Below All Subsets for Minimal Connected Dominating Set

Abstract

A vertex subset S in a graph G is a dominating set if every vertex not contained in S has a neighbor in S. A dominating set S is a connected dominating set if the subgraph G[S] induced by S is connected. A connected dominating set S is a minimal connected dominating set if no proper subset of S is also a connected dominating set. We prove that there exists a constant ε>10−50 such that every graph G on n vertices has at most O(2(1−ε)n) minimal connected dominating sets. For the same ε we also give an algorithm with running time 2(1−ε)n⋅nO(1) to enumerate all minimal connected dominating sets in an input graph G.