Bounding the mim‐width of hereditary graph classes

Nick Brettell,Andrea Munaro,Daniël Paulusma,Giacomo Paesani,Jake Horsfield

doi:10.1002/jgt.22730

Abstract

AbstractA large number of ‐hard graph problems are solvable in time when parameterized by some width parameter. Hence, when solving problems on special graph classes, it is helpful to know if the graph class under consideration has bounded width. In this paper we consider maximum‐induced matching width (mim‐width), a particularly general width parameter that has a number of algorithmic applications whenever a decomposition is “quickly computable” for the graph class under consideration. We start by extending the toolkit for proving (un)boundedness of mim‐width of graph classes. By combining our new techniques with known ones we then initiate a systematic study into bounding mim‐width from the perspective of hereditary graph classes, and make a comparison with clique‐width, a more restrictive width parameter that has been well studied. We prove that for a given graph , the class of ‐free graphs has bounded mim‐width if and only if it has bounded clique‐width. We show that the same is not true for ‐free graphs. We identify several general classes of ‐free graphs having unbounded clique‐width, but bounded mim‐width; moreover, we show that a branch decomposition of constant mim‐width can be found in polynomial time for these classes. Hence, these results have algorithmic implications: when the input is restricted to such a class of ‐free graphs, many problems become polynomial‐time solvable, including classical problems, such as ‐ Colouring and Independent Set, domination‐type problems known as Locally Checkable Vertex Subset and Vertex Partitioning (LC‐VSVP) problems, and distance versions of LC‐VSVP problems, to name just a few. We also prove a number of new results showing that, for certain and , the class of ‐free graphs has unbounded mim‐width. Boundedness of clique‐width implies boundedness of mim‐width. By combining our results with the known bounded cases for clique‐width, we present summary theorems of the current state of the art for the boundedness of mim‐width for ‐free graphs. In particular, we classify the mim‐width of ‐free graphs for all pairs with . When and are connected graphs, we classify all pairs except for one remaining infinite family and a few isolated cases.

Highlights

Many computationally hard graph problems can be solved efficiently after placing appropriate restrictions on the input graph
We prove a number of new results showing that, for certain H1 and H2, the class of (H1, H2)-free graphs has unbounded mim-width
Questions we aim to address in this paper are: Does there exist a hereditary graph class characterized by a finite set F that has bounded mim-width but unbounded clique-width? Can we use the same techniques as when dealing with clique-width? In particular we will focus on the case where |F| = 2, say F = {H1, H2}

Summary

Introduction

Many computationally hard graph problems can be solved efficiently after placing appropriate restrictions on the input graph. The ultimate goal in this type of research is to obtain complexity dichotomies for large families of graph problems. Such dichotomies tell us for which graph classes a certain problem or set of problems can or cannot be solved efficiently (under standard complexity assumptions). One reason that might explain the jump from computational hardness to tractability after restricting the input to some graph class G is that G has bounded “width”, that is, every graph in G has width at most c for some constant c. The width parameters boolean-width, clique-width, module-width, NLC-width and rank-width are all equivalent [15, 36, 42, 44], but more powerful than the equivalent parameters branch-width and treewidth [19, 45, 47]

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Results

Conclusion