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Induced Cycles of Many Lengths

Maria Chudnovsky, Ilya Maier

TL;DR

This work investigates the spectrum of induced cycle lengths in graphs and its interaction with treewidth. It proves a structural dichotomy: for fixed $c$ and $t$, a graph that is $K_{t+1}$- and $K_{t,t}$-free either has large treewidth or contains at least $c$ distinct induced cycle lengths; in particular, bounded treewidth plus a bound on $\mathrm{cl}(G)$ implies strong structural control. The authors develop a rich set of gadgets—domes, bananas, thetas, and kites—and show how to pass from dense bananas to domes, enabling a linear-time subroutine to decide $\mathrm{cl}(G) \ge c$ for fixed $c$ and $k$, and a high-degree algorithm (polynomial-time for $c=3$, $n^{22}$ in general) to detect three distinct induced cycle lengths. The results yield practical algorithms for detecting multiple induced-cycle lengths and deepen the understanding of how induced-cycles interact with forbidden induced minors/minors and treewidth.

Abstract

Let $G$ be a graph and let $\mathrm{cl}(G)$ be the number of distinct induced cycle lengths in $G$. We show that for $c,t\in \mathbb N$, every graph $G$ that does not contain an induced subgraph isomorphic to $K_{t+1}$ or $K_{t,t}$ and satisfies $\mathrm{cl}(G) \le c$ has bounded treewidth. As a consequence, we obtain a polynomial-time algorithm for deciding whether a graph $G$ contains induced cycles of at least three distinct lengths.

Induced Cycles of Many Lengths

TL;DR

This work investigates the spectrum of induced cycle lengths in graphs and its interaction with treewidth. It proves a structural dichotomy: for fixed and , a graph that is - and -free either has large treewidth or contains at least distinct induced cycle lengths; in particular, bounded treewidth plus a bound on implies strong structural control. The authors develop a rich set of gadgets—domes, bananas, thetas, and kites—and show how to pass from dense bananas to domes, enabling a linear-time subroutine to decide for fixed and , and a high-degree algorithm (polynomial-time for , in general) to detect three distinct induced cycle lengths. The results yield practical algorithms for detecting multiple induced-cycle lengths and deepen the understanding of how induced-cycles interact with forbidden induced minors/minors and treewidth.

Abstract

Let be a graph and let be the number of distinct induced cycle lengths in . We show that for , every graph that does not contain an induced subgraph isomorphic to or and satisfies has bounded treewidth. As a consequence, we obtain a polynomial-time algorithm for deciding whether a graph contains induced cycles of at least three distinct lengths.
Paper Structure (14 sections, 26 theorems, 8 equations, 6 figures)

This paper contains 14 sections, 26 theorems, 8 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Key structures
  3. Proof outline and organization
  4. Definitions
  5. Some good cases
  6. Removing basic obstructions
  7. Handling domes
  8. Reducing to a banana
  9. Analyzing bananas
  10. From a banana to a kite
  11. From kites to a dome
  12. Detecting induced cycles
  13. Finding three induced cycles
  14. Acknowledgments

Key Result

Theorem 1.1

Let $G$ be a graph. There exists an algorithm with running time $O(|V(G)|^{22})$ that decides whether $\mathop{\mathrm{cl}}\nolimits(G)\ge 3$ and outputs a list of three induced cycles of distinct lengths if they exist.

Figures (6)

  • Figure 1: A $(4,3)$-dome that is not aligned.
  • Figure 2: A 4-theta (left) and a rigid 4-banana (right).
  • Figure 3: A $(3,2)$-kite (left) and a clean $(3,2)$-kite (right).
  • Figure 4: A proper subdivision of $W_{1\times 4}$ (top left), its line graph (top right), the graph $W_{2\times 4}$ (bottom left), and a proper subdivision of $W_{1\times 2}$ (bottom right). Note that $W_{2\times 4}$ contains a proper subdivision of $W_{1\times 2}$.
  • Figure 5: An ample $(3,4)$-constellation.
  • ...and 1 more figures

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1: Erdös, Moser tournament
  • Theorem 2.2: Ramsey Ramsey
  • Theorem 3.1: Chudnovsky, Hajebi, Spirkl tw16
  • Theorem 3.2: Erdös, Szekeres ErdosSzekeres
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 28 more