Induced Cycles of Many Lengths

TL;DR

This paper proves that graphs with a limited number of induced cycle lengths and excluding certain subgraphs have bounded treewidth, enabling efficient algorithms for detecting multiple induced cycle lengths.

Contribution

It establishes a link between the number of induced cycle lengths, forbidden subgraphs, and bounded treewidth, leading to new algorithmic results.

Findings

Graphs with bounded induced cycle lengths and forbidden subgraphs have bounded treewidth.

Polynomial-time algorithm for detecting multiple induced cycle lengths.

Characterization of graph classes based on induced cycle length constraints.

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.