Bounded diameter tree-decompositions
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arXiv
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| Formato: | Preprint |
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2023
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| _version_ | 1866914651856437248 |
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| author | Berger, Eli Seymour, Paul |
| author_facet | Berger, Eli Seymour, Paul |
| contents | When does a graph admit a tree-decomposition in which every bag has small diameter? For finite graphs, this is a property of interest in algorithmic graph theory, where it is called having bounded ``tree-length''. We will show that this is equivalent to being ``boundedly quasi-isometric to a tree'', which for infinite graphs is a much-studied property from metric geometry. One object of this paper is to tie these two areas together. We will prove that there is a tree-decomposition in which each bag has small diameter, if and only if there is a map $ϕ$ from $V(G)$ into the vertex set of a tree $T$, such that for all $u,v\in V(G)$, the distances $d_G(u,v), d_T(ϕ(u),ϕ(v))$ differ by at most a constant.
A ``geodesic loaded cycle'' in $G$ is a pair $(C,F)$, where $C$ is a cycle of $G$ and $F\subseteq E(C)$, such that for every pair $u,v$ of vertices of $C$, one of the paths of $C$ between $u,v$ contains at most $d_G(u,v)$ $F$-edges, where $d_G(u,v)$ is the distance between $u,v$ in $G$. We will show that a graph $G$ admits a tree-decomposition in which every bag has small diameter, if and only if $|F|$ is small for every geodesic loaded cycle $(C,F)$. Our proof is an extension of an algorithm to approximate tree-length in finite graphs by Dourisboure and Gavoille.
In metric geometry, there is a similar theorem that characterizes when a graph is quasi-isometric to a tree, ``Manning's bottleneck criterion''. The goal of this paper is to tie all these concepts together, and add a few more related ideas. For instance, we prove a conjecture of Rose McCarty, that $G$ admits a tree-decomposition in which every bag has small diameter, if and only if for all vertices $u,v,w$ of $G$, some ball of small radius meets every path joining two of $u,v,w$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_13282 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Bounded diameter tree-decompositions Berger, Eli Seymour, Paul Combinatorics 05C10, 05C62, 05C83 When does a graph admit a tree-decomposition in which every bag has small diameter? For finite graphs, this is a property of interest in algorithmic graph theory, where it is called having bounded ``tree-length''. We will show that this is equivalent to being ``boundedly quasi-isometric to a tree'', which for infinite graphs is a much-studied property from metric geometry. One object of this paper is to tie these two areas together. We will prove that there is a tree-decomposition in which each bag has small diameter, if and only if there is a map $ϕ$ from $V(G)$ into the vertex set of a tree $T$, such that for all $u,v\in V(G)$, the distances $d_G(u,v), d_T(ϕ(u),ϕ(v))$ differ by at most a constant. A ``geodesic loaded cycle'' in $G$ is a pair $(C,F)$, where $C$ is a cycle of $G$ and $F\subseteq E(C)$, such that for every pair $u,v$ of vertices of $C$, one of the paths of $C$ between $u,v$ contains at most $d_G(u,v)$ $F$-edges, where $d_G(u,v)$ is the distance between $u,v$ in $G$. We will show that a graph $G$ admits a tree-decomposition in which every bag has small diameter, if and only if $|F|$ is small for every geodesic loaded cycle $(C,F)$. Our proof is an extension of an algorithm to approximate tree-length in finite graphs by Dourisboure and Gavoille. In metric geometry, there is a similar theorem that characterizes when a graph is quasi-isometric to a tree, ``Manning's bottleneck criterion''. The goal of this paper is to tie all these concepts together, and add a few more related ideas. For instance, we prove a conjecture of Rose McCarty, that $G$ admits a tree-decomposition in which every bag has small diameter, if and only if for all vertices $u,v,w$ of $G$, some ball of small radius meets every path joining two of $u,v,w$. |
| title | Bounded diameter tree-decompositions |
| topic | Combinatorics 05C10, 05C62, 05C83 |
| url | https://arxiv.org/abs/2306.13282 |