Finding Induced Subgraphs from Graphs with Small Mim-Width

TL;DR

This paper explores the complexity of finding dense induced subgraphs in graphs with small mim-width, showing NP-hardness for many problems but providing polynomial-time algorithms for certain graph classes with mim-width at most 1.

Contribution

It establishes NP-hardness results for induced subgraph problems on bounded mim-width graphs and offers polynomial algorithms for specific graph classes with mim-width at most 1.

Findings

NP-hardness of various induced subgraph problems for bounded mim-width graphs

Polynomial-time algorithms for certain problems on graphs with mim-width at most 1

Applicability of algorithms to classes like block graphs, interval graphs, and cographs

Abstract

In the last decade, algorithmic frameworks based on a structural graph parameter called mim-width have been developed to solve generally NP-hard problems. However, it is known that the frameworks cannot be applied to the Clique problem, and the complexity status of many problems of finding dense induced subgraphs remains open when parameterized by mim-width. In this paper, we investigate the complexity of the problem of finding a maximum induced subgraph that satisfies prescribed properties from a given graph with small mim-width. We first give a meta-theorem implying that various induced subgraph problems are NP-hard for bounded mim-width graphs. Moreover, we show that some problems, including Clique and Induced Cluster Subgraph, remain NP-hard even for graphs with (linear) mim-width at most 2. In contrast to the intractability, we provide an algorithm that, given a graph and its branch decomposition with mim-width at most 1, solves Induced Cluster Subgraph in polynomial time. We emphasize that our algorithmic technique is applicable to other problems such as Induced Polar Subgraph and Induced Split Subgraph. Since a branch decomposition with mim-width at most 1 can be constructed in polynomial time for block graphs, interval graphs, permutation graphs, cographs, distance-hereditary graphs, convex graphs, and their complement graphs, our positive results reveal the polynomial-time solvability of various problems for these graph classes.