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The Complexity of Minor-Ancestral Graph Properties with Forbidden Pairs

  • Eli Fox-Epstein
  • Danny Krizanc
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7353)

Abstract

Robertson and Seymour (in work starting with [15]) demonstrated that any minor-ancestral graph property can be decided in polynomial time. Lewis and Yannakakis [14] showed that for any nontrivial node-hereditary graph property, the problem of given a graph, finding the size of the largest induced subgraph of the graph that has the property, is NP-hard. In this paper, we completely characterize those minor-ancestral properties for which the problem of deciding if a given graph contains a subgraph with the property that respects a given set of forbidden vertex pairs (i.e., if one vertex from a pair is in the subgraph then the other isn’t) is in P and for which such properties the problem is NP-complete. In particular, we show that if a given minor-ancestral property can be characterized by the containment of one of a finite set of graphs as a subgraph, the corresponding decision problem with forbidden vertex pairs is in P, otherwise its NP-complete. Unfortunately, we further show that the problem of deciding if a minor-ancestral property (presented as a set of characteristic minors) can be so characterized is NP-hard. Finally we observe that a similar characterization holds for the case of finding subgraphs satisfying a set of forbidden edge pairs and that our problems are all fixed parameter tractable.

Keywords

Polynomial Time Hamiltonian Path Graph Property Internal Edge Satisfying Assignment 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Eli Fox-Epstein
    • 1
  • Danny Krizanc
    • 2
  1. 1.Tufts UniversityUSA
  2. 2.Wesleyan UniversityUSA

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