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A characterization for topologically integer additive set-indexer of graphs

The aim of this paper is to introduce a new function that satisfies the property of topological integer additive set-indexer (Top-IASI) with minimum cardinality for pan, tadpole, path and shovel graphs. Let G be a graph and X is a nonempty set. If an injective function f:V (G) → P(X) induced a new injective function f⊕:E(G) → P(X) −{∅} defined by f⊕(uv) = f(u) ⊕ f(v) for every uv ∈ E(G) then f is called set-indexer. If an injective function f:V (G) → P(N0) produced another injective function f+:E(G) → P(N 0) −{∅} defined by f+(uv) = f(u) + f(v) for every uv ∈ E(G), where N0 is the set of all non-negative integers and P(N0) is its power set then f is called an integer additive set-indexer (IASI). An IASI is called a Top-IASI if f(V (G)) ∪{∅} forms a topology.