On testing single connectedness in directed graphs and some related problems
Information Processing Letters(2015)
摘要
We give improved algorithms to test single-connectedness of a graph.An example shows our runtime bound is tight. 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) 1] gave an algorithm for testing whether G is singly-connected in O ( n 2 ) 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 r 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.
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关键词
algorithms,depth first search,directed graphs,connectivity,np complete
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