Composition of stable set polyhedra
نویسندگان
چکیده
Barahona and Mahjoub found a defining system of the stable set polytope for a graph with a cut-set of cardinality 2. We extend this result to cut-sets composed of a complete graph minus an edge and use the new theorem to derive a class of facets. 1 Preliminaries and Introduction Let G = (V, E) be a simple undirected graph. A set S ⊆ V of pairwise nonadjacent vertices defines a stable set. Let S := {S ⊆ V | S is a stable set}. Each S ∈ S has an incidence vector x ∈ R , where x(v) = 1 if v ∈ S and x(v) = 0 otherwise. Let P (G) be the convex hull of all x such that S ∈ S. P (G) is a full-dimensional polytope and has a unique (up to positive scalar multiples) minimal defining system. Let V ′ ⊂ V and E(V ) := {uv ∈ E | u, v ∈ V }. The subgraph induced by V ′ is G[V ] := (V , E(V )). For v ∈ V , define NG(v) := {u ∈ V | uv ∈ E} and N̄G(v) := NG(v) ∪ {v}. A complete graph consists of pairwise adjacent vertices. A vertex set K ⊆ V is complete whenever G[K] is a complete subgraph. A maximal complete subgraph defines a clique. A vertex v ∈ V is simplicial if G[N(v)] is complete. A cut-set C ⊂ V decomposes G into a pair of proper subgraphs (G1, G2) such that C = V (G1) ∩ V (G2) and all paths from G1 to G2 intersect C. Chvátal [2] showed that the union of defining systems of P (G1) and P (G2) defines P (G) when G[C] is a clique. Barahona and Mahjoub [1] defined P (G) based on systems related to P (G1) and P (G2) when |C| ≤ 2. We extend this result to the case where G[C] is a complete graph minus an edge. Section 2 contains results necessary to extend Barahona and Mahjoub’s theorem. Section 3 generalizes their theorem. Section 3 refers to results from [1]. Section 4 applies the new theorem to derive a class of facets for the stable set polytope called diamonds. Section 5 summarizes the results and discusses future research. [email protected], [email protected] Department of Computational and Applied Mathematics Rice University Houston, 77005, USA. 1
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عنوان ژورنال:
- Oper. Res. Lett.
دوره 36 شماره
صفحات -
تاریخ انتشار 2008