Complexity Framework for Forbidden Subgraphs I: The Framework

Abstract

For a set of graphs \({\mathcal {H}}\), a graph G is \({\mathcal {H}}\)-subgraph-free if G does not contain any graph from \({{{\mathcal {H}}}}\) as a subgraph. We propose general and easy-to-state conditions on graph problems that explain a large set of results for \({\mathcal {H}}\)-subgraph-free graphs. Namely, a graph problem must be efficiently solvable on graphs of bounded treewidth, computationally hard on subcubic graphs, and computational hardness must be preserved under edge subdivision of subcubic graphs. Our meta-classification says that if a graph problem \(\Pi \) satisfies all three conditions, then for every finite set \({{{\mathcal {H}}}}\), it is “efficiently solvable” on \({{{\mathcal {H}}}}\)-subgraph-free graphs if \({\mathcal {H}}\) contains a disjoint union of one or more paths and subdivided claws, and \(\Pi \) is “computationally hard” otherwise. We apply our meta-classification on many well-known partitioning, covering and packing problems, network design problems and width parameter problems to obtain a dichotomy between polynomial-time solvability and NP-completeness. For distance-metric problems, we obtain a dichotomy between almost-linear-time solvability and having no subquadratic-time algorithm (conditioned on some hardness hypotheses). Apart from capturing a large number of explicitly and implicitly known results in the literature, we also prove a number of new results. Moreover, we perform an extensive comparison between the subgraph framework and the existing frameworks for the minor and topological minor relations, and pose several new open problems and research directions.