Neighborhood inclusions for minimal dominating sets enumeration: linear and polynomial delay algorithms in $P_7$-free and $P_8$-free chordal graphs

TL;DR

This paper extends neighborhood inclusion techniques to efficiently enumerate minimal dominating sets in certain chordal graphs, providing linear and polynomial delay algorithms for $P_7$-free and $P_8$-free cases, but shows limitations for larger $k$.

Contribution

It introduces $O(n+m)$ and $O(n^3 m)$ delay algorithms for enumerating minimal dominating sets in $P_7$-free and $P_8$-free chordal graphs, extending previous methods.

Findings

Efficient enumeration algorithms for $P_7$-free and $P_8$-free chordal graphs.

Neighborhood inclusion techniques can be generalized to larger classes.

The approach becomes NP-complete for $P_k$-free chordal graphs with $k geq 9$.

Abstract

In [M. M. Kant\'e, V. Limouzy, A. Mary, and L. Nourine. On the enumeration of minimal dominating sets and related notions. SIAM Journal on Discrete Mathematics, 28(4):1916-1929, 2014] the authors give an $O(n+m)$ delay algorithm based on neighborhood inclusions for the enumeration of minimal dominating sets in split and $P_6$-free chordal graphs. In this paper, we investigate generalizations of this technique to $P_k$-free chordal graphs for larger integers $k$. In particular, we give $O(n+m)$ and $O(n^3\cdot m)$ delays algorithms in the classes of $P_7$-free and $P_8$-free chordal graphs. As for $P_k$-free chordal graphs for $k\geq 9$, we give evidence that such a technique is inefficient as a key step of the algorithm, namely the irredundant extension problem, becomes NP-complete.