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Finding Induced Subgraphs from Graphs with Small Mim-Width

Yota Otachi, Akira Suzuki, Yuma Tamura

TL;DR

The complexity of the problem of finding a maximum induced subgraph that satisfies prescribed properties from a given graph with small mim-width is investigated and the polynomial-time solvability of various problems for these graph classes is revealed.

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.

Finding Induced Subgraphs from Graphs with Small Mim-Width

TL;DR

The complexity of the problem of finding a maximum induced subgraph that satisfies prescribed properties from a given graph with small mim-width is investigated and the polynomial-time solvability of various problems for these graph classes is revealed.

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.
Paper Structure (19 sections, 29 theorems, 7 equations, 3 figures)

This paper contains 19 sections, 29 theorems, 7 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Previous work on mim-width
  4. Preliminaries
  5. Graph classes
  6. Mim-width
  7. Graph properties and problems
  8. NP-hardness
  9. General case
  10. Special cases
  11. Polynomial-time algorithms for graphs with mim-width at most 1
  12. Cluster Vertex Deletion
  13. Proof of \ref{['lem:chainorder']}
  14. Algorithm
  15. Correctness of our algorithm
  16. ...and 4 more sections

Key Result

Proposition 1

For a graph $G$ and an induced subgraph $G^\prime$ of $G$, it holds that $\mathsf{mimw}(G^\prime) \le \mathsf{mimw}(G)$ and $\mathsf{lmimw}(G^\prime) \le \mathsf{lmimw}(G)$.

Figures (3)

  • Figure 1: Let $F$ be the graph depicted in (a). The cut vertex $c_1$ of the left connected component $F_1$ of $F$ gives $\alpha_1 = \langle 4, 2 \rangle$ and the cut vertex $c_2$ of the right connected component $F_2$ of $F$ gives $\alpha_2 = \langle 2, 2 \rangle$, where $\alpha_1 >_L \alpha_2$. Thus, if $F$ is selected as the base of $\overline{\Pi}$-forbidden subgraphs, $F_{1,1}$ and $F^\prime$ are defined as the graphs depicted in (b) and (c), respectively.
  • Figure 2: A transformation of an edge $uv$ with $F_{1,1}$ and $F^\prime$, which are the graphs depicted in \ref{['fig:forbidden_subgraph']}(b) and (c), respectively.
  • Figure 4: Let $\Pi$ be a collection of polar graphs. (a) The base $F$ of $\overline{\Pi}$-forbidden subgraphs, where $F_{1,1}$ is the graph induced by $\{v_1,v_2,v_3,v_4\}$, $c_1 = v_1$, and $d = v_2$; (b) the complement graph $\overline{F_{1,1}}$; and (c) the good linear branch decomposition of $\overline{F_{1,1}}$.

Theorems & Definitions (29)

  • Proposition 1
  • Proposition 2
  • Proposition 3: Vatshelle12
  • Proposition 4: Vatshelle12
  • Theorem 5
  • Lemma 6
  • Lemma 7
  • Theorem 8
  • Theorem 9
  • Theorem 10
  • ...and 19 more