expanded $\mathcal{T}_1$.
\end{definition}

\begin{prop}
\label{SumProp2}

Suppose $N$, $N^1$ and $N^2$ are all in $\mathcal{N}$
and $M(N^1)\oplus M(N^2)=M(N)$.  Then if
$\mathcal{T}_1$ and $\mathcal{T}_2$ are Tutte computation
trees for $N^1$ and $N^2$ respectively with values given
by \eqref{DS1} and \eqref{DS2}, then there is a Tutte
computation tree for $N$ that yields the value given
by \eqref{DS}.

\begin{equation}
\label{DS1}
\tag{DS1}
\sum_{Q^1_i}I(Q^1_{i})c_1(Q^1_{i})\text{ where }Q^1_i=N/B^1_i|P_1.
\end{equation}
\begin{equation}
\label{DS2}
\tag{DS2}
\sum_{Q^2_j}I(Q^2_{j})c_2(Q^2_{j})\text{ where }Q^2_j=N/B^2_j|P_2.
\end{equation}
\begin{equation}
\tag{DS}
\label{DS}
\sum_{Q^1_i,Q^2_j}I(Q_{i,j})c_1(Q^1_{i})c_2(Q^2_{j})
\text{ where }Q_{i,j}=N/B^1_i|(P_1\cup S(N^2))/B^2_j|P_2.
\end{equation}

Furthermore, if $T$ is a Tutte function on $\mathcal{N}$
and $T(N^1)$ and $T(N^2)$ equal the Tutte polynomials
given by \eqref{DS1} and \eqref{DS2} then
$T(N)$ equals the polynomial given by \eqref{DS}.
\end{prop}

\begin{proof}

We show how to relabel $\mathcal{T}_1\cdot\mathcal{T}_2$ to obtain
a Tutte computation tree for $N$.  $M(N^1)\oplus M(N^2)=M(N)$ is defined
means $S(M(N^1))\cap S(M(N^2))=\emptyset$ and
$S(M(N))=S(M(N^1))\cup S(M(N^2))$.   Each node of
$\mathcal{T}_1\cdot\mathcal{T}_2$ is determined by
by deleting and/or contracting some elements of
$E(M(N^1))\cup E(M(N^2))$.  Relabel that node with the
$P$-minor of $N$ obtaining deleting and/or contracting the
same elements respectively in the same order, those in $N^1$
preceding those in $N^2$.  The result is a computation tree for $N$
because $M(N^1)\oplus M(N^2)=M(N)$.  Assume $P\subseteq S(M(N))$
(otherwise, take a smaller $P$) and let $P^1=S(M(N^1))\cap P$
and $P^2=S(M(N^2))\cap P$.
At a leaf of the relabelled
tree, there will be the $P$-quotient $N/B_1/B_2|P$ where
$B_1$ is a $P^1$-subbasis of $M(N^1)$ and $B_2$ is a $P^2$-subbasis
of $M(N^2)$.

\end{proof}

\subsection{Strong Tutte Functions}

Let us extend the definition of strong parametrized Tutte function
to $P$-families $\mathcal{N}$ of objects with matroids and oriented matroids,
in the way that abstracts the known notion of strong Tutte functions
on minor closed families of graphs\cite{Ellis-Monaghan-Traldi}.
We can then specialize
$\mathcal{N}$ to $\mathcal{C}$, a $P$-family  of
matroids or oriented matroids.
There might still be indecomposibles besides or instead of $\emptyset$.

\begin{definition}
A $P$-ported separator-strong Tutte function $T$ on a $P$-family of
objects $\mathcal{N}$
with matroids is called \textbf{strong} if
whenever $\{N^1, N^2, N\}\subseteq\mathcal{N}$
and
$M(N^1)\oplus M(N^2)=M(N)$,
then $T(N^1)T(N^2)=T(N)$.
\end{definition}

We now generalize to $P\neq\emptyset$
the $T(\emptyset)T(\emptyset)=T(\emptyset)$
characterization of strong Tutte functions.

\begin{thm}
\label{StrongTheorem}
A $P$-ported separator-strong Tutte function $T$ on a $P$-family of
objects
with matroids or oriented matroids $\mathcal{N}$
is strong if and only if
$T$ restricted to the indecomposibles of
$\mathcal{N}$ is strong; i.e.,
whenever $Q^1$, $Q^2$ and
$Q$ are indecomposibles and $M(Q^1)\oplus M(Q^2)=M(Q)$ then
$T(Q^1)T(Q^2)=T(Q)$.
\end{thm}

\begin{proof}
Every $P$-quotient is in $\mathcal{N}$, so clearly $T$ restricted
to the $P$-quotients is strong.

Conversely, suppose $N^1$, $N^2$ and $N$ are in $\mathcal{N}$ and
$M(N^1)\oplus M(N^2)=M(N)$, so Proposition \ref{SumProp2} applies.

Since $M(N^1)\oplus M(N^2)=M(N)$,
$M(N/(B_1\cup B_2)|P)$ $=$ $(M(N^1)/B_1|P)\oplus (M(N^2)/B_2|P)$
$=$ $M(N^1/B_1|P)\oplus M(N^2/B_2|P)$.
We now use the fact
that $Q_{ij}=N/B_1/B_2|P$, $Q^1_i=N^1/B_1|P$ and $Q^2_j=N^2/B_2|P$ are
$P$-quotients and the hypothesis to
write $I(Q_{ij})=I(Q^1_i)I(Q^2_j)$.

We therefore conclude $T(N)=T(N^1)T(N^2)$
from \eqref{DS1}, \eqref{DS2} and  \eqref{DS}.

\end{proof}

\subsection{Multiplicative Tutte Functions}

Graphs can be combined in several ways all so
the matroid of the combination is the direct
sum of the matroids of the parts. This motivates:

\begin{definition}
A partially defined binary operation ``$*$'' on
a $P$-family of objects with matroids or oriented matroids
$\mathcal{N}$ is \textbf{a matroidal direct sum} if
whenever $N^1*N^2\in\mathcal{N}$ is defined for
$\{N^1,N^2\}\subseteq\mathcal{N}$, the
matroids or oriented matroids satisfy
$M(N^1)\oplus M(N^2) = M(N^1*N^2)$.
\end{definition}

Proposition \ref{SumProp2} applies
when $N^1*N^2=N$ is defined.
It gives a general recipe for $T(N^1*N^2)$ which generalizes
the identity \cite{Ellis-Monaghan-Traldi}
$T(M^1\oplus M^2)T(\emptyset)=T(M^1)T(M^2)$
for separator-strong Tutte functions of
matroids.
The $P$-ported generalization is more complicated and generally cannot
be expressed by a product in the domain ring of $T$.

\begin{prop}
\label{SumProp}
Suppose $*$ is a matroidal direct sum and
$N^1$, $N^2$ and $N^1*N^2$ are each members of a $P$-family
for which $T$ is a Tutte function.

If for $P$-quotients $Q^j_i$ and $R$-coefficients $c_j(Q^j_i)$, $j=1$ and $2$,
\begin{equation*}
%\label{MD1}
%\tag{MD1}
T(N^1) = \sum_{Q^1_i}T(Q^1_{i})c_1(Q^1_{i})
\end{equation*}
and
\begin{equation*}
%\label{MD2}
%\tag{MD2}
T(N^2) = \sum_{Q^2_j}T(Q^2_{i})c_2(Q^2_{j})
\end{equation*}
then
\begin{equation*}
%\label{MD}
%\tag{MD}
T(N^1 * N^2) = \sum_{Q^1_i,Q^2_j}T(Q^1_{i}*Q^2_{j})c_1(Q^1_{i})c_2(Q^2_{j}).
\end{equation*}
\end{prop}

\begin{proof}
Substitute $Q_{i,j}=Q^1_i*Q^2_j$ in \eqref{DS} of Proposition \ref{SumProp2}.
\end{proof}

When $\mathcal{N}$ is a $P$-family of matroids or oriented matroids,
direct matroid or oriented matroid sum is obviously a matroidal direct
sum operation, and so Proposition \ref{SumProp} is applicable.

\begin{cor}\cite{Ellis-Monaghan-Traldi}
Let $P=\emptyset$.
$T(M^1\oplus M^2)T(\emptyset) = T(M^1)T(M^2)$ for Tutte
function $T$ of matroids.
\end{cor}

\begin{proof}
Our proof demonstrates how Proposition \ref{SumProp}
generalizes this formula to $P$-families.
The expansions \ref{DS1} and \ref{DS2} take the one-term form
$T(M^j) = T(\emptyset) c_j(\emptyset)$, $j$ $=$ $1,2$,
so $T(M^1)T(M^2)=T(\emptyset)^2 c_1(\emptyset) c_2(\emptyset)$.
Expansion \ref{DS} is then
$T(M^1\oplus T^2)=T(\emptyset) c_1(\emptyset) c_2(\emptyset)$.
\end{proof}

Following the definitions for graphs in \cite{Ellis-Monaghan-Traldi}, we write:

\begin{definition}
Given a matroidal direct sum $*$ on $\mathcal{N}$,
a Tutte function $T$ on $\mathcal{N}$ is
\textbf{multiplicative} (with respect to ``$*$'')
if whenever $N^1*N^2$ is defined
for $\{N^1,N^2\}\subseteq \mathcal{N}$, the Tutte
function values satisfy
$T(N^1)T(N^2)=T(N^1*T^2)$.
\end{definition}

A strong Tutte function is certainly multiplicative
for any ``$*$'', but not conversely.

\begin{cor}
A $P$-ported Tutte function $T$ on $P$-family $\mathcal{N}$
is multiplicative with respect to matroidal direct product
``$*$'' if and only if
for every pair of indecomposibles $\{Q_i, Q_j\}\in\mathcal{N}$ for which
$Q_i*Q_j\in\mathcal{N}$ is defined, $T(Q_i)T(Q_j)=T(Q_i*Q_j)$.
\end{cor}

\begin{proof}
When $N^1*N^2$ is defined, Proposition \ref{SumProp2}
applies because $M(N^1)\oplus  M(N^2)=M(N^1*N^2)$.
$T(N^1*N^2)=T(N^1)T(N^2)$ is then a consequence
of $T(Q_i*Q_j)=T(Q_i)T(Q_j)$.
\end{proof}

\begin{cor}
Suppose $T$ is parametrized Tutte function on a minor-closed class
of graphs $\mathcal{G}$ with unlabelled vertices
and $P=\emptyset$.
$T$ is strong if and only if
there is an idempotent $\alpha=\alpha^2\in R$ and $T(E_k)=\alpha$
whenever $\mathcal{G}_k\neq\emptyset$.
\end{cor}

\begin{proof}
$M(E_k)=\emptyset$ for all $k\geq 1$ and
$\emptyset\oplus\emptyset=\emptyset$, so
$T(E_i)T(E_j)=T(E_k)$ whenever $\mathcal{G}_i$,
$\mathcal{G}_j$ and
$\mathcal{G}_k$ are all non-empty.  Hence, if
$\mathcal{G}_k\neq\emptyset$ then $T(E_k)T(E_k)=T(E_k)=\alpha$.
Further, if $\mathcal{G}_j\neq\emptyset$ with $j\neq k$,
$\alpha=T(E_k)=T(E_k)T(E_k)=T(E_j)$.
\end{proof}

This
strengthens part of Corollary 3.13 in \cite{Ellis-Monaghan-Traldi}.
The other part does require its
additional conditions to establish strongness of
a Tutte function on graphs given as multiplicative on
both disjoint union $\amalg$ and one-point unions.
Consider $\mathcal{N}=\{E_3, E_4, E_5,  \ldots \}$,
$T(E_k)=1$ for $k\geq 3$, $k\neq 5$ and
$T(E_5)=0$.  $T(E_3)T(E_4)=1\neq T(E_5)$, so
$T$ is not strong, but $T$ is multiplicative
on disjoint and one-point unions because $E_5$ cannot be expressed
as either kind of union of graphs in $\mathcal{G}$.  The other conditions
are that $\mathcal{G}_k\neq\emptyset$ for all $k$ and that
$\mathcal{G}$ is closed under one-point unions and removal of isolated
vertices.

\part{Remarks, Background, Problems, and References}

\section{Non-commutativity}
We presented in  \cite{TutteEx}
a new kind of strong Tutte-like function on $P$-ported oriented graphic
matroids (more generally, unimodular, i.e. regular oriented matroids)
whose values vary with the orientation.
Each function value $F(G)$ is
in the exterior algebra over $R^{2p}$, where $R$ is
the reals extended by the $x_e, y_e$
and $|P|=p$.  The function
obeys an \emph{anti-commutative} variant of \eqref{TSM}
with exterior multiplication $\wedge$.
(When $P=\emptyset$, $F$ is
the reduced Laplacian determinant in the
famous Matrix Tree Theorem \cite{HararyBook,sdcMTT}.)
It is the first example we know
of
``the possibility of making use of a noncommutative generalization of the Tutte
polynomial at some point in the future.'' mentioned by Bollob\'{a}s and
Riordan in \cite{BollobasRiordanTuttePolyColored}.  We won't say
more beyond that (1) each of the $\binom{2p}{p}$
Pl\"{u}cker coodinates of $F(G)$ is a $P$-ported Tutte function
of the kind we covered; and (2) that quadratic inequalities among
some of them express negative correlation between edges in spanning trees,
results also known as
Rayleigh's inequality\cite{ChoeRayleigh,SempleWelshNegCorr08,HalfPlaneStuff}.

A second non-commutative possibility might be found in
section \ref{DirectSec}.
The abstraction applies to situations where
an object represents an initial matroid, graph, etc. plus
a history of deletions and contractions.
The key feature is that the Tutte decompositions
of an object are identical to the Tutte decompositions
of that object's matroid.
The initial value
on indecomposible object $Q$
might then depend on the order of the deletion and contractions
reductions to
to obtain it from $N$.  This helps us understand the
theory, but whether objects with minors that depend on reduction
order have useful applications remains to be seen.

\section{Background and Other Related Work}
\label{BackgroundSec}

Besides Brylawski's
work,  another early appearance of Tutte decomposition
of a matroid or graph with a basepoint is \cite{SmithPatroids}.
Ellis-Monaghan and Traldi \cite{Ellis-Monaghan-Traldi}
explain that by leaving the reduction
by $e_0$ to last so $e_0$ is always contracted as a coloop or deleted
as a loop, the Tutte function value can be expressed by
$T(M) = (rX_{e_o} + sY_{e_0})T(\emptyset)$ were $r, s$ are
not-necessarilly-unique elements in $R$.  As one application,
they
give a formula for the parametrized Tutte polynomial
for the parallel connection across $e_0$
which generalized Brylawski's.
These $r,s$ appear in
the $P$-ported Tutte function expression
$rT(U^{e_0}_1)$ $+$ $sT(U^{e_0}_0)$ when $P=\{e_0\}$.  They
are parametrized generalizations of the coefficients of
$z'$ and $x'$ in Brylawski's four variable Tutte polynomial.

Las Vergnas defined and gave basic properties of ``set-pointed'' Tutte
polynomials (with no parameters) and used them to study matroid perspectives.
The polynomial given in \cite{MR0419272,SetPointedLV} has a variable
$\xi_l$ for each subset in a collection of $k$ subsets
$P_l\subseteq P$, $l=1,\ldots,k$.  Each term in \eqref{PAE} had
$\prod\xi_l^{r_i(P_l)}$ for $[Q_i]$ where $r_i$ is the rank function
of matroid $Q_i$.  Therefore \eqref{TSM} was satisfied and the association of the
term to (non-oriented) $Q_i$ could be assured by taking all
$2^{|P|}$ subsets for the $P_l$.  The matroid perspective is the
strong map $M\setminus E(M)\rightarrow M/E(M)$ given by the identity on
$P$.

In \cite{sdcPorted}, we
reproduced Las Vergnes' theory with explicit $P$-quotient (matroid)
variables (see the $[Q_i]$ symbols in Corollary \ref{UniversalCor}
in sec. \ref{UniversalSec}) in place of
$\prod\xi_l^{r_i(P_l)}$.  We then
gave formulas for the $P$-ported Tutte polynomial for the
union and its dual of matroids whose common elements are in $P$.
These formulas  work in a way similar to what appears in
sec. \ref{UniversalSec}.
We extend to algebras
the $\mathbb{Z}[u,w]$-module
generated by the $[Q_i]$ by defining multiplications
$\tilde{*}$ with the rules
$[Q_i]\tilde{*}[Q_j]=r_{ij}[Q_{i,j}]$, with $r_{i,j}\in\mathbb{Z}[u,w]$ and
$Q_{i,j}=Q_i*Q_j$ depending on $(Q_i,Q_j)$ and whether $*$ represents
union or its dual.  It is not often recognized that series and parallel
connection of matroids across basepoint $p$
is equivalent to matroid union and its dual on matroids with only
element $p$ in common.  We plan to investigate whether the formulas
for parametrized Tutte polynomials of parallel connections
in \cite{Ellis-Monaghan-Traldi} can be generalized to the dual
of union when $|P|>1$, and to detail the relationship when $|P|=1$.

Recent work on a different generalization, weak Tutte functions
(see sec. \ref{Complications}), has been done by
Ellis-Monaghan and Zaslavsky \cite{ZaslavskyOct18}.
The distinction between weak Tutte functions (satisfying
an additive identity only)
and strong Tutte functions
(which satisfy \eqref{TSM} and \eqref{TA}) seems first to
have been made by Zaslavsky\cite{MR93a:05047}, for matroids.
That paper also defined weak and strong Tutte functions of
graphs.  However, we use using Ellis-Monaghan and Traldi's
definition of strong Tutte functions of
graphs\cite{Ellis-Monaghan-Traldi}.
The latter restricts \eqref{TA} to non-separators (not just
non-loops);  and it
requires $T(G^1)T(G^2)=T(G)$ whenever
matroids $M(G^1)\oplus M(G_2)=M(G)$
(which we sometimes interpret as oriented), not just when
$G$ is the disjoint union of $G^1$ and $G^2$.
The term separator-strong
(I learned\cite{JoAndTom}
after \cite{Ellis-Monaghan-Traldi} appeared.)
is used for Tutte functions of matroids and graphs
as defined in \cite{Ellis-Monaghan-Traldi}; recall that
they satisfy \eqref{TA} and \eqref{TSSM}.
Normal is used in the same way as in\cite{MR93a:05047}.

We introduced $P$-ported parametrized Tutte polynomials
for normal strong Tutte functions
in \cite{TutteEx}, i.e.,
those with
corank-nullity polynomial expressions.
Most of the results in the current paper, when so restricted
appeared in \cite{TutteEx}
or can be derived by adding parameters and oriented matriod considerations
to material in \cite{sdcPorted}.
These include computation tree\cite{GordonMcMachonGreedoid} based activities
expansions
with terms corresponding to $P$-subbases (which are called
``contracting sets'' in \cite{RelTuttePoly}).
%The one non-elementary fact about
%oriented matroids needed is that two subsets $B,B'\in E(M)$ that span
%the same flat define the same $\emph{oriented}$ minor
%$M/B|P=M/B'|P$.  See \cite{OMBOOK}.
In the normal case, the initial values can be assigned arbitrarilly.
We used this to show the our extensor-valued Tutte-like function
\cite{sdcPorted} is expressible
by assigning extensors as the initial values.  In this
electrical network application, the indecomposibles are oriented
graphic matroids
and different values \emph{are} assigned to different orientations of
the same matroid.

\subsection{Questions}
\label{Qsubsection}
Diao and Hetyei's work \cite{RelTuttePoly} is part of a
larger literature on knot invariants which
leads us to ask if the $P$-ported objects with matroids abstraction,
and its related Tutte computation tree expansions for parametrized
Tutte functions (sec. \ref{DirectSec}), can be usefully
applied to objects besides graphs or directed graphs, such
as various kinds of knot diagrams.

Another open project is to classify the solutions to
the conditions of Theorem \ref{BigTheorem} (about separator-strong
$P$-ported Tutte functions) and
Theorem  \ref{StrongTheorem} (about the strong ones) for rings and
for fields along the lines of
\cite{MR93a:05047} and \cite{BollobasRiordanTuttePolyColored}.

Finally, the theory of objects with matroids or oriented
matroids compared with $P$-ported Tutte functions of
oriented matroids
raises the question of what role orientation plays in
$P$-ported Tutte
function theory.  An oriented matroid can be considered to
be an object with a non-oriented matroid.  In other
words, the orientation is a property of the object
that is constrained by, but not uniquely determined by
the object's non-oriented
matroid.  However, our results show that when the initial
values
depend only on the orientations in addition
to depending on the matroids, the Tutte function is
characterized by
the equations of Theorem \ref{BigTheorem}, not
Theorem \ref{ZBRWellBehaved}.
The reason is that when interchanging the elements
when deleting and contracting a series or parallel pair,
both the resulting matroids and
their orientations are the same.  The question we raise
is whether there are interesting objects with matroids
and certain other properties besides matroid orientation
such that Theorem \ref{BigTheorem}
characterizes the Tutte functions, when the initial
values depend only on the matroids and those other properties.

\section{Acknowledgements}

I thank
the Newton Institute for Mathematical Sciences
of Cambridge University for hospitality and support of my
participation in the Combinatorics and Statistical Mechanics
Programme, January to July 2008, during which some of this
work and many related subjects were reviewed and discussed.

I thank the organizers of
the Thomas H. Brylawsky Memorial Conference, Mathematics Department of
The University of North Carolina, Chapel Hill, October 2008,
Henry Crapo, Gary Gordon and James Oxley
for
fostering collaboration beween people who carry
on the memory of Prof. Brylawsky and editing this journal
issue.

I thank Lorenzo Traldi for bringing to my attention and
discussing Diao and Hetyei's work, as well as
Joanne Ellis-Monaghan, Gary Gordon, Elizabeth McMahon
and Thomas Zaslavsky for helpful conversations and communications
at the Newton Institute and elsewhere, and
my University at Albany colleague Eliot Rich for mutual assistance with
writing.

This work is also supported by a Sabbatical leave granted
by the University at Albany, Sept. 2008 to Sept. 2009.

\bibliographystyle{abbrv}
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\title{Some Basic SIR Models with Variations}
\author{Hsiang Wang, Shenghan Mei, Xinyu Shi}

\begin{document}
\maketitle

\begin{abstract}

\noindent
 This paper introduces SIR models of Covid-19 with some variations. First, we introduce some notations for SIR models. Then we discuss agent-based and ODE/PDE models in one-dimension and two-dimension respectively with some proper simulations. Last but not least, we show our extensions to the SIR models including simulating elementary schools disease spreading, reinfection and masks wearing situations.

\end{abstract}

\section*{Introduction}

   SIR model shows how disease spreads through a population. People are in one of three states at time t:

$s(t)$: the number of susceptible population in the total population at time t.

$i(t)$: the number of infectious population in the total population at time t.

$r(t)$: the number of removed population in the total population at time t.

The model is characterized by the following system of differential equations:

$\frac{ds}{dt} = -b \times s(t) \times i(t)$

$\frac{dr}{dt} = k \times i(t)$

$\frac{di}{dt} = b \times s(t) \times i(t) - k \times i(t)$

$b$ represents the number of interactions for each individual per day

$k$ represents the recovery rate from the diesease

There are some more parameters:

$N$ : Total number of people in a population

$T$ : Time duration

$ii$ : initial fraction of population been infected

\medskip \noindent
The ultimate purpose for a model is to serve for real world application, and coronavirus is the best example we can use and extremely care recently. One extension model we propose is to apply our basic SIR model into simulating elementary schools situation to decide whether the school should reopen or not. What I propose is to input the N(population) as the average elementary school students, I(Infectious people) and R(recovered people) to the model, as well as the average k(recover rates) from research as a baseline. After that, we can change the b(contact people) to observe what is the trend for number of infectious people. At last, we will give our conclusions and some improvements to our model.

\medskip \noindent
Secondly, it is likely for people to be reinfected if people are not immune to the disease after they are recovered. Therefore, we could implement reinfection into the SIR model. For Agent-Based model, this could be implemented by adding probability of reinfection, where reinfected people would be infected again. Alternatively, we could let people who are recovered becomes susceptible with certain probability. In addition, we could add a new variable immunization. This variable stores the probability of been immunized from the disease. For example, immunization = 0.2 means 80\% chance of been reinfection. The value increases if individuals recover from the disease. For ODE model, this could be implement by adding probability of reinfection r. This has the form:

$$\frac{ds}{dt} = -b \times s(t) \times i(t)$$

$$\frac{dr}{dt} = k \times i(t) - r \times r(t)$$

$$\frac{di}{dt} = b \times s(t) \times i(t) - k \times i(t) + r \times r(t)$$

\medskip \noindent
It is obvious that people wearing masks can slow down the disease spread. We can know that most of Chinese people wore face masks and the actual numbers of infections have decreased at a greater rate (Cooper et al). Therefore, we could implement the effect of the use of masks into the SIR model. For ODE model, we can implement it by reducing the parameter b. We can obtain data from to calculate the efficiency of the use of masks on the number of interactions each day that could spread the disease. Thus, we can judge whether the use of masks can slow down the spread of the virus.

\section*{Some SIR models }
\subsection*{Basic SIR models}
{Agent-based model}:

\medskip \noindent
The key idea is that we have total number of N people and some of them are infected initially. These N people interact with b other people per day. If the person is infectious, the person who interacts with him will get infected, which can be done by the discrete method. Later, we introduce k, which is the probability of a person gets recovered. We can choose the duration we want to simulate in the model.

\medskip \noindent
ODE model:

\medskip \noindent
This model treats SIR model as initial value problem. s(t), i(t), r(t) as fraction of population with $s(t)+i(t)+r(t)=1$ for this model. We could apply $solve_{ivp}$ from scipy to solve. The parameters are $b$,$k$, $N$ and $ii$.

\medskip \noindent
We fixed parameters for these simulation:
\newline
- number of days = 40
\newline
- percentage of initial infected people = 0.01
\newline
- number of people = 1000

\newpage
  \begin{center}
 	Figure 1: First Simulation models with b=1, k=1/3
 \end{center}

 \begin{center}

 	\begin{tabular}{c c}
 		\textbf{\underline{Continuous}} &
 		\textbf{\underline{Discrete}} \\
 		\includegraphics[width=7cm, height=5.5cm]{C_SIR_1.png} & \includegraphics[width=7cm,height=5.5cm]{D_SIR_2.png}
 	\end{tabular}

 \end{center}

\medskip \noindent
For these models, the number of susceptible people decreases exponentially until day 18 at about 70 and then keeps no change. The number of infectious people peaks at about day 9 and decrease to 0 after day 26, which is also the time that recovery people stays at about 930 after an exponential raise.

%**The second simulation is b=5, k=0.2**

  \begin{center}
 	Figure 2: Second Simulation models with b=5, k=0.2
 \end{center}

 \begin{center}

 	\begin{tabular}{c c}
 		\textbf{\underline{Continuous}} &
 		\textbf{\underline{Discrete}} \\
 		\includegraphics[width=.5\textwidth]{C_SIR_2.png} & \includegraphics[width=.5\textwidth]{D_SIR_1.png}
 	\end{tabular}

 \end{center}

\medskip \noindent
In this case, because of number of contact people increase to 5, the number of infectious people reaches its peak at the first few days with approximately 820 people. This leads to number of susceptible people drops to 0 within 3 days. The rate of recovery is relatively slow compare with the rate of infected. It takes around 24 days to recover every patient.

\medskip \noindent
These simulations demonstrate discrete and continuous models are very similar. There are few points in the graph are not smooth is due to calculation errors from the randomness in the discrete model.

\newpage
  \begin{center}
 	Figure 3: Phase for Susceptible
 \end{center}

 \begin{center}

 	\begin{tabular}{c c}
 		\textbf{\underline{Continuous}} &
 		\textbf{\underline{Discrete}} \\
 		\includegraphics[width=.5\textwidth]{Phase_Continuous_S.png} & \includegraphics[width=.5\textwidth]{Phase_Discrete_S.png}
 	\end{tabular}

 \end{center}
\medskip \noindent
It is no hard to observe that at day 30, both models illustrate that the susceptible people are about zero in most situations with some exceptions, mainly case b = 1 in continuous model and b = k = 1 in discrete model. This should not be surprising since more people they interact with (b increases), it takes shorter time to spread the disease. At day 30, almost the entire population has already gotten.

  \begin{center}
 	Figure 4: Phase for Infectious
 \end{center}
 \begin{center}

 	\begin{tabular}{c c}
 		\textbf{\underline{Continuous}} &
 		\textbf{\underline{Discrete}} \\
 		\includegraphics[width=.5\textwidth]{Phase_Continuous_I.png} & \includegraphics[width=.5\textwidth]{Phase_Discrete_I.png}
 	\end{tabular}

 \end{center}

\medskip \noindent
From the diagram for number of infected people at day 30, as recovery rate (k) increases, number of infected people drops. This result makes common sense because people will be removed from disease if they have higher recovery rates.

  \begin{center}
 	Figure 5: Phase for Recovery
 \end{center}
 \begin{center}

 	\begin{tabular}{c c}
 		\textbf{\underline{Continuous}} &
 		\textbf{\underline{Discrete}} \\
 		\includegraphics[width=.5\textwidth]{Phase_Continuous_R.png} & \includegraphics[width=.5\textwidth]{Phase_Discrete_R.png}
 	\end{tabular}

 \end{center}

\medskip \noindent
In terms of the recovery people at day 30, the diagram should be nearly the complement of the infected people diagram since people are only on this three stages in our model. From the first phase diagram, almost the no people are susceptible except the some cases mentioned.

\subsection*{Spatial SIR model}
The idea for introducing spatial models is that in some cases, disease spreading is more localized, which means people usually interact with a specific group of people (in the work or family communities). Proposing spatial models can resolve this problem to some extent.
\medskip \noindent \newline
Agent-based model:

\medskip \noindent
Based on the basic agent-based model we had above, we did some slight modification to get the spatial models. We first create a 2D position vector, then a person with random position can take a step of length $p$ in a random direction. After that, the people they interact with is all people surrounding by him with radius $q$. We used $KD$Tree from $Scipy$ package to achieve our goal. This way, we replaced parameters $b$ from our previous model since the use of $q$ describes numbers of interactions.

\medskip \noindent \newline
PDE model:
Alternatively, we can tackle spatial component in SIR model through partial differential equation(PDE). Solving PDE first we create the 2-dimensional finite difference Laplacian matrix L, and we discretize the square into an M * M grid. Then we turn this into a system of ODEs by vectorising the 2-dimensional arrays s, i and r. And we can get the following equations:

$\partial s(x, t)/ \partial t = -b \times s(x, t) \times i(x, t) + p \times L \times s(x, t)$

$\partial r(x, t)/ \partial t = k \times i(x, t) + p \times L \times r(x, t)$

$\partial i(x, t)/ \partial t = b \times s(x, t) \times i(x, t) - k \times i(x, t) + p \times L \times i(x, t)$

  \begin{center}
 	Figure 6: Relation between p and infected people(rate)
 \end{center}
 \begin{center}

 	\begin{tabular}{c c}
 		\textbf{\underline{Discrete}} &
 		\textbf{\underline{Continuous}} \\
 		\includegraphics[width=.5\textwidth]{Discrte_spatial(center)_p.png} & \includegraphics[width=.5\textwidth]{PDE(center)_p.png}
 	\end{tabular}

 \end{center}

\medskip \noindent \newline
The above figure shows the evolution of infected people(rate) changes as time increases, when $b=1, k=\frac{2}{3}$ and people initial position is at center. Both Continuous and Discrete model shows as p increases, the maximum infected people in a day decreases. This makes sense because as p increases, people are more likely to spread out, which lead to lower infected probability.

  \begin{center}
 	Figure 7: Discrete spatial SIR with different initial position
 \end{center}
 \begin{center}

 	\begin{tabular}{c c c}
 		\textbf{\underline{Discrete(corner)}} &
 		\textbf{\underline{Discrete(center)}} &
 		\textbf{\underline{Discrete(random)}} \\
 		\includegraphics[width=.33\textwidth]{D_SIR_S_corner.png} & \includegraphics[width=.33\textwidth]{D_SIR_S_center.png}& \includegraphics[width=.33\textwidth]{D_SIR_S_random.png}
 	\end{tabular}

 \end{center}

%\newpage
  \begin{center}
 	Figure 8: Continuous spatial SIR with different initial position
 \end{center}
 \begin{center}

 	\begin{tabular}{c c c}
 		\textbf{\underline{Continuous(corner)}} &
 		\textbf{\underline{Continuous(center)}} &
 		\textbf{\underline{Continuous(random)}} \\
 		\includegraphics[width=.33\textwidth]{PDE(corner)_s.png} & \includegraphics[width=.33\textwidth]{PDE(center)_s.png}& \includegraphics[width=.33\textwidth]{PDE(random)_s.png}
 	\end{tabular}

 \end{center}

\medskip \noindent
Both continuous and discrete model shows the peak of infected people(rate) is highest when the initial position is corner, then it follows by center and lastly random. This is because when people cluster at corner, they are more likely to be infected since the area is more dense and movements are restricted. people cluster at center has less restriction on movement, which lead to lower peak for the infected people(rate). The random starting position is not compact. Therefore, it has the lowest peak of infected people(rate).

\subsection*{Variations investigations}
\subsubsection*{Model 1}
Assuming we want to investigate if reopening the elementary schools across the country is a wise choice or not, our basic SIR model can help us to determine the problem. According to the research, the average number of students per school for all elementary schools is 473 (Public Elementary Schools). This is used as our reference for the studies. The recovery rates is about 13.425 days for young people from 0 to 19 years old, which is calculated roughly by average of 13.61 day(male) and 13.24 days(female) (Voinsky et al.). By following a Poisson distribution with $\lambda$ = 0.074, we calculate the average daily recover rates $k$. In terms of number of contact people, this is the variable we are varying.

\medskip \noindent
Consider the real situation that little children have limited self-disciplinary so we are not assuming they can obey the CDC guidelines such as wearing masks properly, or keeping the 6-feet away from others. Therefore, we used our basic SIR model to do this simulation. Since both continuous and discrete models are the basically the same, we randomly choose to implement the continuous model:

  \begin{center}
 	Figure 9: SIR Model vs Maximum Infected People as b Varies
 \end{center}
 \begin{center}

 	\begin{tabular}{c c}
 		\textbf{\underline{SIR for b = 1, k = 0.074}} &
 		\textbf{\underline{Max infected people vs b}} \\
 		\includegraphics[width=7cm, height=5.2cm]{cont_b1.png} & \includegraphics[width=7cm, height=5.2cm]{cont_peak.png}
 	\end{tabular}

 \end{center}

\medskip \noindent
From the left figure, when ii = 0.01, b = 1, k = 0.074, one clear observation is that we have a very tall, skinny curve for infected people with peak at around Day 7 for about 350 students, which is very bad. Why? Because this means many people get sick simultaneously and we do not have enough time to prevent to spread the disease (Gavin, 2020).

\medskip \noindent
From the figure on the right, we can see that maximum number of infected people has an increasing trend as b increases, but as b gets larger, the curve becomes more stable. This result is not surprising since when we b = 1, the maximum infected people is already 350, which is 74\% of the total students, therefore, as b raises, the maximum infected people is limited, which shows an increasing trend. This plot is useful in a way to give information to schools about how many people will get infected at maximum if they do not do any actions when reopening the schools.

%At the same time, it will also increase the burden for the hospital (ref).

\subsubsection*{Model 2}
Reinfection is a situation when people are re-infected by a disease. This can occur when people are not immune to the disease. For some viruses, infection can provide lifelong immunity. For the other viruses, infection only provides temporary immunity, which could lead to potential reinfection. \emph{Tillett et. al} states the first confirmed case of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) reinfection in the USA. This shows that reinfection could occur for some diseases. COVID-19 reinfection is still under investigation. \emph{Akiko} shows a evidence of COVID-19 reinfection, while \emph{Osmam et. al} shows previous negative results might be attributed to false-negative laboratory results and prolonged viral shedding, rather than to re-infection. Reinfection for COVID-19 is still debatable.

\medskip \noindent
In this model, we would investigate the possibility of COVID 19 reinfection. We would fit COVID-19 data to SIR ODE model, which includes reinfection component.

\medskip \noindent
After we fit the data to the model, we could estimate the reinfected rate $r$. We would conclude there is evidence of COVID 19 reinfection if $r$ is greater than 0.05.

\medskip \noindent
The implementation is completed in two steps: loss function and optimization. The loss function is defined as:
$$l(r)=||Susceptible_{data}-Susceptible_{simulation}||_2+||IandR_{data}-IandR_{simulation}||_2$$
\medskip \noindent
$Susceptible_{data}$ and $IandR_{data}$ is collected from United States Centers for Disease Control and Prevention. The data only contain total number of cases in United States, which is equivalent to total number of infected and removed in the SIR model. Therefore, we need to combine I and R for the loss function. Optimization is implemented through \emph{Scipy}. We fix number of days to be 319, initial infected percentage to be 0.01, then we can estimate $b,k,r$ through optimization.

\medskip \noindent
Notice that for unconstrained optimization, the estimated values are $b=-0.00000966, k= -0.985, r=0.977$, which is impractical as $b$ and $k$ should be non-negative. One solution for this is to fix $k=0.074$ from model 1 and $b=0.0001$ for simplification.

  \begin{center}
 	Figure 10: Figures for reinfection model
 \end{center}
 \begin{center}

 	\begin{tabular}{c c}
 		\textbf{\underline{Loss Function}} &
 		\textbf{\underline{Covid SIR model vs data}} \\
 		\includegraphics[width=7cm, height=5.2cm]{rvsloss.png} & \includegraphics[width=7cm, height=5.2cm]{modelvsdata.png}
 	\end{tabular}

 \end{center}

%\begin{comments}
%\begin{center}
 %	Figure 10: Loss Function
% \end{center}
 %\begin{center}
%	\includegraphics[scale=0.5]{rvsloss.png}
%\end{center}
%(put figure rvsloss here)
%\end{comments}

\medskip \noindent
When we fix $b=0.0001$ and $k=0.074$, we can get the above plot by obtaining the loss through manipulate $r$. The figures show as reinfection rate decreases, the loss function also decreases. This implies there is no evidence of COVID 19 reinfection.
\medskip \noindent

 %\begin{center}
 %	Figure 11: Covid SIR model vs data
 %\end{center}
%\begin{center}
%	\includegraphics[scale=0.5]{modelvsdata.png}
% \end{center}
%(put figure modelvsdata here)

\medskip \noindent
This figure shows SIR simulation with$b=0.0001$, $k=0.074$ and $r=0.01$ vs COVID 19 data in the United States. The simulated SIR model fits well with the data before 200 days. The model fails to capture increase trend of infected and removed people in the United States after 200 days. Therefore, there are some room of improvement for SIR model.

\subsubsection*{Model 3}
It is obvious that people wearing masks can slow down the disease spread (Cooper et al). Therefore, we could implement the effect of the use of masks into the SIR model. For ODE model, we can implement it by reducing the parameter b. We can obtain data to calculate the effect of the use of masks on the number of interactions each day that could spread the disease (C, MacIntyre, et al). Thus, we can judge whether the use of masks can slow down the spread of the virus.

\medskip \noindent
We know that in Cooper’s paper, the authors conducted a prospective cluster-randomized trial(Cooper et al). During the 2006 and 2007 winter seasons, 286 exposed adults from 143 households were recruited. They found that during a severe pandemic when use of face masks is greater, pandemic transmission in households could be reduced.

\medskip \noindent
We introduce a new parameter q (0<q<1) to reduce the parameter b, then we can get the following equations:

$$\frac{ds}{dt} = -b \times s(t) \times i(t)$$
$$\frac{dr}{dt} = k \times i(t)$$
$$\frac{di}{dt} =  b \times q \times s(t) \times i(t) - k \times i(t)$$
\noindent
We have the parameter b = 1/2, k = 1/3 and the initial condition s = 0.99, r = 0, i = 0.01. From the figure shown in Appendix, we know that when we use surgical mask, q $\approx$ 0.83. And when we use P2 mask, q $\approx$ 0.74.

  \begin{center}
 	Figure 11: Different mask conditions
 \end{center}
 \begin{center}

 	\begin{tabular}{c c c}
 		\textbf{\underline{Surgical masks}} &
 		\textbf{\underline{P2 masks}}  \\
 		\includegraphics[width=.5\textwidth]{surgmask.png} & \includegraphics[width=.5\textwidth]{p2mask.png}
 	\end{tabular}

 \end{center}

\begin{center}
    \begin{tabular}{c}
    	\textbf{\underline{No masks}}  \\
	\includegraphics[scale=0.8]{nomask.png}
\end{tabular}
\end{center}
From the plot above, we know that the use of masks can slow down the spread of the virus from seeing there is a small peak at Day 20 while we do not observe that in the cases of wearing mask. In addition, we can compare with the P2 masks and surgical masks and we can get the conclusion that people wearing P2 masks is better than wearing surgical masks since the infectious curve is more flat in P2 mask situation.

\section*{Conclusion}
%\subsection*{Summary to the results}
To conclude, our basic SIR models in 1D and 2D provide us with clear understanding about how many people will be infected/recovered during a specific period of time by manipulating variables. Both models indicate that as b increases, the peak for infected people takes place much earlier. When recovery rate raises, infected people drops since more sick people recovered from disease so they are not infectious anymore as well.

\medskip \noindent
Spatial models add more dynamic to the SIR model. We observe from different starting position, the resulting model will be noticeably different. Parameter p also affects the model significantly. As p increases, the maximum infected people(rate) decreases. One limitation of the spatial models is there are some randomness involve in the discrete model, which leads to some fluctuation in the figures. One could address this by implement Monte Carlo method, which can tackle randomness effectively.

\medskip \noindent
According to what we investigated above for model 1, it is dangerous to reopen the school at this point, if students cannot strictly follow the public health guidelines. Even with just the initial infectious people is 0.01, and each student interact with 1 other, 74\% of the students in the school will get sick at Day 7, which is a very scary result. One improvement can be considered with this model is that we haven't consider the factor that sick students are not allowed to go back to school. This will slow down our results of infected people since they will not interact with other students.

\medskip \noindent
For SIR model with reinfection component, we show that there is no evidence of COVID 19 reinfection. For continuous model, the reinfection rate is fixed. This neglects that fact that immune system enhance after people are recovered from diseases. Vaccine is am example for this. Discrete model addresses this issue by decreasing reinfection rate when people recover. However, there is no simple solution for continuous model.

\medskip \noindent
In terms of the third model, we can generate the following two conclusions. First, the use of masks can slow down the spread of the virus. Moreover, people wearing P2 masks is better than wearing surgical masks. A limitation here is that this trial only compared the effects of two masks and we can also compare the effects of various masks to determine which mask works best. In addition, we can introduce a new parameter d to increase the social distance between people to slow down the spread of the virus, which can make the model more complete.

%\subsection*{Limitations}

\newpage

\section*{Bibliography}

1. C, MacIntyre, et al. “Face mask use and control of respiratory virus transmission in households.” National Library of Medicine. Published by National Center for Biotechnology Information.,
Feb. 2009, https://pubmed.ncbi.nlm.nih.gov/19193267/.

\medskip \noindent
2. Cooper, Ian, et al. “A SIR model assumption for the spread of COVID-19 in different communities.” Chaos Solitons Fractals. Published by Elsevier Ltd., Oct. 2020, https://www.ncbi.nlm.nih.gov/pmc/articles/
PMC7321055/.

\medskip \noindent
3. Gavin, K., 2020. Flattening The Curve For COVID-19: What Does It Mean And How Can You Help?. [online] Healthblog.uofmhealth.org. Available at: <https://healthblog.uofmhealth.org/wellness-prevention/flattening-curve-for-covid-19-what-does-it-mean-and-how-can-you-help> [Accessed 8 December 2020].

\medskip \noindent
4. Osman A.A. and Al Daajani M.M. and Alsahafi A.J. (2020). Re-positive coronavirus disease 2019 PCR test: could it be a reinfection? {\it New Microbes and New Infections} (37)

\medskip \noindent
5. Public Elementary Schools, by Grade Span, Average School Size, and State or Jurisdiction: 2005-06. National Center for Education Statistics, U.S. Department of Education,

\medskip \noindent
6. Tillett R.L., Sevinsky J.R., Hartley P.D., et al. (2020). Genomic evidence for reinfection with SARS-CoV-2: a case study. {\it Lancet Infect Diseases};

\medskip \noindent
7. Iwasaki A. (2020). What reinfections mean for COVID-19. {\it The Lancet Infectious Diseases}.

\medskip \noindent
8. Voinsky, Irena, et al. “Effects of Age and Sex on Recovery from COVID-19: Analysis of 5769 Israeli Patients.” The Journal of Infection, The British Infection Association. Published by Elsevier Ltd., Aug. 2020, www.ncbi.nlm.nih.gov/pmc/articles/PMC7229949/.

\newpage
\section*{Appendix}

  \begin{center}
 	Figure 1: Summary statistics of wearing/no wearing masks
 \end{center}
\begin{center}
	\includegraphics[scale=0.9]{maskov.png}
\end{center}
Citied: C, MacIntyre, et al. “Face mask use and control of respiratory virus transmission in households.” National Library of Medicine. Published by National Center for Biotechnology Information.,
Feb. 2009, https://pubmed.ncbi.nlm.nih.gov/19193267/.

\end{document}