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D. K. Thakkar1 , Neha P. Jamvecha2
Section:Research Paper, Product Type: Isroset-Journal
Vol.5 ,
Issue.4 , pp.374-379, Aug-2018
CrossRef-DOI: https://doi.org/10.26438/ijsrmss/v5i4.374379
Online published on Aug 31, 2018
Copyright © D. K. Thakkar , Neha P. Jamvecha . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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IEEE Style Citation: D. K. Thakkar , Neha P. Jamvecha, “On m-independence in Graphs,” International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.4, pp.374-379, 2018.
MLA Style Citation: D. K. Thakkar , Neha P. Jamvecha "On m-independence in Graphs." International Journal of Scientific Research in Mathematical and Statistical Sciences 5.4 (2018): 374-379.
APA Style Citation: D. K. Thakkar , Neha P. Jamvecha, (2018). On m-independence in Graphs. International Journal of Scientific Research in Mathematical and Statistical Sciences, 5(4), 374-379.
BibTex Style Citation:
@article{Thakkar_2018,
author = {D. K. Thakkar , Neha P. Jamvecha},
title = {On m-independence in Graphs},
journal = {International Journal of Scientific Research in Mathematical and Statistical Sciences},
issue_date = {8 2018},
volume = {5},
Issue = {4},
month = {8},
year = {2018},
issn = {2347-2693},
pages = {374-379},
url = {https://www.isroset.org/journal/IJSRMSS/full_paper_view.php?paper_id=751},
doi = {https://doi.org/10.26438/ijcse/v5i4.374379}
publisher = {IJCSE, Indore, INDIA},
}
RIS Style Citation:
TY - JOUR
DO = {https://doi.org/10.26438/ijcse/v5i4.374379}
UR - https://www.isroset.org/journal/IJSRMSS/full_paper_view.php?paper_id=751
TI - On m-independence in Graphs
T2 - International Journal of Scientific Research in Mathematical and Statistical Sciences
AU - D. K. Thakkar , Neha P. Jamvecha
PY - 2018
DA - 2018/08/31
PB - IJCSE, Indore, INDIA
SP - 374-379
IS - 4
VL - 5
SN - 2347-2693
ER -
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Abstract :
In this paper, we have defined the concepts of m-independent set, maximal m-independent set and maximum m-independent set. In order to define these concepts we have used the notion of m-adjacent vertices. Adjacent vertices are always m-adjacent vertices. This notion also gives rise to a concept called m-domination in graphs. We prove that a set is maximal m-independent set if and only if it is a minimal m-dominating set. We define m-independence number of a graph to be the maximum cardinality of an m-independent set. We prove a necessary and sufficient condition under which the m-independence number decreases when a vertex is removed from the graph. Further, we have also introduced a new operation in graph called m-removal of a vertex. The subgraph obtained by m-removing a vertex is a subgraph of the subgraph obtained by removing the vertex from the graph. We prove that a vertex is an isolated vertex if and only if the m-independence number of the graph decreases when the vertex is m-removed from the graph. Some related examples have been given to illustrate these concepts.
Key-Words / Index Term :
m-independent set, maximal m-independent set, maximum m-independent set, m-independence number, m-dominating set, minimal m-dominating set, m-removal of a vertex
References :
[1] D. K. Thakkar and Neha P. Jamvecha, “About m-domination number of Graphs” (communicated).
[2] D. K. Thakkar and Neha P. Jamvecha, “A New Variant of Edge Stability in Graphs”, International Journal of Pure and Engineering Mathematics, Vol.5, Issue.3, pp.87-97, 2017.
[3] R. Laskar and K. Peters, “Vertex and edge domination parameters in graphs”, Congressus Numerantium, Vol.48, pp.291-305, 1985.
[4] E. Sampathkumar, and P. S. Neeralagi, “The neighbourhood number of a graph”, Journal of Pure and Applied Mathematics, Vol.16, Issue.2, pp.126-136, 1985.
[5] E. Sampathkumar, and S. S. Kamath, “Mixed Domination in Graphs”, The Indian Journal of Statistics, 1992.
[6] T. W. Haynes, S. T. Hedetniemi and P. J. Slater, “Domination in Graphs Advanced Topics”, Marcel Dekker, Inc., New-York, 1998.
[7] T. W. Haynes, S. T. Hedetniemi, and P. J. Slater, “Fundamentals of domination in graphs”, Marcel Dekker, Inc., New-York, 1998.
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