Trees in Graphs with Conflict Edges or Forbidden Transitions

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.