A Note on Open Support of a Graph under
Addition II

S. Balamurugan; M. Anitha; P. Aristotle; C. Karnan

doi:https://doi.org/10.14445/22315373/IJMTT-V65I5P517

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 65 | Issue 5 | Year 2019 | Article Id. IJMTT-V65I5P517 | DOI : https://doi.org/10.14445/22315373/IJMTT-V65I5P517

A Note on Open Support of a Graph under Addition II

S. Balamurugan , M. Anitha , P. Aristotle , C. Karnan

Abstract

In this work we consider finite, undirected, simple graphs 𝐺 = (𝑉,𝐸) with 𝑛 vertices and 𝑚 edges. The neighbourhood of a vertex 𝑣 ∈ 𝑉(𝐺) is the set 𝑁𝐺(𝑣) of all the vertices adjacent to 𝑣 in 𝐺. For a set 𝑋 ⊆ 𝑉(𝐺), the open neighbourhood𝑁𝐺(𝑋) is defined to be ∪𝑣∈𝑋 𝑁𝐺(𝑣) and the closed neighbourhood 𝑁𝐺[𝑋] = 𝑁𝐺(𝑋) ∪ 𝑋. The degree of a vertex 𝑣 ∈ 𝑉(𝐺) is the number of edges of 𝐺 incident with 𝑣 and is denoted by 𝑑𝑒𝑔𝐺(𝑣) or 𝑑𝑒𝑔(𝑣). The maximum and the minimum degrees of the vertices of 𝐺 are respectively denoted by Δ(𝐺) and 𝛿(𝐺). A vertex of a degree 0 in 𝐺 is called an isolated vertex and a vertex of degree 1 is called a pendent vertex or an end vertex of 𝐺. A vertex of a graph 𝐺 is said to be a vertex of full degree if it is adjacent to all other vertices in 𝐺. A graph 𝐺 is said to be regular of degree 𝑟 if every vertex of 𝐺 has degree 𝑟. Such graphs are called r-regular graphs. The Dutch windmill graph𝐷𝑛 (𝑚) , is the graph obtained by taking 𝑚 copies of the cycle graph 𝐶𝑛 with a vertex in common. The Butterfly graph(also called the bowtie graph and the hourglass graph) is a planar undirected graph with 5 vertices and 6 edges. It can be constructed by joining 2 copies of the cycle graph 𝐶3 with a common vertex. It is denoted by 𝐵𝑛 . The ladder graph𝐿𝑛 is a planar undirected graph with 2n vertices and 𝑛 + 2(𝑛 − 1) edges. The Ladder graph obtained as the cartesian product of two graphs one of which has only one edge: 𝐿𝑛,1 = 𝑃𝑛 × 𝑃1.

Keywords

Vertex Degree, Open Support of a Vertex, Open Support of a Graph

References

[1] M. Anitha, S. Balamurugan: Strong (Weak) Efficient Open Domination on Product Graphs, International Journal of Pure and Applied Mathematics, (J.No. 23425), IISN 1311 - 8080, communicated.
[2] S. Balamurugan, M. Anitha, C. Karnan and P. Aristotle A Note on Open Support of a Graph under Addition I, Communicated.
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[4] S. Balamurugan, A study on chromatic strong domination in graphs, Ph.D Thesis, Madurai Kamaraj University, India 2008.
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Citation :

S. Balamurugan , M. Anitha , P. Aristotle , C. Karnan, "A Note on Open Support of a Graph under Addition II," International Journal of Mathematics Trends and Technology (IJMTT), vol. 65, no. 5, pp. 115-119, 2019. Crossref, https://doi.org/10.14445/22315373/IJMTT-V65I5P517