On testing single connectedness in directed graphs and some related problems

Let G = (V,E) be a directed graph with n vertices and m edges. The graph G is called singly-connected if for each pair of vertices v,w ∈ V there is at most one simple path from v to w in G. Buchsbaum and Carlisle (1993) gave an algorithm for testing whether G is singly-connected in O(n) time. In this paper we describe a refined version of this algorithm with running time O(s·t+m), where s and t are the number of sources and sinks, respectively, in the reduced graph G obtained by first contracting each strongly connected component of G into one vertex and then eliminating vertices of indegree or outdegree 1 by a contraction operation. Moreover, we show that the problem of finding a minimum cardinality edge subset C ⊆ E (respectively, vertex subset F ⊆ V ) whose removal from G leaves a singly-connected graph is NP-hard.