On Maximum Common Subgraph Problems in Series-Parallel Graphs

The complexity of the maximum common connected subgraph problem in partial $k$-trees is still not fully understood. Polynomial-time solutions are known for degree-bounded outerplanar graphs, a subclass of the partial $2$-trees. On the other hand, the problem is known to be ${\bf NP}$-hard in vertex-labeled partial $11$-trees of bounded degree. We consider series-parallel graphs, i.e., partial $2$-trees. We show that the problem remains ${\bf NP}$-hard in biconnected series-parallel graphs with all but one vertex of degree $3$ or less. A positive complexity result is presented for a related problem of high practical relevance which asks for a maximum common connected subgraph that preserves blocks and bridges of the input graphs. We present a polynomial time algorithm for this problem in series-parallel graphs, which utilizes a combination of BC- and SP-tree data structures to decompose both graphs.

[1]  D. Matula.  Subtree Isomorphism in O(n5/2) , 1978 .

[2]  Maciej M. SysŁ.  The subgraph isomorphism problem for outerplanar graphs , 1982 .

[3]  Tatsuya Akutsu,et al.  On the Complexity of the Maximum Common Subgraph Problem for Partial k-Trees of Bounded Degree , 2012, ISAAC.

[4]  A. Brandstädt,et al.  Graph Classes: A Survey , 1987 .

[5]  Jens Vygen,et al.  The edge-disjoint paths problem is NP-complete for series-parallel graphs , 2001, Discret. Appl. Math..

[6]  David S. Johnson,et al.  `` Strong '' NP-Completeness Results: Motivation, Examples, and Implications , 1978, JACM.

[7]  Harold W. Kuhn,et al.  The Hungarian method for the assignment problem , 1955, 50 Years of Integer Programming.

[8]  Robin Thomas,et al.  On the complexity of finding iso- and other morphisms for partial k-trees , 1992, Discret. Math..

[9]  Jan Ramon,et al.  Efficient frequent connected subgraph mining in graphs of bounded tree-width , 2010, LWA.

[10]  Maurice Bruynooghe,et al.  A polynomial-time maximum common subgraph algorithm for outerplanar graphs and its application to chemoinformatics , 2013, Annals of Mathematics and Artificial Intelligence.

[11]  Naomi Nishimura,et al.  Sequential and Parallel Algorithms for Embedding Problems on Classes of Partial k-Trees , 1994, SWAT.

[12]  Markus Chimani,et al.  A Tighter Insertion-Based Approximation of the Crossing Number , 2011, ICALP.

[13]  Maurice Bruynooghe,et al.  A Polynomial-time Metric for Outerplanar Graphs , 2007, MLG.

[14]  Naomi Nishimura,et al.  The Complexity of Subgraph Isomorphism for Classes of Partial k-Trees , 1996, Theor. Comput. Sci..

[15]  Tatsuya Akutsu,et al.  A Polynomial-Time Algorithm for Computing the Maximum Common Connected Edge Subgraph of Outerplanar Graphs of Bounded Degree , 2013, Algorithms.

[16]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[17]  Maurice Bruynooghe,et al.  An Efficiently Computable Graph-Based Metric for the Classification of Small Molecules , 2008, Discovery Science.

[18]  T. Akutsu.  A Polynomial Time Algorithm for Finding a Largest Common Subgraph of almost Trees of Bounded Degree , 1993 .

[19]  Luc De Raedt,et al.  Effective feature construction by maximum common subgraph sampling , 2010, Machine Learning.

[20]  Petra Mutzel,et al.  A Linear Time Implementation of SPQR-Trees , 2000, GD.

[21]  Nils M. Kriege,et al.  Finding Maximum Common Biconnected Subgraphs in Series-Parallel Graphs , 2014, MFCS.