On super edge-magic deficiency of volvox and dumbbell graphs

Let G=(V,E)G=(V,E) be a finite, simple and undirected graph of order pp and size qq. A super edge-magic total labeling of a graph GG is a bijection λ:V(G)∪E(G)→{1,2,…,p+q}λ:V(G)∪E(G)→{1,2,…,p+q}, where the vertices are labeled with the numbers 1,2,…,p1,2,…,p and there exists a constant tt such that f(x)+f(xy)+f(y)=tf(x)+f(xy)+f(y)=t, for every edge xy∈E(G)xy∈E(G). The super edge-magic deficiency of a graph GG, denoted by μs(G)μs(G), is the minimum nonnegative integer nn such that G∪nK1G∪nK1 has a super edge-magic total labeling, or it is ∞∞ if there exists no such nn.In this paper, we are dealing with the super edge-magic deficiency of volvox and dumbbell type graphs.