Let G be a non-bipartite strongly regular graph on n vertices of valency k. We prove that if G has a distance-regular antipodal cover of diameter 4, then k ≤ 2(n + 1) / 5, unless G is the complement of triangular graph T(7), the folded Johnson graph J(8, 4) or the folded halved 8-cube. However, for these three graphs the bound k ≤ Original Uggs Uk
(n − 1) / 2 holds. This result implies that only one of a complementary pair of strongly regular graphs can be the antipodal quotient of an antipodal distance-regular graph.
A graph G is said to be k–γ-critical if the size of any minimum dominating set of vertices is k, but if any edge is added to G the resulting graph can be dominated with k−1 vertices. A graph G is factor-critical if G−v has a perfect matching for every Ugg Cheap Uk
vertex v∈V(G) and is bicritical if G−u−v has a perfect matching for every pair of distinct vertices u,v∈V(G). In the present paper, it is shown that under certain assumptions regarding connectivity and minimum degree, a 3-γ-critical graph G will be either factor-critical (if |V(G)| is odd) or bicritical (if |V(G)| is even).