Abstract
In a recent paper [Paths, trees and matchings under disjunctive constraints, Darmann et. al., Discr. Appl. Math., 2011] the authors add to a graph G a set of conflicts, i.e. pairs of edges of G that cannot be both in a subgraph of G. They proved hardness results on the problem of constructing minimum spanning trees and maximum matchings containing no conflicts. A forbidden transition is a particular conflict in which the two edges of the conflict must be incident. We consider in this paper graphs with forbidden transitions. We prove that the construction of a minimum spanning tree without forbidden transitions is still \({\ensuremath{\mathcal{NP}}}\)-Hard, even if the graph is a complete graph. We also consider the problem of constructing a maximum tree without forbidden transitions and prove that it cannot be approximated better than n 1/2 − ε for all ε > 0 even if the graph is a star. We strengthen in this way the results of Darmann et al. concerning the minimum spanning tree problem. We also describe sufficient conditions on forbidden transitions (conflicts) to ensure the existence of a spanning tree in complete graphs. One of these conditions uses graphic sequences.
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Kanté, M.M., Laforest, C., Momège, B. (2013). Trees in Graphs with Conflict Edges or Forbidden Transitions. In: Chan, TH.H., Lau, L.C., Trevisan, L. (eds) Theory and Applications of Models of Computation. TAMC 2013. Lecture Notes in Computer Science, vol 7876. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38236-9_31
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DOI: https://doi.org/10.1007/978-3-642-38236-9_31
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