A width parameter useful for chordal and co-comparability graphs

TL;DR

This paper introduces a new graph width parameter called sim-width, explores its properties in relation to mim-width, and applies these concepts to various graph classes including chordal, co-comparability, and circle graphs, with algorithmic implications.

Contribution

The paper defines sim-width, demonstrates its advantages over mim-width, and analyzes its bounds in different graph classes, extending the understanding of graph decompositions and algorithmic applications.

Findings

Chordal and co-comparability graphs have bounded sim-width.

Certain graph classes have unbounded mim-width and sim-width.

Algorithmic solutions are feasible for specific classes with bounded mim-width.

Abstract

We investigate new graph classes of bounded mim-width, strictly extending interval graphs and permutation graphs. The graphs $K_t \boxminus K_t$ and $K_t \boxminus S_t$ are graphs obtained from the disjoint union of two cliques of size $t$, and one clique of size $t$ and one independent set of size $t$ respectively, by adding a perfect matching. We prove that : (1) interval graphs are $(K_3\boxminus S_3)$-free chordal graphs; and $(K_t\boxminus S_t)$-free chordal graphs have mim-width at most $t-1$, (2) permutation graphs are $(K_3\boxminus K_3)$-free co-comparability graphs; and $(K_t\boxminus K_t)$-free co-comparability graphs have mim-width at most $t-1$, (3) chordal graphs and co-comparability graphs have unbounded mim-width in general. We obtain several algorithmic consequences; for instance, while Minimum Dominating Set is NP-complete on chordal graphs, it can be solved in time $n^{\mathcal{O}(t)}$ on $(K_t\boxminus S_t)$-free chordal graphs. The third statement strengthens a result of Belmonte and Vatshelle stating that either those classes do not have constant mim-width or a decomposition with constant mim-width cannot be computed in polynomial time unless $P=NP$. We generalize these ideas to bigger graph classes. We introduce a new width parameter sim-width, of stronger modelling power than mim-width, by making a small change in the definition of mim-width. We prove that chordal graphs and co-comparability graphs have sim-width at most 1. We investigate a way to bound mim-width for graphs of bounded sim-width by excluding $K_t\boxminus K_t$ and $K_t\boxminus S_t$ as induced minors or induced subgraphs, and give algorithmic consequences. Lastly, we show that circle graphs have unbounded sim-width, and thus also unbounded mim-width.