On the super edge-magic deficiency of some families related to ladder graphs

A graph G is called edge-magic if there exists a bijective function φ : V (G)∪E(G) → {1, 2,. .. , |V (G)|+ |E(G)|} such that φ(x)+φ(xy)+φ(y) is a constant c(φ) for every edge xy ∈ E(G); here c(φ) is called the valence of φ. A graph G is said to be super edge-magic if φ(V (G)) = {1, 2,. .. , |V (G)|}. The super edge-magic deficiency, denoted by μ s (G), is the minimum nonnegative integer n such that G ∪ nK 1 has a super edge-magic labeling; if such an integer does not exist we define μ s (G) to be +∞. In this paper we study the super edge-magic deficiency of some families of graphs related to ladder graphs.