Chính T. Hoàng

Four classes of perfectly orderable graphs

A graph is called “perfectly orderable” if its vertices can be ordered in such a way that, for each induced subgraph F, a certain “greedy” coloring heuristic delivers an optimal coloring of F. No polynomial-time algorithm to recognize these graphs is known. We present four classes of perfectly orderable graphs: Welsh–Powell perfect graphs, Matula perfect graphs, graphs of Dilworth number at most three, and unions of two threshold graphs. Graphs in each of the first three classes are recognizable in a polynomial time. In every graph that belongs to one of the first two classes, we can find a largest clique and an optimal coloring in a linear time.

P 4 -comparability graphs

In 1981, Chvatal defined the class of perfectly orderable graphs. This class of perfect graphs contains the comparability graphs. In this paper, we introduce a new class of perfectly orderable graphs, the P4-comparability graphs. This class generalizes comparability graphs in a natural way. We also prove a decomposition theorem which leads to a structural characterization of P4-comparability graphs. Using this characterization, we develop a polynomial-time recognition algorithm and polynomial-time algorithms for the clique and colouring problems for P4-comparability graphs.

Optimizing weakly triangulated graphs

A graph is weakly triangulated if neither the graph nor its complement contains a chordless cycle with five or more vertices as an induced subgraph. We use a new characterization of weakly triangulated graphs to solve certain optimization problems for these graphs. Specifically, an algorithm which runs inO((n + e)n3) time is presented which solves the maximum clique and minimum colouring problems for weakly triangulated graphs; performing the algorithm on the complement gives a solution to the maximum stable set and minimum clique covering problems. Also, anO((n + e)n4) time algorithm is presented which solves the weighted versions of these problems.

On brittle graphs

Chvatal defined a graph G to be brittle if each induced subgraph F of G contains a vertex that is not a midpoint of any P4 or not an endpoint of any P4. Every brittle graph is perfectly orderable. In this paper, we prove that a graph is brittle whenever it is HHD-free (containing no chordless cycle with at least five vertices, no cycle on six vertices with a long chord, and no complement of the chordless path on five vertices). We also design an O(n4) algorithm to recognize HHD-free graphs, and also an O(n4) algorithm to construct a perfect order of an HHD-free graph. It follows from this result that an optimal coloring and a largest clique of an HHD-free graph can be found in O(n4) time.

A Certifying Algorithm for 3-Colorability of P5-Free Graphs

We provide a certifying algorithm for the problem of deciding whether a P 5-free graph is 3-colorable by showing there are exactly six finite graphs that are P 5-free and not 3-colorable and minimal with respect to this property.

Efficient algorithms for minimum weighted colouring of some classes of perfect graphs

Abstract We design an O(nm) algorithm to find a minimum weighted colouring and a maximum weighted clique of a perfectly ordered graph. We also present two O(n2) algorithms to find a minimum weighted colouring of a comparability graph and of a triangulated graph. Our colouring algorithms use an algorithm to find a stable set meeting all maximal (with respect to set inclusion) cliques of a perfectly ordered graph. We show that the problem of finding such stable set in an arbitrary graph is NP-hard. We shall also describe a polynomial algorithm to find a minimum weighted colouring of a clique separable graph.

The Complexity of the List Partition Problem for Graphs

The

Stability number of bull- and chair-free graphs revisited

k

Some properties of minimal imperfect graphs

-partition problem is as follows: Given a graph

A Characterization of b-Perfect Graphs

G