Finding Induced Subgraphs from Graphs with Small Mim-Width

Authors Yota Otachi , Akira Suzuki , Yuma Tamura

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Author Details

Yota Otachi
  • Graduate School of Informatics, Nagoya University, Japan
Akira Suzuki
  • Graduate School of Information Sciences, Tohoku University, Sendai, Japan
Yuma Tamura
  • Graduate School of Information Sciences, Tohoku University, Sendai, Japan

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Yota Otachi, Akira Suzuki, and Yuma Tamura. Finding Induced Subgraphs from Graphs with Small Mim-Width. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 38:1-38:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • mim-width
  • graph algorithm
  • NP-hardness
  • induced subgraph problem
  • cluster vertex deletion


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