5 research outputs found
Circumference and Pathwidth of Highly Connected Graphs
Birmele [J. Graph Theory, 2003] proved that every graph with circumference t
has treewidth at most t-1. Under the additional assumption of 2-connectivity,
such graphs have bounded pathwidth, which is a qualitatively stronger result.
Birmele's theorem was extended by Birmele, Bondy and Reed [Combinatorica, 2007]
who showed that every graph without k disjoint cycles of length at least t has
bounded treewidth (as a function of k and t). Our main result states that,
under the additional assumption of (k + 1)- connectivity, such graphs have
bounded pathwidth. In fact, they have pathwidth O(t^3 + tk^2). Moreover,
examples show that (k + 1)-connectivity is required for bounded pathwidth to
hold. These results suggest the following general question: for which values of
k and graphs H does every k-connected H-minor-free graph have bounded
pathwidth? We discuss this question and provide a few observations.Comment: 11 pages, 4 figure
Treedepth vs circumference
The circumference of a graph is the length of a longest cycle in , or
if has no cycle. Birmel\'e (2003) showed that the treewidth of a
graph is at most its circumference minus one. We strengthen this result for
-connected graphs as follows: If is -connected, then its treedepth is
at most its circumference. The bound is best possible and improves on an
earlier quadratic upper bound due to Marshall and Wood (2015)
Treedepth vs circumference
The circumference of a graph is the length of a longest cycle in , or if has no cycle. Birmel\'e (2003) showed that the treewidth of agraph is at most its circumference minus one. We strengthen this result for-connected graphs as follows: If is -connected, then its treedepth isat most its circumference. The bound is best possible and improves on anearlier quadratic upper bound due to Marshall and Wood (2015).<br