Characterizing the Complexity of Subgraph Isomorphism for Graphs of Bounded Path-Width

We show that the complexity of the subgraph isomorphism problem on graphs of bounded path-width is inherently dependent on the connectivity of the source and target graphs. In particular, for the problem of determining whether a source graph G of path-width k is a subgraph of a target graph H of path-width k, we present an O(n3) algorithm for G and H both k-connected, for n the sum of the sizes of the graphs, and NP-completeness results for connectivity less than k. In previous polynomial-time algorithms, the degree of the polynomial in the running time was a function of k. In contrast, we show that when neither G nor H is k-connected, the problem becomes NP-complete. The same result also holds if one of the graphs has at least k vertices of unbounded degree. Since bounded path-width graphs are also bounded tree-width graphs, our hardness results immediately extend to this larger class. A further NP-completeness result applies to the situation in which both graphs have tree-width k, but only the target graph is k-connected. This provides a complete characterization of the subgraph isomorphism problem on bounded tree-width graphs, thus answering an open question of Matousek and Thomas.