Automated Testing and Interactive Construction of Unavoidable Sets for Graph Classes of Small Path-width
Abstract
We present an interactive framework that, given a membership test for a graph class $\mathcal{G}$ and a number $k$, finds and tests unavoidable sets for the class of graphs in $\mathcal{G}$ of path-width at most $k$. We put special emphasis on the case that $\mathcal{G}$ is the class of cubic graphs and tailor the algorithm to this case. In particular, we introduce the new concept of high-degree-first path-decompositions, which yields highly efficient pruning techniques. Using this framework we determine all extremal girth values of cubic graphs of path-width $k$ for all $k \in \{3,\dots, 10\}$. Moreover, we determine all smallest graphs which take on these extremal girth values. As a further application of our framework we characterise the extremal cubic graphs of path-width 3 and girth 4.
- Publication:
-
arXiv e-prints
- Pub Date:
- October 2020
- DOI:
- 10.48550/arXiv.2010.08373
- arXiv:
- arXiv:2010.08373
- Bibcode:
- 2020arXiv201008373B
- Keywords:
-
- Mathematics - Combinatorics;
- 05C85;
- 05C75;
- 68R10;
- 05C38
- E-Print:
- 26 pages, 4 figures