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Sparse graphs with bounded induced cycle packing number have logarithmic treewidth

Marthe Bonamy, Édouard Bonnet, Hugues Déprés, Louis Esperet, Colin Geniet, Claire Hilaire, Stéphan Thomassé, Alexandra Wesolek

TL;DR

The paper investigates induced cycle packing in graphs through the lens of $\mathcal{O}_k$-freeness and sparsity, proving that $\mathcal{O}_k$-free graphs excluding a $K_{t,t}$ subgraph have a feedback vertex set of size $O_{t,k}(\log n)$ and hence logarithmic treewidth. Central to the approach is the cycle rank $r(G)$ and a rich-vertex argument that iteratively removes high-degree vertices to reduce the graph to a forest, yielding strong structural bounds. This leads to quasi-polynomial algorithms for Maximum Independent Set and 3-Coloring on $\mathcal{O}_k$-free graphs and, more broadly, polynomial-time solvability for many NP-hard problems in sparse $\mathcal{O}_k$-free graphs, along with polynomial-time testing of $\mathcal{O}_k$-freeness in sparse graphs. The results tie into induced grid-minor theory and provide tight lower bounds via explicit constructions, charting a path toward practical algorithms on broad graph families.

Abstract

A graph is $\mathcal{O}_k$-free if it does not contain $k$ pairwise vertex-disjoint and non-adjacent cycles. We prove that "sparse" (here, not containing large complete bipartite graphs as subgraphs) $\mathcal{O}_k$-free graphs have treewidth (even, feedback vertex set number) at most logarithmic in the number of vertices. This is optimal, as there is an infinite family of $\mathcal{O}_2$-free graphs without $K_{2,3}$ as a subgraph and whose treewidth is (at least) logarithmic. Using our result, we show that Maximum Independent Set and 3-Coloring in $\mathcal{O}_k$-free graphs can be solved in quasi-polynomial time. Other consequences include that most of the central NP-complete problems (such as Maximum Independent Set, Minimum Vertex Cover, Minimum Dominating Set, Minimum Coloring) can be solved in polynomial time in sparse $\mathcal{O}_k$-free graphs, and that deciding the $\mathcal{O}_k$-freeness of sparse graphs is polynomial time solvable.

Sparse graphs with bounded induced cycle packing number have logarithmic treewidth

TL;DR

The paper investigates induced cycle packing in graphs through the lens of -freeness and sparsity, proving that -free graphs excluding a subgraph have a feedback vertex set of size and hence logarithmic treewidth. Central to the approach is the cycle rank and a rich-vertex argument that iteratively removes high-degree vertices to reduce the graph to a forest, yielding strong structural bounds. This leads to quasi-polynomial algorithms for Maximum Independent Set and 3-Coloring on -free graphs and, more broadly, polynomial-time solvability for many NP-hard problems in sparse -free graphs, along with polynomial-time testing of -freeness in sparse graphs. The results tie into induced grid-minor theory and provide tight lower bounds via explicit constructions, charting a path toward practical algorithms on broad graph families.

Abstract

A graph is -free if it does not contain pairwise vertex-disjoint and non-adjacent cycles. We prove that "sparse" (here, not containing large complete bipartite graphs as subgraphs) -free graphs have treewidth (even, feedback vertex set number) at most logarithmic in the number of vertices. This is optimal, as there is an infinite family of -free graphs without as a subgraph and whose treewidth is (at least) logarithmic. Using our result, we show that Maximum Independent Set and 3-Coloring in -free graphs can be solved in quasi-polynomial time. Other consequences include that most of the central NP-complete problems (such as Maximum Independent Set, Minimum Vertex Cover, Minimum Dominating Set, Minimum Coloring) can be solved in polynomial time in sparse -free graphs, and that deciding the -freeness of sparse graphs is polynomial time solvable.
Paper Structure (9 sections, 32 theorems, 21 equations, 6 figures)

This paper contains 9 sections, 32 theorems, 21 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Organization of the paper.
  3. Sparse $\mathcal{O}_2$-free graphs with unbounded treewidth
  4. Algorithmic applications
  5. Preliminary results
  6. Logarithmic treewidth of sparse $\mathcal{O}_k$-free graphs
  7. Cutting trees into stars and paths
  8. Trees, stars, and paths
  9. Proof of Theorem \ref{['thm:richvertex']}

Key Result

Theorem 1.1

Every $\mathcal{O}_k$-free graph on $n$ vertices that does not contain $K_{t,t}$ as a subgraph has a feedback vertex set of size $O_{t,k}(\log n)$.

Figures (6)

  • Figure 1: The graph $G_k$ for $k=5$: an $\mathcal{O}_2$-free graph without $K_{3,3}$ subgraph, $k+2^k-1$ vertices, and treewidth $k$.
  • Figure 2: Subgraph of an $\mathcal{O}_4$-free graph $G$. $V(G)$ is partitioned into three sets $C,N,R$, where $C$ is a shortest cycle, $N$ is an independent set and first neighborhood of $C$, and $R$ is $\mathcal{O}_{3}$-free. $S$ is the second neighborhood of $N$. Gray lines correspond to induced paths where all internal vertices have degree 2.
  • Figure 3: A visual summary of Section \ref{['sec:cutting']}.
  • Figure 4: The graphs $G_1$ (left) and $G_2$ (right) in the proof of \ref{['cl:case2']}.
  • Figure 5: The graphs $G_3$ (left) and $G_4$ (right) in the proof of \ref{['claim:smalldegree']}.
  • ...and 1 more figures

Theorems & Definitions (83)

  • Theorem 1.1
  • Corollary 1.1
  • Conjecture 1.2
  • Theorem 1.3
  • Conjecture 1.4: Le Khang17
  • Theorem 2.1
  • Corollary 3.0
  • Corollary 3.1
  • proof
  • Theorem 3.2: Pilipczuk11
  • ...and 73 more