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 Wₓ ₓ 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 Wₓ ₓ. 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) (|V (G) |). ii) If G has at least two vertices, then G has a tree decomposition where every bag has independence number at most ^c₂ (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) (|V (G) |) vertices. ii) a can be separated from b by removing a set of vertices with independence number at most ^c₄ (t) (|V (G) |).