Strong tree-cographs are birkhoff graphs

Abstract

In this paper a class of undirected graphs is considered which can be derived from trees by a finite sequence of operations involving disjoint union and complementation. The graphs obtained in this way are called tree-cographs. The isomorphism problem for tree-cographs is obviously polynomial, due to their similarity to cographs.The notion of a Birkhoff graph is introduced. We represent graphs by their adjacency matrices. Any automorphism of a graph A is then representable by a permutation matrix P which commutes with A. But there may exist also doubly stochastic matrices X which commute with A, i.e. satisfy XA=AX, and which are not permutation matrices. A graph A is called a Birkhoff graph iff every doubly stochastic matrix X which commutes with A is a convex sum of automorphisms of A.If in the derivation of tree-cographs disjoint union is restricted to pairwise non-isomorphic components, then a class of “strong” tree-cographs is obtained. It is shown that strong tree-cographs are Birkhoff graphs. Furthermore, two strong tree-cographs A and B are isomorphic iff there exists a doubly stochastic matrix X satisfying XA=BX. Every such X is a convex sum of isomorphisms of A and B.The results of this paper may be considered as first steps towards a polyhedral theory of graph isomorphism.