We complement our study of 2-connectivity in directed graphs, by considering
the computation of the following 2-vertex-connectivity relations: We say that
two vertices v and w are 2-vertex-connected if there are two internally
vertex-disjoint paths from v to w and two internally vertex-disjoint paths from
w to v. We also say that v and w are vertex-resilient if the removal of any
vertex different from v and w leaves v and w in the same strongly connected
component. We show how to compute the above relations in linear time so that we
can report in constant time if two vertices are 2-vertex-connected or if they
are vertex-resilient. We also show how to compute in linear time a sparse
certificate for these relations, i.e., a subgraph of the input graph that has
O(n) edges and maintains the same 2-vertex-connectivity and vertex-resilience
relations as the input graph, where n is the number of vertices.Comment: arXiv admin note: substantial text overlap with arXiv:1407.304
