Vertex elimination orderings for hereditary graph classes

Abstract

We provide a general method to prove the existence and compute efficiently elimination orderings in graphs. Our method relies on several tools that were known before, but that were not put together so far: the algorithm LexBFS due to Rose, Tarjan and Lueker, one of its properties discovered by Berry and Bordat, and a local decomposition property of graphs discovered by Maffray, Trotignon and Vu\vskovi\'c. We use this method to prove the existence of elimination orderings in several classes of graphs, and to compute them in linear time. Some of the classes have already been studied, namely even-hole-free graphs, square-theta-free Berge graphs, universally signable graphs and wheel-free graphs. Some other classes are new. It turns out that all the classes that we study in this paper can be defined by excluding some of the so-called Truemper configurations. For several classes of graphs, we obtain directly bounds on the chromatic number, or fast algorithms for the maximum clique problem or the coloring problem

The structure of (theta, pyramid, 1‐wheel, 3‐wheel)‐free graphs

“…The first algorithm is obtained by a direct search for certain Truemper configurations, so, although it is slower than the second one, we believe that its intermediate steps are of independent interest. The second algorithm has running time and is based on the description of the local structure of graphs in that is obtained in . (Throughout the section, for a graph , we let and .)…”

“…To describe our second algorithm, we first recall some definitions from . Let be a set of graphs.…”

“…Let denote the stable graph on three vertices and the path on three vertices. The following theorem is the key for our second algorithm (we note that the class is denoted by in —see also Table 1 from the same paper).…”

“…This class contains all chordal graphs and all line graphs of triangle‐free graphs (or equivalently, [claw, diamond]‐free graphs ). This class was first studied in where it was shown that every graph in has a vertex, whose neighborhood is a disjoint union of two (possibly empty) cliques, and furthermore an ordering of such vertices can be found by lexicographic breadth first search (LexBFS). A consequence of this is a linear‐time algorithm for the maximum weight clique problem on , as well as a linear‐time coloring algorithm that colors the graph with at most colors, where denotes the size of the largest clique in .…”

Clique‐cutsets beyond chordal graphs

“…As shown in the table, an time algorithm solving the MWCP for the class was given in ; that algorithm relies on LexBFS . In the present paper, we give a different algorithm solving the MWCP for the class (our algorithm has the same complexity as the one from , but it relies on our structural results for the class ). Further, we note that the complexity of the ColP for the class was left open in ; here, we give a polynomial‐time algorithm that solves this problem.…”