About Enclave Inclusive Sets In Graphs

D.K.Thakkar; N.J.Savaliya

doi:https://doi.org/10.14445/22315373/IJMTT-V65I5P514

International Journal of Mathematics Trends and Technology

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

Citation :

D.K.Thakkar, N.J.Savaliya, "About Enclave Inclusive Sets In Graphs," International Journal of Mathematics Trends and Technology (IJMTT), vol. 65, no. 5, pp. 100-105, 2019. Crossref, https://doi.org/10.14445/22315373/IJMTT-V65I5P514

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.

Keywords

enclave point, enclave inclusive set, minimum enclave inclusive set, minimal enclave inclusive set, upper enclave inclusive number.

References

[1] D. K. Thakkar and N. J. Savaliya , “About Isolate Domination in Graphs “, International Journal of Mathematical Archive Vol.8 Issue 11(2017).
[2] D. K. Thakkar and N. J. Savaliya , “On Isolate Inclusive Sets in Graphs “, International Journal of Innovation in Science and Mathematics Vol.5,Issue 3(2017).
[3] T. W. Haynes, S. T. Hedetniemi, P. J. Slater, “Fundamental of Domination In graphs”, Marcel Dekker, New York, (1998).
[4] T. W. Haynes, S. T. Hedetniemi, P. J. Slater, “Domination In graphs Advanced Topics”, Marcel Dekker, New York, (1998).