The world of hereditary graph classes viewed through Truemper configurations

In 1982 Truemper gave a theorem that characterizes graphs whose edges can be labeled so that all chordless cycles have prescribed parities. The characterization states that this can be done for a graph G if and only if it can be done for all induced subgraphs of G that are of few speci c types, that we will call Truemper con gurations. Truemper was originally motivated by the problem of obtaining a co-NP characterization of bipartite graphs that are signable to be balanced (i.e. bipartite graphs whose node-node incidence matrices are balanceable matrices). The con gurations that Truemper identi ed in his theorem ended up playing a key role in understanding the structure of several seemingly diverse classes of objects, such as regular matroids, balanceable matrices and perfect graphs. In this survey we view all these classes, and more, through the excluded Truemper con gurations, focusing on the algorithmic consequences, trying to understand what structurally enables e cient recognition and optimization algorithms

Perfect Graphs, Partitionable Graphs and Cutsets

We prove a theorem about cutsets in partitionable graphs that generalizes earlier results on amalgams, 2-amalgams and homogeneous pairs. 1 Introduction A graph G is perfect if, for all induced subgraphs of G, the size of a largest clique is equal to the chromatic number. A graph is minimally imperfect if it is not perfect but all its proper induced subgraphs are. A hole is a chordless cycle of length at least four. The strong perfect graph conjecture of Berge [1] states that G is minimally imperfect if and only if G or its complement is an odd hole. (The complement G of G is a graph with same node set as G and two nodes are adjacentin G if and only if they are not adjacentinG). A graph G is partitionable if there exist integers ff and ! greater than one such that G has exactly ff! + 1 nodes and, for eachnodev 2 V , G n v can be partitioned into both ff cliques of size ! and ! stable sets of size ff.Lov'asz [13] has proven that every minimally imperfect graph is partitionable. A..

Perfect graphs, partitionable graphs and cutsets

We prove a theorem about cutsets in partitionable graphs that generalizes earlier results on amalgams, 2-amalgams and homogeneous pairs