planar graph is minor-closed. Every minor-closed property is monotone, but not necessarily vice versa; for instance, minors of triangle-free graphs are not necessarily themselves triangle-free. These definitions may be extended from properties to numerical invariants of graphs: a graph invariant is hereditary, monotone, or minor-closed if the function formalizing the invariant forms a monotonic function from the corresponding partial order on graphs to the real numbers. Additionally, graph invariants have been studied with respect to their behavior with regard to disjoint unions of graphs: A graph invariant is additive if, for all two graphs G and H, the value of the invariant on the disjoint union of G and H is the sum of the values on G and on H. For instance, the number of vertices is additive. A graph invariant is multiplicative if, for all two graphs G and H, the value of the invariant on the disjoint union of G and H is the product of the values on G and on H. For instance, the Hosoya index (number of matchings) is multiplicative. A graph invariant is maxing if, for all two graphs G and H, the value of the invariant on the disjoint union of G and H is the maximum of the values on G and on H. For instance, the chromatic number is maxing. In addition, graph properties can be classified according to the type of graph they describe: whether the graph is undirected or directed, whether the property applies to multigraphs, etc. Values of invariants The target set of a function that defines a graph invariant may be one of: A truth-value, true or false, for the indicator function of a graph property. An integer, such as the number of vertices or chromatic number of a graph. A real number, such as the fractional chromatic number of a graph. A sequence of integers, such as the degree sequence of a graph. A polynomial, such as the Tutte polynomial of a graph. Graph invariants and graph isomorphism Easily computable graph invariants are instrumental for fast recognition of graph isomorphism, or rather non-isomorphism, since for any invariant at all, two graphs with different values cannot (by definition) be isomorphic. Two graphs with the same invariants may or may not be isomorphic, however. A graph invariant I(G) is called complete if the identity of the invariants I(G) and I(H) implies the isomorphism of the graphs G