# Characterizing the complexity of subgraph isomorphism for graphs of bounded path-width

## Abstract

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*(n^{3}) 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 Matoušek and Thomas.

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