Local edge domination critical graphs Original Research Article

Sumner and Blitch defined a graph G to be k-y-critical if y(G) = k and y(G + uv) = k - 1 for each pair u, v of nonadjacent vertices of G. We define a graph to be k-(y,d)-critical if y(G) = k and y(G + uv) = k - 1 for each pair u, v of nonadjacent vertices of G that are at distance at most d apart. The 2-(y, 2)-critical graphs are characterized. Sharp upper bounds on the diameter of 3-(γ, 2)- and 4-(y, 2)-critical graphs are established and partial characterizations of 3-(γ, 2)-critical graphs are obtained.