Abstract. We deal with non-rank facets of the stable set polytope of claw-free graphs. We extend results of Giles and Trotter [7] by (i) showing that for any nonnegative integer a there exists a circulant graph whose stable set polytope has a facet-inducing inequality with (a,a þ 1)-valued coefficients (rank facets have only coefficients 0, 1), and (ii) providing new facets of the stable set polytope with up to five different non-zero coefficients for claw-free graphs. We prove that coefficients have to be consecutive in any facet with exactly two different non-zero coefficients (assuming they are relatively prime). Last but not least, we present a complete description of the stable set polytope for graphs with stability number 2, already observed by Cook [3] and Shepherd [18].

This paper deals with the problem of representing the matching independence system in a graph as the intersection of finitely many matroids. After characterizing the graphs for which the matching independence system is the intersection of two matroids, we study the function µ(G), which is the minimum number of matroids that need to be intersected in order to obtain the set of matchings on a graph G, and examine the maximal value, µ(n), for graphs with n vertices. We describe an integer programming formulation for deciding whether µ(G) ≤ k. Using combinatorial arguments, we prove that µ(n) ∈ Ω(log log n). On the other hand, we establish that µ(n) ∈ O(log n/ log log n). Finally, we prove that µ(n) = 4 for n = 5, . . . , 12, and sketch a proof of µ(n)=5 for n = 13, 14, 15.

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