On the b-Stable Set Polytope of Graphs without Bad K4

We prove that for a graph G = (V,E) without bad K4 subdivision, and for b ∈ Z ∪E + , the b-stable set polytope is determined by the system of constraints determined by the vertices, edges and odd circuits. We also prove that this system is totally dual integral. This relates to t-perfect graphs. Let G = (V,E) be a graph and let b ∈ ZV ∪E + . Then a b-stable set in G is a vector x ∈ Z+ satisfying xv ≤ bv for every vertex v and xu + xv ≤ buv for every edge uv. The b-stable set polytope of G is defined as the convex hull of the b-stable sets in G. We will use the following notation. For sets B ⊆ A and a vector x ∈ RA, let χB be the characteristic vector of B and let x(B) := xTχB. For an edge {u, v} we will use the shorthand notation uv. The vectors in the b-stable set polytope obviously satisfy the following system of inequalities. (i) 0 ≤ xv ≤ bv for each v ∈ V ; (1) (ii) xu + xv ≤ buv for each edge uv ∈ E; (iii) x(V C) ≤ b 2b(EC)c for each odd circuit C. We call a graph G t-perfect with respect to b if the b-stable set polytope is determined by (1). Since each integral vector satisfying (1) is a b-stable set, the polytope determined by (1) equals the b-stable set polytope if and only if it is integral. We call a graph G strongly t-perfect with respect to b if system (1) is totally dual integral. For any weight function w ∈ Z+ and any b ∈ ZV ∪E + , denote by α(G, b, w) the maximum w-weight wTx of a b-stable set x in G. Define a w-cover as a family of vertices, edges and odd circuits in G that covers each vertex v at least wv times. The b-cost of a w-cover is defined as the sum of the costs of its elements, where the cost of a vertex v equals bv, the cost of an edge e equals be and the cost of an odd circuit C equals b 2b(EC)c. Denote by ∗Department of Mathematics, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands ([email protected]). †CWI, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands, and Department of Mathematics, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands ([email protected]).