Detecting induced subgraphs

  • Authors:

  • Benjamin Lévêque;David Y. Lin;Frédéric Maffray;Nicolas Trotignon

  • Affiliations:

  • CNRS, Laboratoire G-SCOP, 46 Avenue Félix Viallet, 38031 Grenoble Cedex, France;Princeton University, Princeton, NJ, 08544, United States;CNRS, Laboratoire G-SCOP, 46 Avenue Félix Viallet, 38031 Grenoble Cedex, France;CNRS, Université Paris 7, Paris Diderot, LIAFA, Case 7014, 75205 Paris Cedex 13, France

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2009

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Abstract

An s-graph is a graph with two kinds of edges: subdivisible edges and real edges. A realisation of an s-graph B is any graph obtained by subdividing subdivisible edges of B into paths of arbitrary length (at least one). Given an s-graph B, we study the decision problem @P"B whose instance is a graph G and question is ''Does G contain a realisation of B as an induced subgraph?''. For several B's, the complexity of @P"B is known and here we give the complexity for several more. Our NP-completeness proofs for @P"B's rely on the NP-completeness proof of the following problem. Let S be a set of graphs and d be an integer. Let @C"S^d be the problem whose instance is (G,x,y) where G is a graph whose maximum degree is at most d, with no induced subgraph in S and x,y@?V(G) are two non-adjacent vertices of degree 2. The question is ''Does G contain an induced cycle passing through x,y?''. Among several results, we prove that @C"0"@?^3 is NP-complete. We give a simple criterion on a connected graph H to decide whether @C"{"H"}^+^~ is polynomial or NP-complete. The polynomial cases rely on the algorithm three-in-a-tree, due to Chudnovsky and Seymour.