On ideal sumset labelled graphs

The sumset of two sets A and B of integers, denoted by A + B, is defined as A+B = {a+b : a ∈ A, b ∈ B}. Let X be a non-empty set of non-negative integers. A sumset labelling of a graph G is an injective function f : V (G) → P(X) − {∅} such that the induced function f+ : E(G) → P(X)−{∅} is defined by f+(uv) = f(u) +f(v) ∀uv ∈ E(G). In this paper, we introduce the notion of ideal sumset labelling of graph and discuss the admissibility of this labelling by certain graph classes and discuss some structural characterization of those graphs.