Abstract
Given a connected graph G, a vertex v of G is said to be a branch vertex if its degree is strictly greater than 2. The Minimum Branch Vertices Spanning Tree problem (MBVST) consists in finding a spanning tree of G with the minimum number of branch vertices. This problem has been well studied in the literature and has applications specially for routing in optical networks. In this paper we propose a generalization of this problem. We introduce the notion of k-branch vertex which is a vertex with degree strictly greater than k + 2. The parameter k can be seen as the limit of the capacity of optical splitters to divide the light signal. In order to respect as far as possible this limit, we propose to search a spanning tree of G with the minimum number of k-branch vertices (k-MBVST problem). We propose a proof of NP-hardness of this new problem whatever the value of k. We also propose an ILP formulation of the k-MBVST by generalising the MBVST one. Experimental tests on random graphs show that the number of k-branch vertices increases with graph size but decreases with k as well as with the density. They also show that when k ≥ 4, the number of k-branch vertices is close to zero whatever the size and the density of the tested graph.