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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2510.15074 |
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| _version_ | 1866914533045436416 |
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| author | Chudnovsky, Maria S, Ajaykrishnan E Lokshtanov, Daniel |
| author_facet | Chudnovsky, Maria S, Ajaykrishnan E Lokshtanov, Daniel |
| contents | An independent set in a graph $G$ is a set of pairwise non-adjacent vertices. A tree decomposition of $G$ is a pair $(T, χ)$ where $T$ is a tree and $χ: V(T) \rightarrow 2^{V(G)}$ is a function satisfying the following two axioms: for every edge $uv \in V(G)$ there is a $x \in V(T)$ such that $\{u,v\} \subseteq χ(x)$, and for every vertex $u \in V(G)$ the set $\{x \in V(T) ~:~ u \in χ(X)\}$ induces a non-empty and connected subtree of $T$. The sets $χ(x)$ for $x \in V(T)$ are called the bags of the tree decomposition. The tree-independence number of $G$ is the minimum taken over all tree decompositions of $G$ of the maximum size of an independent set of the graph induced by a bag of the tree decomposition.
The study of graph classes with bounded tree-independence number has attracted much attention in recent years, in part due its improtant algorithmic implications. A conjecture of Dallard, Milanič and Štorgel, connecting tree-independence number to the classical notion of treewidth, was one of the motivating problems in the area. This conjecture was recently disproved, but here we prove a slight variant of it, that retains much of the algorithmic significance. As part of the proof we introduce the notion of independence-containers, which can be viewed as a generalization of the set of all maximal cliques of a graph, and is of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_15074 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | (Treewidth, Clique)-Boundedness and Poly-logarithmic Tree-Independence Chudnovsky, Maria S, Ajaykrishnan E Lokshtanov, Daniel Combinatorics An independent set in a graph $G$ is a set of pairwise non-adjacent vertices. A tree decomposition of $G$ is a pair $(T, χ)$ where $T$ is a tree and $χ: V(T) \rightarrow 2^{V(G)}$ is a function satisfying the following two axioms: for every edge $uv \in V(G)$ there is a $x \in V(T)$ such that $\{u,v\} \subseteq χ(x)$, and for every vertex $u \in V(G)$ the set $\{x \in V(T) ~:~ u \in χ(X)\}$ induces a non-empty and connected subtree of $T$. The sets $χ(x)$ for $x \in V(T)$ are called the bags of the tree decomposition. The tree-independence number of $G$ is the minimum taken over all tree decompositions of $G$ of the maximum size of an independent set of the graph induced by a bag of the tree decomposition. The study of graph classes with bounded tree-independence number has attracted much attention in recent years, in part due its improtant algorithmic implications. A conjecture of Dallard, Milanič and Štorgel, connecting tree-independence number to the classical notion of treewidth, was one of the motivating problems in the area. This conjecture was recently disproved, but here we prove a slight variant of it, that retains much of the algorithmic significance. As part of the proof we introduce the notion of independence-containers, which can be viewed as a generalization of the set of all maximal cliques of a graph, and is of independent interest. |
| title | (Treewidth, Clique)-Boundedness and Poly-logarithmic Tree-Independence |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2510.15074 |