Abstract

We introduce the concept of enclave inclusive set in graphs in this paper. A set of vertices of a graph is called an enclave inclusive set if contains an enclave point. We prove that a set of vertices is a minimal enclave set if and only if its compliment is a maximal non dominating set. We observe that the close neighbourhood of a vertex with minimum degree is a minimum enclave inclusive set. We also prove that if 𝑣is a vertex of a graph. Then enclave inclusive number of 𝐺 − 𝑣is less than the enclave inclusive number of 𝐺if and only if there is a neighbour 𝑢 of 𝑣such that 𝑑 𝑢 is minimum. We deduce that for a graph 𝐺 without isolated vertices there are at least 𝛿 𝐺 vertices such that removal of any one of them reduces the enclave inclusive number of the graph. We further prove that if 𝑒 = 𝑢𝑣 is an edge of the graph 𝐺. Then enclave inclusive number of 𝐺 − 𝑒 is less than the enclave inclusive number of 𝐺 if and only if 𝑑 𝑢 is minimum or 𝑑 𝑣 is minimum. Finally, we observe that if 𝐺 is a 𝑘-regular graph 𝑘 ≥ 1 . Then removal of any vertex or any edge reduce the enclave inclusive number of the graph.

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References

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