Induced subgraphs and tree decompositions X. Towards logarithmic treewidth for even-hole-free graphs

TL;DR

The paper proves that for every integer $t$, there exists a constant $c_t$ such that every even-hole-free graph $G$ with no clique of size $t$ and no generalized $t$-pyramid satisfies $ w(G) \le c_t \log |V(G)|$, advancing a special case of the Sintiari–Trotignon conjecture. The authors introduce $k$-lean tree decompositions and a central-bag framework built around a subgraph $\beta^A(S)$ to manage hubs and non-hubs, avoiding prior obstacles associated with balanced vertices. This yields not only a tight logarithmic bound but also immediate algorithmic consequences, making several NP-hard problems polynomial-time on this class and informing subsequent papers in the series on the full conjecture. The approach hinges on a refined interplay of PMCs, lean decompositions, separators, and connectifiers to structure even-hole-free graphs with bounded clique number and no generalized pyramids, producing a robust toolkit for future structural and algorithmic results.

Abstract

A generalized $t$-pyramid is a graph obtained from a certain kind of tree (a subdivided star or a subdivided cubic caterpillar) and the line graph of a subdivided cubic caterpillar by identifying simplicial vertices. We prove that for every integer $t$ there exists a constant $c(t)$ such that every $n$-vertex even-hole-free graph with no clique of size $t$ and no induced subgraph isomorphic to a generalized $t$-pyramid has treewidth at most $c(t)\log{n}$. This settles a special case of a conjecture of Sintiari and Trotignon; this bound is also best possible for the class. It follows that several \textsf{NP}-hard problems such as \textsc{Stable Set}, \textsc{Vertex Cover}, \textsc{Dominating Set} and \textsc{Coloring} admit polynomial-time algorithms on this class of graphs. Results from this paper are also used in later papers of the series, in particular to solve the full version of the Sintiari-Trotignon conjecture.

Figures (2)

• Figure 1: Theta, prism, pyramid and an even wheel. Dashed lines represent paths of length at least one.
• Figure 2: Examples of generalized $3$-pyramids. Dashed lines represent paths of length at least one.

Theorems & Definitions (69)

• Conjecture 1.1: Sintiari and Trotignon ST
• Conjecture 1.2: Sintiari and Trotignon ST
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1: Menger Menger
• Theorem 2.2: Menger Menger
• Theorem 2.3: Bouchitté and Todinca BouchitteT01
• Theorem 2.4: Bouchitté and Todinca BouchitteT01
• Theorem 2.5: see Diestel diestel
• Theorem 2.6