Just excellence and very excellence in graphs with respect to strong domination

Authors

  • C. V. R. Harinarayanan
  • C. Y. Ponnappan
  • S. P. Subbiah
  • R. Sundareswaran
  • V. Swaminathan

DOI:

https://doi.org/10.5556/j.tkjm.38.2007.87

Abstract

A graph $ G $ is said to be excellent with respect to strong domination if each $ u\in V(G) $, belongs to some $ \gamma_s $-set of $G$. $ G $ is said to be just excellent with respect to strong domination if each $u\in V(G)$ is contained in a unique $\gamma_s$-set of $G$. A graph $ G $ which is excellent with respect to strong domination is said to be very excellent with respect to strong domination if there is a $ \gamma_s $-set $ D $ of $ G $ such that to each vertex $ u\in V-D $, there exists a vertex $ v\in D $ such that $ (D-\{v\})\cup\{u\} $ is a $ \gamma_s $-set of $ G $. In this paper we study these two classes of graphs. A strong very excellent graph is said to be rigid very excellent with respect to strong domination if the following condition is satisfied. Let $ D $ be a very excellent $ \gamma_s $-set of $ G $. To each $ u\not\in D $, let $ E(u, D)=\{v\in D: (D-\{v\})\cup\{u\} $ is a $ \gamma_s $-set of $ G\} $. If $ |E(u, D)|=1 $ for all $ u\not\in D $ then $ D $ is said to be a rigid very excellent $ \gamma_s $-set of $ G $. If $ G $ has at least one rigid very excellent $ \gamma_s $-set of $ G $ then $ G $ is said to be a rigid very excellent graph with respect to strong domination (or) a strong rigid very excellent graph. Some results regarding strong very excellent graphs are obtained.

Author Biographies

C. V. R. Harinarayanan

Department of Mathematics, K.L.N College of Information Technology, Sivagangai, India.

C. Y. Ponnappan

Department of Mathematics, K.L.N College of Engineering, Sivagangai, India.

S. P. Subbiah

Department of Mathematics, Mannar Thirumalai Naicker College, Madurai, India.

R. Sundareswaran

Department of Mathematics, Rajalakshmi Engineering College, Chennai, India.

V. Swaminathan

Coordinator and Head, Ramanujan Research center, Saraswathi Narayanan College, Madurai, India.

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Published

2007-06-30

How to Cite

Harinarayanan, C. V. R., Ponnappan, C. Y., Subbiah, S. P., Sundareswaran, R., & Swaminathan, V. (2007). Just excellence and very excellence in graphs with respect to strong domination. Tamkang Journal of Mathematics, 38(2), 167-175. https://doi.org/10.5556/j.tkjm.38.2007.87

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Section

Papers