The stable set polytope is a fundamental object in combinatorial optimization. Among the many valid inequalities that are known for it, clique-family play an important role. Pêcher and Wagler showed can be strengthened under certain conditions. We show they even further, using surprisingly simple mixed-integer rounding argument.

Let G = (V, E) be a simple undirected graph. A forest F ⊆ E of G is said to be clique-connecting if each tree of F spans a clique of G. This paper adresses the clique-connecting forest polytope. First we give a formulation and a polynomial time separation algorithm. Then we show that the nontrivial nondegenerate facets of the stable set polytope are facets of the clique-connecting polytope. Fin...

There can, of course, be no concessions as regards existence. If it should turn out that our requirements concerning a solution. . . are, in any special case, unfulfillable,—this would certainly necessitate a fundamental change in the theory. Thus a general proof of the existence of solutions. . . for all particular cases1 is most desirable. It will appear from our subsequent investigations tha...

In this article, we provide an overview on how the maximum weighted stable set problem can be solved exactly with Branch & Cut techniques. In addition, we provide selected references to other exact methods. We start with a brief introduction of the stable set problem and a few basic definitions but assuming that the reader is already familiar with the basic concepts. The main stress of this art...

The 2-bond is a generalization of the 2-join where the subsets of nodes that are connected on each shore of the partition are not necessarily disjoint. If all the subsets are cliques we say that the 2-bond is a 2-clique-bond. We consider a graph G obtained as the 2-clique-bond of two graphs G1 and G2 and we study the polyhedral properties of the stable set polytope associated with this graph. I...

The stable set of von Neumann and Morgenstern imposes credibility on coalitional deviations. Their credibility notion can be extended to cover farsighted coalitional deviations, as proposed by Harsanyi (1974), and more recently reformulated by Ray and Vohra (2015). However, the resulting farsighted stable set suffers from a conceptual drawback: while coalitional deviations improve on existing o...

We present a new graph composition that produces a graph G from a given graph H and a fixed graph B called gear and we study its polyhedral properties. This composition yields counterexamples to a conjecture on the facial structure of STAB(G) when G is claw-free.

A facet of the stable set polytope of a graph G can be viewed as a generalization of the notion of an-critical graph. We extend several results from the theory of-critical graphs to facets. The defect of a nontrivial, full-dimensional facet P v2V a(v)x v b of the stable set polytope of a graph G is deened by = P v2V a(v) ?2b. We prove the upper bound a(u) + for the degree of any vertex u in a c...

We give a simple algorithm for the weighted stable set problem of an arbitrary graph which yields an extended formulation for its stable set polytope The algorithm runs in polynomial time for the class of distance claw free graphs These are the graphs such that for each node neither its neighbour set nor the set of nodes at distance two contain a stable set of size three The extended formulatio...

A facet of the stable set polytope of a graph G can be viewed as a generalization of the notion of an α-critical graph. We extend several results from the theory of α-critical graphs to facets. The defect of a nontrivial, full-dimensional facet ∑ v∈V a(v)xv ≤ b of the stable set polytope of a graph G is defined by δ = ∑ v∈V a(v)−2b. We prove the upper bound a(u) + δ for the degree of any node u...