Chapter 11 Not so perfect graphs

On polygon numbers of circle graphs and distance hereditary graphs

Consider a class of graphs $$\mathcal{G}$$ having a polynomial time algorithm computing the set of all minimal separators for every graph in $$\mathcal{G}$$ . We show that there is a polynomial time algorithm for treewidth and minimum fill-in, respectively, when restricted to the class $$\mathcal{G}$$ . Many interesting classes of intersection graphs have a polynomial time algorithm computing all minimal separators, like permutation graphs, circle graphs, circular arc graphs, distance hereditary graphs, chordal bipartite graphs etc. Our result generalizes earlier results for the treewidth and minimum fill-in for several of these classes. We also consider the related problems pathwidth and interval completion when restricted to some special graph classes.

A minimal separator of a graph is a set such that there exist vertices with the property that separates from in , but no proper subset of does. For an integer , we say that a minimal separator is ‐simplicial if it can be covered by cliques and denote by the class of all graphs in which each minimal separator is ‐simplicial. We show that for each , the class is closed under induced minors, and we use this to show that the Maximum Weight Stable Set problem can be solved in polynomial time for . We also give a complete list of minimal forbidden induced minors for . Next, we show that, for , every nonnull graph in has a ‐simplicial vertex, that is, a vertex whose neighborhood is a union of cliques; we deduce that the Maximum Weight Clique problem can be solved in polynomial time for graphs in . Further, we show that, for , it is NP‐hard to recognize graphs in ; the time complexity of recognizing graphs in is unknown. We also show that the Maximum Clique problem is NP‐hard for graphs in . Finally, we prove a decomposition theorem for diamond‐free graphs in (where the diamond is the graph obtained from by deleting one edge), and we use this theorem to obtain polynomial‐time algorithms for the Vertex Coloring and recognition problems for diamond‐free graphs in , and improved running times for the Maximum Weight Clique and Maximum Weight Stable Set problems for this class of graphs.

This is a summary of the author’s Ph.D. thesis supervised by Sara Nicoloso and Gianpaolo Oriolo and defended on 3 April 2008 at Sapienza Universita di Roma. The thesis is written in English and is available from the author upon request. This work deals with three classical combinatorial problems, namely the isomorphism, the vertex-coloring and the stable set problem, restricted to two graph classes, namely circulant and claw-free graphs. In the first part (joint work with Sara Nicoloso), we derive a necessary and sufficient condition to test isomorphism of circulant graphs, and give simple algorithms to solve the vertex-coloring problem on this class of graphs. In the second part (joint work with Gianpaolo Oriolo and Gautier Stauffer), we propose a new combinatorial algorithm for the maximum weighted stable set problem in claw-free graphs, and devise a robust algorithm for the same problem in the subclass of fuzzy circular interval graphs, which also provides recognition when the stability number is greater than three.

Stable sets and polynomials

A fast algorithm for the maximum clique problem

String graphs of k-bend paths on a grid

On edge transitivity of directed graphs

Graphs and Algorithms A graph class has few cliques if there is a polynomial bound on the number of maximal cliques contained in any member of the class. This restriction is equivalent to the requirement that any graph in the class has a polynomial sized intersection representation that satisfies the Helly property. On any such class of graphs, some problems that are NP-complete on general graphs, such as the maximum clique problem and the maximum weighted clique problem, admit polynomial time algorithms. Other problems, such as the vertex clique cover and edge clique cover problems remain NP-complete on these classes. Several classes of graphs which have few cliques are discussed, and the complexity of some partitioning and covering problems are determined for the class of all graphs which have fewer cliques than a given polynomial bound.

In this paper, we single out the following particular case of the clique search problem. The vertices of the given graph are legally colored with k colors and we are looking for a clique with k nodes in the graph. In other words, we want to decide if a given k-partite graph contains a clique with k nodes. The maximum clique problem asks for finding a maximum clique in a given finite simple graph. The problem of deciding if the given graph contains a clique with k vertices is called the k-clique problem. The first problem is NP-hard and the second one is NP-complete. The special clique search problem, we propose, is still an NP-complete problem. We will show that the k-clique problem in the special case of k-partite graphs is more tractable than in the general case. In order to illustrate the possible practical utility of this restricted type clique search problem we will show that the job shop scheduling problem can be reduced to such a clique search problem in a suitable constructed graph. We carry out numerical experiments to assess the efficiency of the approach. It is a common practice that before one embarks on a large scale clique search typically one attempts to simplify and tidy up the given graph. This procedure is commonly referred as preconditioning or kernelization of the given graph. Of course, the preconditioning or kernelization is meant with respect to the given type of clique search problem. The other main topic of the paper is to describe a number of kernelization methods tailored particularly to the proposed special k-clique problem. Some of these techniques works in connection with the generic k-clique problem. In these situations, we will see that they are more efficient in the case of k-partite graphs. Some other preconditioning methods applicable only to k-partite graphs. We illustrate how expedient these preconditioning methods can be by solving non-trivial scheduling problems to optimality employing only kernelization techniques dispensing with exhaustive clique search algorithms altogether.

The class of even-hole-free graphs is structurally quite similar to the class of perfect graphs, which was the key initial motivation for their study. The techniques developed in the study of even-hole-free graphs were generalized to other complex hereditary graph classes, and in the case of perfect graphs led to the famous resolution of the Strong Perfect Graph Conjecture and their polynomial time recognition. The class of even-holefree graphs is also of independent interest due to its relationship to ?-perfect graphs. In this survey we describe all the different structural characterizations of even-hole-free graphs, focusing on their algorithmic consequences.

The study of perfectness, via the strong perfect graph conjecture, has given rise to numerous investigations concerning the structure of many particular classes of perfect graphs. In “Perfect Product Graphs” (Discrete Mathematics, Vol. 20, 1977, pp. 177‐‐186), G. Ravindra and K. R. Parthasarathy tried, but unfortunately without success, to characterize the perfectness of the cartesian product of graphs (see also MR No. 58‐‐10567, 1979). In this paper we completely characterize the graphs that are both nontrivial cartesian products and s‐strongly perfect.

We study various optimization problems in t-subtree graphs, the intersection graphs of t-subtrees, where a t-subtree is the union of t disjoint subtrees of some tree. This graph class generalizes both the class of chordal graphs and the class of t-interval graphs, a generalization of interval graphs that has recently been studied from a combinatorial optimization point of view. We present approximation algorithms for the Maximum Independent Set, Minimum Coloring, Minimum Vertex Cover, Minimum Dominating Set, and Maximum Clique problems in t-subtree graphs.

It is a long standing open problem to find an explicit description of the stable set polytope of claw-free graphs. Yet more than 20 years after the discovery of a polynomial algorithm for the maximum stable set problem for claw-free graphs, there is even no conjecture at hand today.Such a conjecture exists for the class of quasi-line graphs. This class of graphs is a proper superclass of line graphs and a proper subclass of claw-free graphs for which it is known that not all facets have 0/1 normal vectors. Ben Rebea’s conjecture states that the stable set polytope of a quasi-line graph is completely described by clique-family inequalities. Chudnovsky and Seymour recently provided a decomposition result for claw-free graphs and proved that Ben Rebea’s conjecture holds, if the quasi-line graph is not a fuzzy circular interval graph.In this paper, we give a proof of Ben Rebea’s conjecture by showing that it also holds for fuzzy circular interval graphs. Our result builds upon an algorithm of Bartholdi, Orlin and Ratliff which is concerned with integer programs defined by circular ones matrices.KeywordsNegative CycleClique InequalityHeavy EdgeRational PolyhedronLight EdgeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The Maximum Weight Stable Set (MWS) Problem is one of the fundamental algorithmic problems in graphs. It is NP-complete in general, and it has polynomial time solutions on many particular graph classes, some of them defined by forbidden subgraphs. A classical example is the result of Minty that the MWS problem is polynomially solvable for the class of claw-free graphs. The complexity of the MWS problem is unknown for the class of P5-free graphs. We give a survey on recently obtained efficient algorithms for the MWS problem for several graph classes defined by forbidden subgraphs where the algorithm avoids to recognize whether the (arbitrary) input graph is in the class i.e. the output is either a correct solution of the MWS problem or the fact that the input graph is not in the class. Such algorithms were called robust by Spinrad. The algorithms use the concepts of modular decomposition and of clique width of graphs.