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 Matoušek and Thomas.
Unable to display preview. Download preview PDF.
- [ACP89]S. ARNBORG, D. CORNEIL, and A. PROSKUROWSKI, Complexity of finding embeddings in a k-tree, SIAM Journal of Algebraic and Discrete Methods 8 (1987), pp. 277–284.Google Scholar
- [APS90]S. ARNBORG, A. PROSKUROWSKI, and D. SEESE, Monadic second order logic, tree automata, and forbidden minors, Proceedings of the 4th Workshop on Computer Science Logic (CSL 90), pp. 1–16, Lecture Notes in Computer Science 533, Springer-Verlag, 1990.Google Scholar
- [Bod93]H. L. BODLAENDER, A linear time algorithm for finding treedecompositions of small treewidth, Proceedings of the 25th Annual ACM Symposium on the Theory of Computing, pp. 226–234, 1993.Google Scholar
- [BH95]H. L. BODLAENDER and T. HAGERUP, Parallel algorithms with optimal speedup for bounded treewidth, Proceedings of the 22nd International Colloquium on Automata, Languages, and Programming, 1995.Google Scholar
- [BK91]H. L. BODLAENDER and T. KLOKS, Better algorithms for the path-width and treewidth of graphs, Proceedings of the 18th International Colloquium on Automata, Languages, and Programming, pp. 544–555, 1991.Google Scholar
- [GJ79]M. R. GAREY and D. S. JOHNSON, “Computers and Intractability: A Guide to the Theory of NP-completeness,” Freeman, San Francisco, 1979.Google Scholar
- [Klo94]T. KLOKS, Treewidth — Computations and approximations, Springer-Verlag, Lecture Notes in Computer Science 842, 1994.Google Scholar
- [LS88]A. LINGAS and M. M. SYSLO, A polynomial-time algorithm for subgraph isomorphism of two-connected series parallel graphs, Proceedings of the 15th International Colloquium on Automata, Languages, and Programming pp. 394–409, 1988.Google Scholar
- [Mat78]D. MATULA, Subtree isomorphism in O(n5/2), Annals of Discrete Mathematics 2 (1978), pp. 91–106.Google Scholar