A Computational Study for the Graph-Theoretic Version of the Union-Closed Sets Conjecture

ABSTRACT An induced subgraph S of a graph G is called a derived subgraph of G if S contains no isolated vertices. An edge e of G is said to be residual if e voccurs in more than half of the derived subgraphs of G . We prove some theorems which calculate the number of derived subgraphs for some special graphs. We also present a new algorithm SDSA that calculates the number of derived subgraphs for a given graph G and determines the residual and non-residual edges. Finally, we introduce a computational study which supports our results. Keywords 2 Union closed sets conjecture, induced graphs, derived subgraphs. 1. INTRODUCTION Example A union-closed family of sets A dis a finite collection of sets not all empty such that the union of any two members of A is also a member of A . The following Conjecture is due to Peter Frankl [1, 2, 3]. Conjecture 1 . Let A = { A 1 , A 2 , . . . , A n } be a union-closed family of n distinct sets. Then there exists an element which belongs to at least