Title: Slimness of graphs,
Abstract: Slimness of a graph measures the local deviation of its metric from a tree
metric. In a graph $G=(V,E)$, a geodesic triangle $\bigtriangleup(x,y,z)$ with
$x, y, z\in V$ is the union $P(x,y) \cup P(x,z) \cup P(y,z)$ of three shortest
paths connecting these vertices. A geodesic triangle $\bigtriangleup(x,y,z)$ is
called $\delta$-slim if for any vertex $u\in V$ on any side $P(x,y)$ the
distance from $u$ to $P(x,z) \cup P(y,z)$ is at most $\delta$, i.e. each path
is contained in the union of the $\delta$-neighborhoods of two others. A graph
$G$ is called $\delta$-slim, if all geodesic triangles in $G$ are
$\delta$-slim. The smallest value $\delta$ for which $G$ is $\delta$-slim is
called the slimness of $G$. In this paper, using the layering partition
technique, we obtain sharp bounds on slimness of such families of graphs as (1)
graphs with cluster-diameter $\Delta(G)$ of a layering partition of $G$, (2)
graphs with tree-length $\lambda$, (3) graphs with tree-breadth $\rho$, (4)
$k$-chordal graphs, AT-free graphs and HHD-free graphs. Additionally, we show
that the slimness of every 4-chordal graph is at most 2 and characterize those
4-chordal graphs for which the slimness of every of its induced subgraph is at
most 1.