Abstract.

The circumference of a graph \(G\) with at least one cycle is the length of a longest cycle in \(G\). A classic result of Birmelé [J. Graph Theory, 43 (2003), pp. 24–25] 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 et al. [Treedepth vs circumference, Combinatorica, 43 (2023), pp. 659–664]). In this paper, we study whether similar bounds hold when replacing the circumference of \(G\) by its cocircumference, defined as the largest size of a bond in \(G\), an inclusionwise 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.

Keywords

  1. treewidth
  2. pathwidth
  3. treedepth
  4. circumference
  5. cocircumference

MSC codes

  1. 05C83

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Acknowledgment.

We thank the two anonymous referees for their helpful remarks on a previous version of this manuscript.

References

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Information & Authors

Information

Published In

cover image SIAM Journal on Discrete Mathematics
SIAM Journal on Discrete Mathematics
Pages: 857 - 866
ISSN (online): 1095-7146

History

Submitted: 14 July 2023
Accepted: 17 November 2023
Published online: 26 February 2024

Keywords

  1. treewidth
  2. pathwidth
  3. treedepth
  4. circumference
  5. cocircumference

MSC codes

  1. 05C83

Authors

Affiliations

Marcin Briański
Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, 30-348, Poland.
Computer Science Department, Université libre de Bruxelles, Brussels, 1050, Belgium.
Michał T. Seweryn
Computer Science Department, Université libre de Bruxelles, Brussels, 1050, Belgium.

Funding Information

Narodowe Centrum Nauki (NCN): 2019/34/E/ST6/00443
Funding: The first author was partially supported by the Polish National Science Centre grant 2019/34/E/ST6/00443. The second and third authors were supported by a PDR grant from the Belgian National Fund for Scientific Research (FNRS).

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