On the (Consecutively) Super Edge-Magic Deficiency of Subdivision of Double Stars

Let G be a finite, simple, and undirected graph with vertex set V(G) and edge set E(G). A super edge-magic labeling of G is a bijection f: V(G) boolean OR E(G). {1, 2,..., vertical bar V(G)vertical bar + vertical bar E(G)vertical bar} such that f(V(G)). {1, 2,..., vertical bar V(G)vertical bar} and f(u) + f(uv) + f(v) is a constant for every edge uv is an element of E(G). +esuper edge-magic labeling f ofG is called consecutively super edge-magic ifG is a bipartite graph with partite sets A and B such that f(A) = {1, 2,..., vertical bar A vertical bar} and f(B) = {vertical bar A vertical bar + 1, vertical bar A vertical bar + 2,..., vertical bar V(G)vertical bar}. A graph that admits (consecutively) super edge-magic labeling is called a (consecutively) super edge-magic graph. The super edge-magic deficiency of G, denoted by mu(s)(G), is either the minimum nonnegative integer n such that G boolean OR nK(1) is super edge-magic or +infinity if there exists no such n. The consecutively super edge-magic deficiency of a graph G is defined by a similar way. In this paper, we investigate the (consecutively) super edge-magic deficiency of subdivision of double stars. We show that, some of them have zero (consecutively) super edge-magic deficiency.