Below all subsets for Minimal Connected Dominating Set

TL;DR

This paper proves an exponential upper bound on the number of minimal connected dominating sets in any graph and provides an efficient algorithm to enumerate all such sets, improving understanding of their complexity.

Contribution

It establishes a non-trivial exponential bound on minimal connected dominating sets and introduces an algorithm with subexponential runtime for enumeration.

Findings

Bound of $O(2^{(1- ext{epsilon})n})$ on minimal connected dominating sets

Existence of an enumeration algorithm with runtime $2^{(1- ext{epsilon})n} imes n^{O(1)}$

Constant epsilon > 10^{-50} ensuring these bounds

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 $\varepsilon > 10^{-50}$ such that every graph $G$ on $n$ vertices has at most $O(2^{(1-\varepsilon)n})$ minimal connected dominating sets. For the same $\varepsilon$ we also give an algorithm with running time $2^{(1-\varepsilon)n}\cdot n^{O(1)}$ to enumerate all minimal connected dominating sets in an input graph $G$.