On α-split domination in graphs

Abstract

In this paper, we present the concept of alpha-split domination in graphs. Besides, we calculate the alpha-split domination number of path, cycle, complete bipartite graphs and discuss the upper bounds for a-split domination number in terms of order p, size q and the maximum degree d. We prove that alpha-split domination number is equal to the vertex covering number with 1 >= alpha > (Delta(G) - 1)/ Delta(G) and propose the upper bound for alpha-split domination number of a tree with diam(G) >= 3 in terms of the leaves among all its vertices and equality holds if and only if the split domination number of a tree is equal to the leaves among all its vertices. Also, we probe the alpha-split domination vertex critical graph with set of all support vertices of G and the lower bound for alpha-split domination number in terms of the order p and alpha satisfies 0 < alpha <= 1. Moreover, we investigate the alpha-split domination edge criticalness and show that G is 2-gamma(-)(alpha)(s)- critical with delta(G) > 1 if and only if G = C-3.