Tree-independence number VI. Thetas and pyramids
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2025
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| author | Chudnovsky, Maria Codsi, Julien |
| author_facet | Chudnovsky, Maria Codsi, Julien |
| contents | Given a family $\mathcal{H}$ of graphs, we say that a graph $G$ is $\mathcal{H}$-free if no induced subgraph of $G$ is isomorphic to a member of $\mathcal{H}$. Let $W_{t\times t}$ be the $t$-by-$t$ hexagonal grid and let $\mathcal{L}_t$ be the family of all graphs $G$ such that $G$ is the line graph of some subdivision of $W_{t \times t}$. We denote by $ω(G)$ the size of the largest clique in $G$. We prove that for every integer $t$ there exist integers $c_1(t)$, $c_2(t)$ and $d(t)$ such that every (pyramid, theta, $\mathcal{L}_t$)-free graph $G$ satisfies: i) $G$ has a tree decomposition where every bag has size at most $ω(G)^{c_1(t)} \log (|V(G)|)$. ii) If $G$ has at least two vertices, then $G$ has a tree decomposition where every bag has independence number at most $\log^{c_2(t)} (|V(G)|)$. 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 $c_3(t), c_4(t)$ such that for every (theta, pyramid)-free graph $G$ and for every non-adjacent pair of vertices $a,b \in V(G)$, i) $a$ can be separated from $b$ by removing at most $w(G)^{c_3(t)}\log(|V(G)|)$ vertices. ii) $a$ can be separated from $b$ by removing a set of vertices with independence number at most $\log^{c_4(t)}(|V(G)|)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_15458 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tree-independence number VI. Thetas and pyramids Chudnovsky, Maria Codsi, Julien Combinatorics Discrete Mathematics Data Structures and Algorithms 05C75 (Primary) 05C40, 05C85 (Secondary) F.2.2; G.2.2 Given a family $\mathcal{H}$ of graphs, we say that a graph $G$ is $\mathcal{H}$-free if no induced subgraph of $G$ is isomorphic to a member of $\mathcal{H}$. Let $W_{t\times t}$ be the $t$-by-$t$ hexagonal grid and let $\mathcal{L}_t$ be the family of all graphs $G$ such that $G$ is the line graph of some subdivision of $W_{t \times t}$. We denote by $ω(G)$ the size of the largest clique in $G$. We prove that for every integer $t$ there exist integers $c_1(t)$, $c_2(t)$ and $d(t)$ such that every (pyramid, theta, $\mathcal{L}_t$)-free graph $G$ satisfies: i) $G$ has a tree decomposition where every bag has size at most $ω(G)^{c_1(t)} \log (|V(G)|)$. ii) If $G$ has at least two vertices, then $G$ has a tree decomposition where every bag has independence number at most $\log^{c_2(t)} (|V(G)|)$. 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 $c_3(t), c_4(t)$ such that for every (theta, pyramid)-free graph $G$ and for every non-adjacent pair of vertices $a,b \in V(G)$, i) $a$ can be separated from $b$ by removing at most $w(G)^{c_3(t)}\log(|V(G)|)$ vertices. ii) $a$ can be separated from $b$ by removing a set of vertices with independence number at most $\log^{c_4(t)}(|V(G)|)$. |
| title | Tree-independence number VI. Thetas and pyramids |
| topic | Combinatorics Discrete Mathematics Data Structures and Algorithms 05C75 (Primary) 05C40, 05C85 (Secondary) F.2.2; G.2.2 |
| url | https://arxiv.org/abs/2509.15458 |