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Sufficient conditions for the non-separating paths

For two vertices in a graph, a path between them is non-separating if the remaining graph is connected. A previous work shows that a nonseparating path always exists if the connectivity is more than two. In this paper, we show that to determine if there is a non-separating path is NPcomplete for graphs of connectivity one or two. Sufficient conditions for the existence of a nonseparating path on biconnected graphs are given, as well as polynomial time algorithms for finding such a path whenever the condition holds.