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)


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.


Polynomial Time Hamiltonian Path Graph Property Internal Edge Satisfying Assignment 
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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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