The stable set polytope of claw-free graphs with stability number at least four. I. Fuzzy antihat graphs are W-perfect

Abstract Fuzzy antihat graphs are graphs obtained as 2-clique-bond compositions of fuzzy line graphs with three different types of three-cliqued graphs. By the decomposition theorem of Chudnovsky and Seymour [2] , fuzzy antihat graphs form a large subclass of claw-free, not quasi-line graphs with stability number at least four and with no 1-joins. A graph is W -perfect if its stable set polytope is described by: nonnegativity, rank, and lifted 5-wheel inequalities. By exploiting the polyhedral properties of the 2-clique-bond composition, we prove that fuzzy antihat graphs are W -perfect and we move a crucial step towards the solution of the longstanding open question of finding an explicit linear description of the stable set polytope of claw-free graphs

Small Chvatal rank

We propose a variant of the Chvatal-Gomory procedure that will produce a sufficient set of facet normals for the integer hulls of all polyhedra {xx : Ax <= b} as b varies. The number of steps needed is called the small Chvatal rank (SCR) of A. We characterize matrices for which SCR is zero via the notion of supernormality which generalizes unimodularity. SCR is studied in the context of the stable set problem in a graph, and we show that many of the well-known facet normals of the stable set polytope appear in at most two rounds of our procedure. Our results reveal a uniform hypercyclic structure behind the normals of many complicated facet inequalities in the literature for the stable set polytope. Lower bounds for SCR are derived both in general and for polytopes in the unit cube.Comment: 24 pages, 3 figures, v3. Major revision: additional author, new application to stable-set polytopes, reorganization of sections. Accepted for publication in Mathematical Programmin

Lift-and-project ranks of the stable set polytope of joined a-perfect graphs

In this paper we study lift-and-project polyhedral operators defined by Lov?asz and Schrijver and Balas, Ceria and Cornu?ejols on the clique relaxation of the stable set polytope of web graphs. We compute the disjunctive rank of all webs and consequently of antiweb graphs. We also obtain the disjunctive rank of the antiweb constraints for which the complexity of the separation problem is still unknown. Finally, we use our results to provide bounds of the disjunctive rank of larger classes of graphs as joined a-perfect graphs, where near-bipartite graphs belong

The Strong Perfect Graph Conjecture: 40 years of Attempts, and its Resolution

International audienceThe Strong Perfect Graph Conjecture (SPGC) was certainly one of the most challenging conjectures in graph theory. During more than four decades, numerous attempts were made to solve it, by combinatorial methods, by linear algebraic methods, or by polyhedral methods. The first of these three approaches yielded the first (and to date only) proof of the SPGC; the other two remain promising to consider in attempting an alternative proof. This paper is an unbalanced survey of the attempts to solve the SPGC; unbalanced, because (1) we devote a signicant part of it to the 'primitive graphs and structural faults' paradigm which led to the Strong Perfect Graph Theorem (SPGT); (2) we briefly present the other "direct" attempts, that is, the ones for which results exist showing one (possible) way to the proof; (3) we ignore entirely the "indirect" approaches whose aim was to get more information about the properties and structure of perfect graphs, without a direct impact on the SPGC. Our aim in this paper is to trace the path that led to the proof of the SPGT as completely as possible. Of course, this implies large overlaps with the recent book on perfect graphs [J.L. Ramirez-Alfonsin and B.A. Reed, eds., Perfect Graphs (Wiley & Sons, 2001).], but it also implies a deeper analysis (with additional results) and another viewpoint on the topic

Characterizing Structurally Cohesive Clusters in Networks: Theory and Algorithms

This dissertation aims at developing generalized network models and solution approaches for studying cluster detection problems that typically arise in networks. More specifically, we consider graph theoretic relaxations of clique as models for characterizing structurally cohesive and robust subgroups, developing strong upper bounds for the maximum clique problem, and present a new relaxation that is useful in clustering applications. We consider the clique relaxation models of k-block, and k-robust 2-club for describing cohesive clusters that are reliable and robust to disruptions, and introduce a new relaxation called s-stable cluster, for modeling stable clusters. First, we identify the structural properties associated with the models, and investigate the computational complexity of these problems. Next, we develop mathematical programming techniques for the optimization problems introduced, and apply them in presenting effective solution approaches to the problems. We present integer programming formulations for the optimization problems of interest, and provide a detailed study of the associated polytopes. Particularly, we develop valid inequalities and identify different classes of facets for the polytopes. Exact solution approaches developed for solving the problems include simple branch and bound, branch and cut, and combinatorial branch and bound algorithms. In addition, we introduce many preprocessing techniques and heuristics to enhance their performance. The presented algorithms are tested computationally on a number of graph instances, that include social networks and random graphs, to study the capability of the proposed solution methods. As a fitting conclusion to this work, we propose new techniques to get easily computable and strong upper bounds for the maximum clique problem. We investigate k-core and its stronger variant k-core/2-club in this light, and present minimization problems to get an upper bound on the maximization problems. Simple linear programming relaxations are developed and strengthened by valid inequalities, which are then compared with some standard relaxations from the literature. We present a detailed study of our computational results on a number of benchmark instances to test the effectiveness of our technique for getting good upper bounds