Pathwidth vs cocircumference
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917603304275968 |
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| author | Briański, Marcin Joret, Gwenaël Seweryn, Michał T. |
| author_facet | Briański, Marcin Joret, Gwenaël Seweryn, Michał T. |
| contents | The {\em circumference} of a graph $G$ with at least one cycle is the length of a longest cycle in $G$. A classic result of Birmelé (2003) states that the treewidth of $G$ is at most its circumference minus $1$. In case $G$ is $2$-connected, this upper bound also holds for the pathwidth of $G$; in fact, even the treedepth of $G$ is upper bounded by its circumference (Briański, Joret, Majewski, Micek, Seweryn, Sharma; 2023). In this paper, we study whether similar bounds hold when replacing the circumference of $G$ by its {\em cocircumference}, defined as the largest size of a {\em bond} in $G$, an inclusion-wise minimal set of edges $F$ such that $G-F$ has more components than $G$. In matroidal terms, the cocircumference of $G$ is the circumference of the bond matroid of $G$.
Our first result is the following `dual' version of Birmelé's theorem: The treewidth of a graph $G$ is at most its cocircumference. Our second and main result is an upper bound of $3k-2$ on the pathwidth of a $2$-connected graph $G$ with cocircumference $k$. Contrary to circumference, no such bound holds for the treedepth of $G$. Our two upper bounds are best possible up to a constant factor. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_03621 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Pathwidth vs cocircumference Briański, Marcin Joret, Gwenaël Seweryn, Michał T. Combinatorics Discrete Mathematics The {\em circumference} of a graph $G$ with at least one cycle is the length of a longest cycle in $G$. A classic result of Birmelé (2003) states that the treewidth of $G$ is at most its circumference minus $1$. In case $G$ is $2$-connected, this upper bound also holds for the pathwidth of $G$; in fact, even the treedepth of $G$ is upper bounded by its circumference (Briański, Joret, Majewski, Micek, Seweryn, Sharma; 2023). In this paper, we study whether similar bounds hold when replacing the circumference of $G$ by its {\em cocircumference}, defined as the largest size of a {\em bond} in $G$, an inclusion-wise minimal set of edges $F$ such that $G-F$ has more components than $G$. In matroidal terms, the cocircumference of $G$ is the circumference of the bond matroid of $G$. Our first result is the following `dual' version of Birmelé's theorem: The treewidth of a graph $G$ is at most its cocircumference. Our second and main result is an upper bound of $3k-2$ on the pathwidth of a $2$-connected graph $G$ with cocircumference $k$. Contrary to circumference, no such bound holds for the treedepth of $G$. Our two upper bounds are best possible up to a constant factor. |
| title | Pathwidth vs cocircumference |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2306.03621 |