3-Vertex Anti-duplication Self Switching of Graphs

Abstract

Anti-duplication of a vertex v in G produces a new graph AD(vG) by adding a new vertex v
′
such
that NG(v
′
) = [NG(v)]c
. A k-vertex anti-duplication of the k vertices vi ∈ σ (1 ≤ i ≤ k) produces a
new graph AD((v1, v2, ..., vk)G) by adding k new vertices v
′
i (1 ≤ i ≤ k) such that NG′ (v
′
i) = [NG[vi]]c
.
σ = {v1, v2, ..., vk} is called as k-vertex anti-duplication self switching of G if AD(σG) ∼= AD(σG)
σ
.
The set of all k-vertex anti-duplication self switching of G is denoted by ADSSk(G) and the number of elements in the set is denoted by adssk(G). When k=3, it is called as 3-Vertex Anti-duplication Self Switching. In this paper, we provide the necessary and sufficient conditions needed for a graph to be 3-vertex anti-duplication self switching. We also find adss3(G) of complete graph.