Recognizing small subgraphs

Although the general problem of subgraph isomorphism is NP-complete, polynomial-time algorithms exist for recognizing any fixed subgraph. However, certain subgraphs appear easier to recognize than others. In this paper, we present general algorithms for fixed-subgraph isomorphism which improve or unify previous results. In particular, we present an O(n f m) algorithm for recognizing a fixed subgraph H with flower number f within a graph G with n vertices and m edges. Special cases of this algorithm match the best algorithms known for recognizing small paths, cycles, and cliques. Further, we improve previous results for recognizing C 5 and small even cycles C 2k , k ≥ 3

[1]  Arthur L. Liestman,et al.  Finding Cycles of a Given Length , 1985 .

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

[3]  Svatopluk Poljak,et al.  On the complexity of the subgraph problem , 1985 .

[4]  B. Monien.  How to Find Long Paths Efficiently , 1985 .

[5]  Michael R. Fellows,et al.  On search, decision, and the efficiency of polynomial-time algorithms , 1994, FOCS 1994.

[6]  Hans L. Bodlaender,et al.  On Linear Time Minor Tests and Depth First Search , 1989, WADS.

[7]  M. Simonovits,et al.  Cycles of even length in graphs , 1974 .

[8]  Julian R. Ullmann,et al.  An Algorithm for Subgraph Isomorphism , 1976, J. ACM.

[9]  P. Erdos,et al.  On maximal paths and circuits of graphs , 1959 .

[10]  Lorna Stewart,et al.  A Linear Recognition Algorithm for Cographs , 1985, SIAM J. Comput..

[11]  David Eppstein,et al.  Connectivity, graph minors, and subgraph multiplicity , 1993, J. Graph Theory.

[12]  Norishige Chiba,et al.  Arboricity and Subgraph Listing Algorithms , 1985, SIAM J. Comput..

[13]  Dana S. Richards,et al.  Finding Short Cycles in Planar Graphs Using Separators , 1986, J. Algorithms.

[14]  Bernd Voigt,et al.  Finding Minimally Weighted Subgraphs , 1991, WG.

[15]  Paul D. Seymour,et al.  Graph Minors. II. Algorithmic Aspects of Tree-Width , 1986, J. Algorithms.

[16]  Michael R. Fellows,et al.  Polynomial-time self-reducibility: theoretical motivations and practical results ∗ , 1989 .

[17]  Alon Itai,et al.  Finding a minimum circuit in a graph , 1977, STOC '77.

[18]  Mihalis Yannakakis,et al.  The complexity of restricted spanning tree problems , 1982, JACM.