AbstractIf 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 u2, v1, u2, and v2 of degree two in G such that u1v1 and u2v2 are edges of G, and u1v2, and u2v2 are edges of the complement of G, and B) the complement of G is a block with no critical vertex of degree two.