Automated Testing and Interactive Construction of Unavoidable Sets for Graph Classes of Small Path-width

Let G ${\mathscr{G}}$ be a class of graphs with a membership test, k∈N $k\in {\mathbb{N}}$ , and let Gk ${{\mathscr{G}}}^{k}$ be the class of graphs in G ${\mathscr{G}}$ of path‐width at most k $k$ . We present an interactive framework that finds an unavoidable set for Gk ${{\mathscr{G}}}^{k}$ , which is a set of graphs U ${\mathscr{U}}$ such that any graph in Gk ${{\mathscr{G}}}^{k}$ contains an isomorphic copy of a graph in U ${\mathscr{U}}$ . At the core of our framework is an algorithm that verifies whether a set of graphs is, indeed, unavoidable for Gk ${{\mathscr{G}}}^{k}$ . While obstruction sets are well‐studied, so far there is no general theory or algorithm for finding unavoidable sets. In general, it is undecidable whether a finite set of graphs is unavoidable for a given graph class. However, we give a criterion for termination: our algorithm terminates whenever G ${\mathscr{G}}$ is locally checkable of bounded maximum degree and U ${\mathscr{U}}$ is a finite set of connected graphs. For example, l $l$ ‐regular graphs, l $l$ ‐colourable graphs, and H $H$ ‐free graphs are locally checkable classes. We put special emphasis on the case that G ${\mathscr{G}}$ 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 enables highly efficient pruning techniques. We exploit our framework to prove a new lower bound on the path‐width of cubic graphs. Moreover, we determine the extremal girth values of cubic graphs of path‐width k $k$ for all k∈{3,…,10} $k\in \{3,\ldots ,10\}$ and all smallest graphs which take on these extremal girth values. Further, we present a new constructive characterisation of the extremal cubic graphs of path‐width 3 and girth 4.