A Charming Class of Perfectly Orderable Graphs

We investigate the following conjecture of Vašek Chvátal: any weakly triangulated graph containing no induced path on five vertices is perfectly orderable. In the process we define a new polynomially recognizable class of perfectly orderable graphs called charming. We show that every weakly triangulated graph not containing as an induced subgraph a path on five vertices or the complement of a path on six vertices is charming

Perfect Graphs

This chapter is a survey on perfect graphs with an algorithmic flavor. Our emphasis is on important classes of perfect graphs for which there are fast and efficient recognition and optimization algorithms. The classes of graphs we discuss in this chapter are chordal, comparability, interval, perfectly orderable, weakly chordal, perfectly contractile, and chi-bound graphs. For each of these classes, when appropriate, we discuss the complexity of the recognition algorithm and algorithms for finding a minimum coloring, and a largest clique in the graph and its complement

A note on perfect orders

AbstractPerfectly orderable graphs were introduced by Chvátal in 1984. Since then, several classes of perfectly orderable graphs have been identified. In this paper, we establish three new results on perfectly orderable graphs. First, we prove that every graph with Dilworth number at most three has a simplical vertex, in the graph or in its complement. Second, weprovide a characterization of graphs G with this property: each maximal vertex ofG is simplical in the complement of G. Finally, we introduce the notion of a locally perfect order and show that every arborescence-comparability graph admits a locally perfect order

Results on perfect graphs

The chromatic number of a graph G is the least number of colours that can be assigned to the vertices of G such that two adjacent vertices are assigned different colours. The clique number of a graph G is the size of the largest clique that is an induced subgraph of G. The notion of perfect graphs was first introduced by Claude Berge in 1960. He defined a graph G to be perfect if the chromatic number of H is equal to the clique number of H for every induced subgraph H C G. He also conjectured that perfect graphs are exactly the class of graphs with no induced odd hole (a chordless odd cycle of greater than or equal to five vertices) or no induced complement of an odd hole, an odd anti-hole. This conjecture, that still remains an open problem, is better known as the Strong Perfect Graph Conjecture (or SPGC). An equivalent statement to SPGC is that minimal imperfect graphs are odd holes and odd anti-holes. Fonlupt conjectured that all minimal imperfect graphs with a minimal cutset that is the union of more than one disjoint clique, must be an odd hole. In this thesis we prove that any hole-free graph G with a minimal cutset C that is the union of vertexdisjoint cliques must have a clique in each component o f G — C that sees all of C. We further prove that minimal imperfect graphs with a minimal cutset that is the union of two disjoint cliques have a hole. Since the introduction of perfectly orderable graphs by Chvdtal in 1984, many classes of perfectly orderable graphs and their recognition algorithms have been identified. Perfectly ordered graphs are those graphs G such that for each induced ordered subgraph H of G, the greedy (or, sequential) colouring algorithm produces an optimal colouring of H. Hohng and Reed previously studied six natural subclasses of perfecdy orderable graphs that are defined by the orientations of the P4 ’s. Four of the six classes can be recognized in polynomial time. The recognition problem for the fifth class has been proven to be NP-complete. In this thesis, we discuss the problem o f recognition for the sixth class, known as one-in-one-out graphs. Also, we consider pyramid-free graphs with the same orientation as one-in-one-out graphs and prove that this class of graphs cannot contain a directed 3-cycle of more than one equivalence class

Induced subgraphs of graphs with large chromatic number. XI. Orientations

Fix an oriented graph H, and let G be a graph with bounded clique number and very large chromatic number. If we somehow orient its edges, must there be an induced subdigraph isomorphic to H? Kierstead and Rodl raised this question for two specific kinds of digraph H: the three-edge path, with the first and last edges both directed towards the interior; and stars (with many edges directed out and many directed in). Aboulker et al subsequently conjectured that the answer is affirmative in both cases. We give affirmative answers to both questions

Recognition of some perfectly orderable graph classes

AbstractThis paper presents new algorithms for recognizing several classes of perfectly orderable graphs. Bipolarizable and P4-simplicial graphs are recognized in O(n3.376) time, improving the previous bounds of O(n4) and O(n5), respectively. Brittle and semi-simplicial graphs are recognized in O(n3) time using a randomized algorithm, and O(n3log2n) time if a deterministic algorithm is required. The best previous time bound for recognizing these classes of graphs is O(m2). Welsh–Powell opposition graphs are recognized in O(n3) time, improving the previous bound of O(n4). HHP-free graphs and maxibrittle graphs are recognized in O(mn) and O(n3.376) time, respectively

Weak Bipolarizable Graphs

We characterize a new class of perfectly orderable graphs and give a polynomial-time recognition algorithm, together with linear-time optimization algorithms for this class of graphs