Efficient Enumeration of Dominating Sets for Sparse Graphs

TL;DR

This paper introduces two algorithms for enumerating all dominating sets in sparse graphs, achieving optimal or constant time enumeration depending on the graph class, thus advancing enumeration methods for these graph types.

Contribution

It presents novel algorithms for enumerating all dominating sets in sparse graphs, including degenerate and high-girth graphs, with optimal and constant time performance.

Findings

First algorithm enumerates all dominating sets in $O(k)$ time per solution for $k$-degenerate graphs.

Second algorithm enumerates all dominating sets in constant time for graphs with girth at least nine.

Algorithms are efficient for classes like trees, planar graphs, and $H$-minor free graphs.

Abstract

A dominating set $D$ of a graph $G$ is a set of vertices such that any vertex in $G$ is in $D$ or its neighbor is in $D$. Enumeration of minimal dominating sets in a graph is one of central problems in enumeration study since enumeration of minimal dominating sets corresponds to enumeration of minimal hypergraph transversal. However, enumeration of dominating sets including non-minimal ones has not been received much attention. In this paper, we address enumeration problems for dominating sets from sparse graphs which are degenerate graphs and graphs with large girth, and we propose two algorithms for solving the problems. The first algorithm enumerates all the dominating sets for a $k$-degenerate graph in $O(k)$ time per solution using $O(n + m)$ space, where $n$ and $m$ are respectively the number of vertices and edges in an input graph. That is, the algorithm is optimal for graphs with constant degeneracy such as trees, planar graphs, $H$-minor free graphs with some fixed $H$. The second algorithm enumerates all the dominating sets in constant time per solution for input graphs with girth at least nine.