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

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

Quasi-Brittle Graphs, a New Class of Perfectly Orderable Graphs

A graph G is quasi-brittle if every induced subgraph H of G contains a vertex which is incident to no edge extending symmetrically to a chordless path with three edges in either Hor its complement H¯. The quasi-brittle graphs turn out to be a natural generalization of the well-known class of brittle graphs. We propose to show that the quasi-brittle graphs are perfectly orderable in the sense of Chvátal: there exists a linear order \u3c on their set of vertices such that no induced path with vertices a, b, c, d and edges ab, bc, cd has a \u3c b and d \u3c c

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

T-colorings of graphs: recent results and open problems

AbstractSuppose G is a graph and T is a set of nonnegative integers. A T-coloring of G is an assignment of a positive integer ƒ(x) to each vertex x of G so that if x and y are joined by an edge of G, then |ƒ(x) - ƒ(y)ƒ| is not in T. T-colorings were introduced by Hale in connection with the channel assignment problem in communications. Here, the vertices of G are transmitters, an edge represents interference, ƒ(x) is a television or radio channel assigned to x, and T is a set of disallowed separations for channels assigned to interfering transmitters. One seeks to find a T -coloring which minimizes either the number of different channels ƒ(x) used or the distance between the smallest and largest channel. This paper surveys the results and mentions open problems concerned with T-colorings and their variations and generalizations

On the perfect orderability of unions of two graphs

A graph G is perfectly orderable if it admits an order < on its vertices such that the sequential coloring algorithm delivers an optimum coloring on each induced subgraph (H, <) of (G, <). A graph is a threshold graph if it contains no P4 , 2K2 or C4 as induced subgraph. A theorem of Chvatal, Hoang, Mahadev and de Werra states that a graph is perfectly orderable if it can be written as the union of two threshold graphs. In this thesis, we investigate possible generalizations of the above theorem. We conjecture that if G is the union of two graphs G1 and G2 then G is perfectly orderable whenever (i) G1 and G2 are both P4 -free and 2K2-free, or (ii) G1 is P4-free, 2K2-free and G2 is P4 -free, C4 -free. We show that the complement of the chordless cycle with at least five vertices cannot be a counter-example to our conjecture and we prove, jointly with Hoang, a special case of (i): if G1 and G2 are two edge disjoint graphs that are P4 -free and 2K2 -free then the union of G1 and G2 is perfectly orderable

On the Recognition of Bipolarizable and P_4-simplicial Graphs

The classes of Raspail (also known as Bipolarizable) and P_4-simplicial graphs were introduced by Hoàng and Reed who showed that both classes are perfectly orderable and admit polynomial-time recognition algorithms HR1. In this paper, we consider the recognition problem on these classes of graphs and present algorithms that solve it in O(n m) time. In particular, we prove properties and show that we can produce bipolarizable and P_4-simplicial orderings on the vertices of the input graph G, if such orderings exist, working only on P_3s that participate in a P_4 of G. The proposed recognition algorithms are simple, use simple data structures and both require O(n + m) space. Additionally, we show how our recognition algorithms can be augmented to provide certificates, whenever they decide that G is not bipolarizable or P_4-simplicial; the augmentation takes O(n + m) time and space. Finally, we include a diagram on class inclusions and the currently best recognition time complexities for a number of perfectly orderable classes of graphs