$H$-critical graphs

Abstract: We are interested in the class of 3-connected graphs with a minor isomorphic to a specific 3-connected graph $H$. A 3-connected graph is minimally 3-connected if deleting any edge destroys 3-connectivity. Suppose that $G$ is a simple 3-connected graph with a simple 3-connected minor $H$. We say $G$ is $H$-critical, if deleting any edge either destroys 3-connectivity or the $H$-minor. If $H$ is minimally 3-connected, then $G$ is also minimally 3-connected, and the class of $H$-critical graphs is the class of minimally 3-connected graphs with an $H$ minor. In general, however, $H$ is not minimally 3-connected, and in this case $H$-critical graphs are not minimally 3-connected graphs. Yet we have obtained splitter-type structural results for $H$-critical graphs that are very similar to Dawes' result on the structure of minimally 3-connected graphs. We also get a result that is very similar to Halin's bound on the size of minimally 3-connected graphs. I will present these results in this talk. The results are joint work with Joao Paulo Costalonga.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2021)

Series comments: This is the nineteenth in a series of annual workshops sponsored by the New York Number Theory Seminar on problems in combinatorial and additive number theory and related parts of mathematics.

Registration for the conference is free. Register at cant2021.eventbrite.com.

The conference website is www.theoryofnumbers.com/cant/ Lectures will be broadcast on Zoom. The Zoom login will be emailed daily to everyone who has registered on eventbrite. To join the meeting, you may need to download the free software from www.zoom.us.

The conference program, list of speakers, and abstracts are posted on the external website.