Induced Cycles of Many Lengths

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.

💡 Research Summary

The paper investigates the relationship between the number of distinct induced‑cycle lengths in a graph, denoted cl(G), and the graph’s treewidth. The central structural result (Theorem 1.3) states that for any fixed integers c and t, there exists a constant f₁.₃(c,t) such that every graph G with treewidth at least f₁.₃(c,t) must either contain a K_{t+1} or a K_{t,t} as an induced subgraph, or else satisfy cl(G) ≥ c. In other words, large treewidth forces the presence of many induced‑cycle lengths unless a large complete or complete bipartite subgraph is present.

To exploit this structural insight algorithmically, the authors develop two complementary procedures. First, for graphs of bounded treewidth (treewidth ≤ k), Theorem 1.2 provides a linear‑time algorithm that decides whether cl(G) ≥ c and, if so, outputs c induced cycles of pairwise distinct lengths. The algorithm builds a tree decomposition, then runs a dynamic program over the decomposition, maintaining for each bag the set of cycle lengths that can be realized within the subgraph induced by the bag’s vertices and its descendants. Because each bag contains at most k + 1 vertices, the state space is bounded by a function of k and c, yielding an O(|V|) running time.

Second, for unrestricted graphs the paper shows that deciding whether cl(G) ≥ 3 can be done in polynomial time (Theorem 1.1). The approach is to first handle the trivial cases where G contains a triangle K₃ or a 4‑cycle K_{2,2}, using the known non‑shortest‑path algorithm of Berger, Seymour, and Spirkl. If those subgraphs are absent, G is {K₃, K_{2,2}}‑free, and Theorem 1.3 guarantees that G has bounded treewidth. The bounded‑treewidth subroutine from Theorem 1.2 then decides the problem in linear time on this reduced instance. The overall complexity is O(|V|²), dominated by the initial detection of the forbidden small subgraphs.

The proof of Theorem 1.3 is the technical heart of the work. The authors introduce a hierarchy of intricate structures—(s,d)‑domes, q‑bananas, q‑thetas, and (k,q)‑kites—each designed to capture a specific pattern that forces many induced‑cycle lengths. A dome consists of a subdivided star whose leaves are linked to a path P; when the dome is “aligned,” the ordering of leaf‑to‑path connections is consistent, enabling the construction of a sequence of induced cycles with strictly increasing lengths. A banana is a collection of internally disjoint x–y paths; a rigid banana has pairwise complete connections among the interiors of distinct paths. The authors prove that any large dense banana either already yields many induced cycles or can be transformed into a large aligned dome (Theorem 4.1). This transformation proceeds via the analysis of rigid bananas (Section 5), the extraction of many tailed thetas (Section 6), and the assembly of a chain of kites (Section 7). Each kite contributes a piece of the eventual dome: the final kite supplies the central path P, while earlier kites provide the subdivided star’s arms. Lemma 3.6 shows that an aligned dome with sufficiently many leaves guarantees cl(G) ≥ c, using the Erdős–Szekeres monotone subsequence theorem to select a monotone set of leaf‑to‑path distances.

Section 3 also handles “basic obstructions”: large grid subdivisions, line graphs of such subdivisions, proper subdivisions of complete graphs, and induced K_{r,r} minors. By invoking known results on constellations and Ramsey theory, the authors demonstrate that any graph containing one of these configurations must already have many induced‑cycle lengths, allowing them to focus on the more subtle banana‑to‑dome reduction.

Finally, Section 8 details the dynamic‑programming subroutine for bounded‑treewidth graphs, and Section 9 combines all pieces to obtain the polynomial‑time algorithm for cl(G) ≥ 3. The paper thus bridges structural graph theory (excluded minors, treewidth, Ramsey‑type arguments) with algorithmic graph theory (tree‑decomposition DP, non‑shortest‑path detection), delivering both a deep combinatorial theorem and concrete computational tools.