Tree-independence number VI. Thetas and pyramids

Given a family H of graphs, we say that a graph G is H-free if no induced subgraph of G is isomorphic to a member of H. Let Wt× t be the t-by-t hexagonal grid and let Lₜ be the family of all graphs G such that G is the line graph of some subdivision of Wt × t. We denote by $ω(G)$ the size of the largest clique in G. We prove that for every integer t there exist integers c₁(t), c₂(t) and $d(t)$ such that every (pyramid, theta, Lₜ)-free graph G satisfies: i) G has a tree decomposition where every bag has size at most ω(G)c₁(t) log (|V(G)|). ii) If G has at least two vertices, then G has a tree decomposition where every bag has independence number at most logc₂(t) (|V(G)|). iii) For any weight function, G has a balanced separator that is contained in the union of the neighborhoods of at most $d(t)$ vertices. These results qualitatively generalize the main theorems of Abrishami et al. (2022) and Chudnovsky et al. (2024). Additionally, we show that there exist integers c₃(t), c₄(t) such that for every (theta, pyramid)-free graph G and for every non-adjacent pair of vertices a,b ∈ V(G), i) a can be separated from b by removing at most w(G)c₃(t)log(|V(G)|) vertices. ii) a can be separated from b by removing a set of vertices with independence number at most logc₄(t)(|V(G)|).