Complexity Framework For Forbidden Subgraphs

For any finite set $\mathcal{H} = \{H_1,\ldots,H_p\}$ of graphs, a graph is $\mathcal{H}$-subgraph-free if it does not contain any of $H_1,\ldots,H_p$ as a subgraph. Similar to known meta-classifications for the minor and topological minor relations, we give a meta-classification for the subgraph relation. Our framework classifies if problems are "efficiently solvable" or "computationally hard" for $\mathcal{H}$-subgraph-free graphs. The conditions are that the problem should be efficiently solvable on graphs of bounded treewidth, computationally hard on subcubic graphs, and computational hardness is preserved under edge subdivision. We show that all problems satisfying these conditions are efficiently solvable if $\mathcal{H}$ contains a disjoint union of one or more paths and subdivided claws, and are computationally hard otherwise. To illustrate the broad applicability of our framework, we study partitioning, covering and packing problems, network design problems and width parameter problems. We apply the framework to obtain a dichotomy between polynomial-time solvability and NP-completeness. For other problems we obtain a dichotomy between almost-linear-time solvability and having no subquadratic-time algorithm (conditioned on some hardness hypotheses). Along the way we unify and strengthen known results from the literature.