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. © 2012 Springer-Verlag.
CITATION STYLE
Fox-Epstein, E., & Krizanc, D. (2012). The complexity of minor-ancestral graph properties with forbidden pairs. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 7353 LNCS, pp. 138–147). https://doi.org/10.1007/978-3-642-30642-6_14
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