Research Article | Open Access | Download PDF
Volume 65 | Issue 5 | Year 2019 | Article Id. IJMTT-V65I5P514 | DOI : https://doi.org/10.14445/22315373/IJMTT-V65I5P514
About Enclave Inclusive Sets In Graphs
D.K.Thakkar, N.J.Savaliya
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 ๐ฃis a
vertex of a graph. Then enclave inclusive number of ๐บ โ ๐ฃis less than the enclave inclusive number of ๐บif and
only if there is a neighbour ๐ข of ๐ฃsuch that ๐ ๐ข is minimum. We deduce that for a graph ๐บ without isolated
vertices there are at least ๐ฟ ๐บ vertices such that removal of any one of them reduces the enclave inclusive
number of the graph. We further prove that if ๐ = ๐ข๐ฃ is an edge of the graph ๐บ. Then enclave inclusive
number of ๐บ โ ๐ is less than the enclave inclusive number of ๐บ if and only if ๐ ๐ข is minimum or ๐ ๐ฃ is
minimum. Finally, we observe that if ๐บ is a ๐-regular graph ๐ โฅ 1 . Then removal of any vertex or any edge
reduce the enclave inclusive number of the graph.
Keywords
enclave point, enclave inclusive set, minimum enclave inclusive set, minimal enclave inclusive set, upper enclave inclusive number.
References
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[2] D. K. Thakkar and N. J. Savaliya , โOn Isolate Inclusive Sets in Graphs โ, International Journal of Innovation in Science and Mathematics Vol.5,Issue 3(2017).
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