Chordless paths through three vertices

Consider the following problem, that we call “Chordless Path through Three Vertices” or Cp3v, for short: Given a simple undirected graph G=(V,E), a positive integer k, and three distinct vertices s, t, and v∈ V, is there a chordless path from s via v to t in G that consists of at most k vertices? In a chordless path, no two vertices are connected by an edge that is not in the path. Alternatively, one could say that the subgraph induced by the vertex set of the path in G is the path itself. The problem has been raised in the context of service deployment in communication networks. We resolve the parametric complexity of Cp3v by proving it W[1]-complete with respect to its natural parameter k. Our reduction extends to a number of related problems about chordless paths. In particular, deciding on the existence of a single directed chordless (s,t)-path in a digraph is also W[1]-complete with respect to the length of the path.

[1]  P. Seymour,et al.  The Strong Perfect Graph Theorem , 2002, math/0212070.

[2]  Daniel Bienstock,et al.  Corrigendum to: D. Bienstock, "On the complexity of testing for odd holes and induced odd paths" Discrete Mathematics 90 (1991) 85-92 , 1992, Discret. Math..

[3]  Michael T. Hallett,et al.  The Parameterized Complexity of Some Problems in Logic and Linguistics , 1994, LFCS.

[4]  D. R. Fulkerson,et al.  Maximal Flow Through a Network , 1956 .

[5]  Luca Trevisan,et al.  On the Efficiency of Polynomial Time Approximation Schemes , 1997, Inf. Process. Lett..

[6]  Neil Robertson,et al.  Graph Minors .XIII. The Disjoint Paths Problem , 1995, J. Comb. Theory B.

[7]  Richard M. Karp,et al.  On the Computational Complexity of Combinatorial Problems , 1975, Networks.

[8]  Robert Haas,et al.  Service deployment in programmable networks , 2003 .

[9]  Marco Cesati.  Perfect Code is W[1]-complete , 2000, Inf. Process. Lett..

[10]  Paul D. Seymour,et al.  Recognizing Berge Graphs , 2005, Comb..

[11]  Gérard Cornuéjols,et al.  A polynomial algorithm for recognizing perfect graphs , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

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

[13]  Liming Cai,et al.  On the parameterized complexity of short computation and factorization , 1997, Arch. Math. Log..

[14]  Michael R. Fellows,et al.  The complexity of induced minors and related problems , 1995, Algorithmica.

[15]  Robert E. Tarjan,et al.  Simple Linear-Time Algorithms to Test Chordality of Graphs, Test Acyclicity of Hypergraphs, and Selectively Reduce Acyclic Hypergraphs , 1984, SIAM J. Comput..

[16]  Daniel Bienstock,et al.  On the complexity of testing for odd holes and induced odd paths , 1991, Discret. Math..

[17]  Robert E. Tarjan,et al.  Algorithmic Aspects of Vertex Elimination on Graphs , 1976, SIAM J. Comput..

[18]  Marco Cesati,et al.  The Turing way to parameterized complexity , 2003, J. Comput. Syst. Sci..

[19]  Leonidas Palios,et al.  Hole and antihole detection in graphs , 2004, SODA '04.

[20]  Michael R. Fellows,et al.  Fixed-Parameter Tractability and Completeness II: On Completeness for W[1] , 1995, Theor. Comput. Sci..

[21]  Bruce A. Reed,et al.  Induced Circuits in Planar Graphs , 1994, J. Comb. Theory, Ser. B.

[22]  John E. Hopcroft,et al.  The Directed Subgraph Homeomorphism Problem , 1978, Theor. Comput. Sci..

[23]  Bruce A. Reed,et al.  Non-Interfering Network Flows , 1992, SWAT.