# 2-blocks in strongly biconnected directed graphs

A directed graph G=(V,E) is called strongly biconnected if G is strongly connected and the underlying graph of G is biconnected. A strongly biconnected component of a strongly connected graph G=(V,E) is a maximal vertex subset L⊆ V such that the induced subgraph on L is strongly biconnected. Let G=(V,E) be a strongly biconnected directed graph. A 2-edge-biconnected block in G is a maximal vertex subset U⊆ V such that for any two distict vertices v,w ∈ U and for each edge b∈ E, the vertices v,w are in the same strongly biconnected components of G∖{ b}. A 2-strong-biconnected block in G is a maximal vertex subset U⊆ V of size at least 2 such that for every pair of distinct vertices v,w∈ U and for every vertex z∈ V∖{ v,w }, the vertices v and w are in the same strongly biconnected component of G∖{ v,w }. In this paper we study 2-edge-biconnected blocks and 2-strong biconnected blocks.

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