Minimal Spanning Trees with Conflict Graphs Andreas Darmann ∗∗ Ulrich Pferschy ∗ Joachim Schauer ∗ Abstract For the classical minimum spanning tree problem we introduce disjunctive constraints for pairs of edges which can not be both included in the span- ning tree at the same time. These constraints are represented by a conflict graph whose vertices correspond to the edges of the original graph. Edges in the conflict graph connect conflicting edges of the original graph. It is shown that the problem becomes strongly NP-hard even if the connected components of the conflict graph consist only of paths of length two. On the other hand, for conflict graphs consisting of disjoint edges (i.e. paths of length one) the problem remains polynomially solvable. Keywords: minimal spanning tree, conflict graph. 1 Introduction In this paper we consider an extension of the minimum spanning tree problem (MST). In addition to the well studied problem of finding a minimum spanning tree in a weighted, undirected connected graph, there exist incompatibilities for certain pairs of edges. This means that from each such conflicting pair at most one edge can occur in the spanning tree. It is natural to represent these symmetric conflict relations by means of an undirected conflict graph, where every vertex of the conflict graph corresponds uniquely to an edge in the original graph and an edge in the conflict graph implies that the two adjacent vertices, i.e. edges in the original graph, cannot occur together in an MST solution. For a formal definition of this minimum spanning tree problem with conflict graph (MSTCG), let G =(V,E) be an undirected connected graph with n vertices and m edges, where each edge e has associated a weight w(e) (w is a weight function w : E → R). Furthermore, an undirected graph ¯ G =(E, ¯ E) represents a conflict graph where each of the m vertices corresponds uniquely to an edge e ∈ E of G. An edge ¯ e =(i, j ) ∈ ¯ E implies that the two vertices incident to ¯ e – that ** University of Graz, Institute of Public Economics, Universitaetsstr. 15, A-8010 Graz, Austria, andreas.darmann@uni-graz.at * University of Graz, Department of Statistics and Operations Research, Universi- taetsstr. 15, A-8010 Graz, Austria, {pferschy, joachim.schauer}@uni-graz.at 1