Independent set and matching permutations

Abstract

Let G be a graph G whose largest independent set has size m. A permutation π of {1, . . . , m} is an independent set permutation of G ifwhere a k (G) is the number of independent sets of size k in G. In 1987 Alavi, Malde, Schwenk and Erdős proved that every permutation of {1, . . . , m} is an independent set permutation of some graph. They raised the question of determining, for each m, the smallest number f (m) such that every permutation of {1, . . . , m} is an independent set permutation of some graph with at most f (m) vertices, and they gave an upper bound on f (m) of roughly m 2m . Here we settle the question, determining f (m) = m m .Alavi et al. also considered matching permutations, defined analogously to independent set permutations. They observed that not every permutation is a matching permutation of some graph, putting an upper bound of 2 m−1 on the number of matching permutations of {1, . . . , m}. Confirming their speculation that this upper bound is not tight, we improve it to O(2 m / √ m).We also consider an extension of independent set permutations to weak orders, and extend Alavi et al.'s main result to show that every weak order on {1, . . . , m} can be realized by the independent set sequence of some graph with largest independent set size m, and with at most m m+2 vertices.

On the independent set sequence of a tree

“…In contrast, the independent set sequence of a graph G -the sequence whose kth term i k = i k (G) is the number of independent sets (sets of mutually non adjacent vertices) of size k in G -is not in general unimodal. Alavi, Malde, Schwenk and Erdős [1] showed, in fact, that it can be arbitrarily far from unimodal, in a precise sense (see also [3]).…”