Product Structure and Tree-Decompositions
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910671094939648 |
|---|---|
| author | Liu, Chun-Hung Norin, Sergey Wood, David R. |
| author_facet | Liu, Chun-Hung Norin, Sergey Wood, David R. |
| contents | This paper explores the structure of graphs defined by an excluded minor or an excluded odd minor through the lens of graph products and tree-decompositions. We prove that every graph excluding a fixed odd minor is contained in the strong product of two graphs each with bounded treewidth. For graphs excluding a fixed minor, we strengthen the result by showing that every such graph is contained in the strong product of two digraphs with bounded indegree and with bounded treewidth. This result has the advantage that the product now has bounded degeneracy.
In the setting of 3-term products, we show that every $K_t$-minor-free graph is contained in $H_1\boxtimes H_2 \boxtimes K_{c(t)}$ where $\text{tw}(H_i)\leq t-2$. This treewidth bound is close to tight: in any such result with $\text{tw}(H_i)$ bounded, both $H_1$ and $H_2$ can be forced to contain any graph of treewidth $t-5$, implying $\text{tw}(H_1)\geq t-5$ and $\text{tw}(H_2)\geq t-5$. Analogous lower and upper bounds are shown for any excluded minor, where the minimum possible bound on $\text{tw}(H_i)$ is tied to the treedepth of the excluded minor.
Subgraphs of the product of two graphs with bounded treewidth have two tree-decompositions where any bag from the first decomposition intersects any bag from the second decomposition in a bounded number, $k$, of vertices, so called $k$-orthogonal tree-decompositions. We show that graphs excluding a fixed odd-minor have a tree-decomposition and a path-decomposition that are $O(1)$-orthogonal. This implies that such graphs have a tree-decomposition in which each bag has bounded pathwidth. This result is best possible in that `pathwidth' cannot be replaced by `bandwidth' or `treedepth'. Moreover, we characterize the minor-closed classes that have a tree-decomposition in which each bag has bounded bandwidth, or each bag has bounded treedepth. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_20333 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Product Structure and Tree-Decompositions Liu, Chun-Hung Norin, Sergey Wood, David R. Combinatorics Discrete Mathematics This paper explores the structure of graphs defined by an excluded minor or an excluded odd minor through the lens of graph products and tree-decompositions. We prove that every graph excluding a fixed odd minor is contained in the strong product of two graphs each with bounded treewidth. For graphs excluding a fixed minor, we strengthen the result by showing that every such graph is contained in the strong product of two digraphs with bounded indegree and with bounded treewidth. This result has the advantage that the product now has bounded degeneracy. In the setting of 3-term products, we show that every $K_t$-minor-free graph is contained in $H_1\boxtimes H_2 \boxtimes K_{c(t)}$ where $\text{tw}(H_i)\leq t-2$. This treewidth bound is close to tight: in any such result with $\text{tw}(H_i)$ bounded, both $H_1$ and $H_2$ can be forced to contain any graph of treewidth $t-5$, implying $\text{tw}(H_1)\geq t-5$ and $\text{tw}(H_2)\geq t-5$. Analogous lower and upper bounds are shown for any excluded minor, where the minimum possible bound on $\text{tw}(H_i)$ is tied to the treedepth of the excluded minor. Subgraphs of the product of two graphs with bounded treewidth have two tree-decompositions where any bag from the first decomposition intersects any bag from the second decomposition in a bounded number, $k$, of vertices, so called $k$-orthogonal tree-decompositions. We show that graphs excluding a fixed odd-minor have a tree-decomposition and a path-decomposition that are $O(1)$-orthogonal. This implies that such graphs have a tree-decomposition in which each bag has bounded pathwidth. This result is best possible in that `pathwidth' cannot be replaced by `bandwidth' or `treedepth'. Moreover, we characterize the minor-closed classes that have a tree-decomposition in which each bag has bounded bandwidth, or each bag has bounded treedepth. |
| title | Product Structure and Tree-Decompositions |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2410.20333 |