On induced subgraphs of a block

If G is a block, then a vertex u of G is called critical if G ‐ u is not a block. In this article, relationships between the localization of critical vertices and the localization of vertices of relatively small degrees (especially, of degree two) are studied. A block is called semicritical if a) each edge is incident with at least one critical vertex and b) each vertex of degree two is critical. Let G be a semicritical block with at least six vertices. It is proved that A) there exist distinct vertices u 2 , v 1 , u 2 , and v 2 of degree two in G such that u 1 v 1 and u 2 v 2 are edges of G , and u 1 v 2 , and u 2 v 2 are edges of the complement of G , and B) the complement of G is a block with no critical vertex of degree two.