A $\beta$ Invariant for Greedoids and Antimatroids

Abstract

We extend Crapo's $\beta $ invariant from matroids to greedoids, concentrating especially on antimatroids. Several familiar expansions for $\beta (G)$ have greedoid analogs. We give combinatorial interpretations for $\beta (G)$ for simplicial shelling antimatroids associated with chordal graphs. When $G$ is this antimatroid and $b(G)$ is the number of blocks of the chordal graph $G$, we prove $\beta (G)=1-b(G)$.

Linear Relations for a Generalized Tutte Polynomial

“…The invariant on the left-hand side is b 1,0 , the beta invariant, and the right-hand side is an alternating sum over convex sets with exactly one interior point. The beta invariant gives interesting combinatorial information about the antimatroid -this is the focus of [13]. We will examine this identity for several classes of antimatroids below.…”

“…A set of vertices is free convex if the graph it induces is a clique in G. The next result gives a combinatorial interpretation to the beta invariant b 1,0 . Theorem 4.9 (Theorem 5.1 [13]). Let G be a chordal graph with b 2-connected blocks, and let f i be the number of cliques of size i.…”

“…Theorem 5.1[13]). Let G be a chordal graph with b 2-connected blocks, and let f i be the number of cliques of size i. Thenn i=0 (−1) i if i = b − 1.…”