Output-Polynomial Enumeration on Graphs of Bounded (Local) Linear MIM-Width

TL;DR

This paper develops output-polynomial algorithms for enumerating minimal dominating sets in graphs with bounded LMIM-width, including unit square graphs, using polynomial delay and space.

Contribution

It introduces new enumeration algorithms for minimal dominating sets on graphs with bounded LMIM-width and local LMIM-width, including unit square graphs.

Findings

All 1-minimal and 1-maximal (,)-dominating sets can be enumerated with polynomial delay.

Minimal dominating sets in graphs of bounded LMIM-width can be enumerated with polynomial delay and space.

Enumeration of minimal dominating sets in unit square graphs is achievable in incremental polynomial time.

Abstract

The linear induced matching width (LMIM-width) of a graph is a width parameter defined by using the notion of branch-decompositions of a set function on ternary trees. In this paper we study output-polynomial enumeration algorithms on graphs of bounded LMIM-width and graphs of bounded local LMIM-width. In particular, we show that all 1-minimal and all 1-maximal (\sigma,\rho)-dominating sets, and hence all minimal dominating sets, of graphs of bounded LMIM-width can be enumerated with polynomial (linear) delay using polynomial space. Furthermore, we show that all minimal dominating sets of a unit square graph can be enumerated in incremental polynomial time.