Abstract
The complexity of the maximum common subgraph problem in partial k-trees is still largely unknown. We consider the restricted case, where the input graphs are k-connected partial k-trees and the common subgraph is required to be k-connected. For biconnected outerplanar graphs this problem is solved and the general problem was reported to be tractable by means of tree decomposition techniques. We discuss key obstacles of tree decompositions arising for common subgraph problems that were ignored by previous algorithms and do not occur in outerplanar graphs. We introduce the concept of potential separators, i.e., separators of a subgraph to be searched that not necessarily are separators of the input graph. We characterize these separators and propose a polynomial time solution for series-parallel graphs based on SP-trees.
Research supported by the German Research Foundation (DFG), priority programme “Algorithms for Big Data” (SPP 1736).
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References
Akutsu, T.: A polynomial time algorithm for finding a largest common subgraph of almost trees of bounded degree. IEICE Trans. Fundamentals E76-A(9) (1993)
Akutsu, T., Tamura, T.: On the complexity of the maximum common subgraph problem for partial k-trees of bounded degree. In: Chao, K.-M., Hsu, T.-S., Lee, D.-T. (eds.) ISAAC 2012. LNCS, vol. 7676, pp. 146–155. Springer, Heidelberg (2012)
Akutsu, T., Tamura, T.: A polynomial-time algorithm for computing the maximum common connected edge subgraph of outerplanar graphs of bounded degree. Algorithms 6(1), 119–135 (2013)
Bachl, S., Brandenburg, F.J., Gmach, D.: Computing and drawing isomorphic subgraphs. J. Graph Algorithms Appl. 8(2), 215–238 (2004)
Dessmark, A., Lingas, A., Proskurowski, A.: Faster algorithms for subgraph isomorphism of k-connected partial k-trees. Algorithmica 27, 337–347 (2000)
Gupta, A., Nishimura, N.: Sequential and parallel algorithms for embedding problems on classes of partial k-trees. In: Schmidt, E.M., Skyum, S. (eds.) SWAT 1994. LNCS, vol. 824, pp. 172–182. Springer, Heidelberg (1994)
Gupta, A., Nishimura, N.: The complexity of subgraph isomorphism for classes of partial k-trees. Theoretical Computer Science 164(1-2), 287–298 (1996)
Hajiaghayi, M., Nishimura, N.: Subgraph isomorphism, log-bounded fragmentation, and graphs of (locally) bounded treewidth. J. Comput. System Sci. 73(5), 755 (2007)
Matoušek, J., Thomas, R.: On the complexity of finding iso- and other morphisms for partial k-trees. Discrete Mathematics 108(1-3), 343–364 (1992)
Matula, D.W.: Subtree isomorphism in O(n 5/2). In: Algorithmic Aspects of Combinatorics. Ann. Discrete Math., vol. 2, p. 91 (1978)
Schietgat, L., Ramon, J., Bruynooghe, M.: A polynomial-time metric for outerplanar graphs. In: Mining and Learning with Graphs, MLG 2007, Firence, Italy, August 1-3 (2007)
Sysło, M.M.: The subgraph isomorphism problem for outerplanar graphs. Theoretical Computer Science 17(1), 91–97 (1982)
Yamaguchi, A., Aoki, K.F., Mamitsuka, H.: Finding the maximum common subgraph of a partial k-tree and a graph with a polynomially bounded number of spanning trees. Inf. Process. Lett. 92(2), 57–63 (2004)
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Kriege, N., Mutzel, P. (2014). Finding Maximum Common Biconnected Subgraphs in Series-Parallel Graphs. In: Csuhaj-Varjú, E., Dietzfelbinger, M., Ésik, Z. (eds) Mathematical Foundations of Computer Science 2014. MFCS 2014. Lecture Notes in Computer Science, vol 8635. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44465-8_43
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