Automated testing and interactive construction of unavoidable sets for graph classes of small path‐width

Let G be a class of graphs with a membership test, k∈N , and let Gk be the class of graphs in G of path-width at most k. We present an interactive framework that finds an unavoidable set for Gk, which is a set of graphs U such that any graph in Gk contains an isomorphic copy of a graph in U. At the core of our framework is an algorithm that verifies whether a set of graphs is, indeed, unavoidable for Gk. 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 is locally checkable of bounded maximum degree and U is a finite set of connected graphs. For example, l-regular graphs, l-colourable graphs, and H-free graphs are locally checkable classes. We put special emphasis on the case that 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 for all 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.