Counting Dominating Sets in Directed Path Graphs

TL;DR

This paper investigates the computational complexity of counting dominating sets in directed path graphs, establishing #P-completeness for general directed path graphs and polynomial-time solvability for rooted variants.

Contribution

It proves #P-completeness for directed path graphs and identifies polynomial-time solvability for rooted directed path graphs.

Findings

Counting dominating sets is #P-complete for directed path graphs.

Counting dominating sets is polynomial-time solvable for rooted directed path graphs.

Abstract

A dominating set of a graph is a set of vertices such that every vertex not in the set has at least one neighbor in the set. The problem of counting dominating sets is #P-complete for chordal graphs but solvable in polynomial time for its subclass of interval graphs. The complexity status of the corresponding problem is still undetermined for directed path graphs, which are a well-known class of graphs that falls between chordal graphs and interval graphs. This paper reveals that the problem of counting dominating sets remains #P-complete for directed path graphs but a stricter constraint to rooted directed path graphs admits a polynomial-time solution.