A characterization for topologically integer additive set-indexer of graphs
Abstract
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 GG be a graph and XX is a nonempty set. If an injective function f:V(G)→P(X)f:V(G)→P(X) induced a new injective function f⊕:E(G)→P(X)−{∅}f⊕:E(G)→P(X)−{∅} defined by f⊕(uv)=f(u)⊕f(v)f⊕(uv)=f(u)⊕f(v) for every uv∈E(G)uv∈E(G) then ff is called set-indexer. If an injective function f:V(G)→P(N0)f:V(G)→P(N0) produced another injective function f+:E(G)→P(N0)−{∅}f+:E(G)→P(N0)−{∅} defined by f+(uv)=f(u)+f(v)f+(uv)=f(u)+f(v) for every uv∈E(G)uv∈E(G), where N0N0 is the set of all non-negative integers and P(N0)P(N0) is its power set then ff is called an integer additive set-indexer (IASI). An IASI is called a Top-IASI if f(V(G))∪{∅}f(V(G))∪{∅} forms a topology.
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