The Middle Equitable Dominating Graphs

Affiliation(s)

Department of Studies in Mathematics, University of Mysore, Mysore 570 006, India.

Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia.

Department of Studies in Mathematics, University of Mysore, Mysore 570 006, India.

Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia.

ABSTRACT

Let*G*= (*V, E*) be a graph and *A(G)* is the collection of all minimal equitable dominating set of *G*. The middle equitable dominating graph of *G* is the graph denoted by *M*_{ed}(*G*) with vertex set the disjoint union of *V∪A(G)* and (*u, v*) is an edge if and only if *u ∩ v ≠ φ* whenever *u, v ∈ A(G)* or *u ∈ v* whenever *u ∈ v* and *v ∈ A(G)* . In this paper, characterizations are given for graphs whose middle equitable dominating graph is connected and *K*_{p}∈M_{ed}(G) . Other properties of middle equitable dominating graphs are also obtained.

Let

Cite this paper

A. Alwardi, N. Soner and A. Al-Kenani, "The Middle Equitable Dominating Graphs,"*Open Journal of Discrete Mathematics*, Vol. 2 No. 3, 2012, pp. 93-95. doi: 10.4236/ojdm.2012.23017.

A. Alwardi, N. Soner and A. Al-Kenani, "The Middle Equitable Dominating Graphs,"

References

[1] K. D. Dharmalingam, “Studies in Graph Theorey-Equitable Domination and Bottleneck Domination,” Ph.D. Thesis, Madurai Kamaraj University, Madurai, 2006.

[2] F. Harary, “Graph Theory,” Addison-Wesley, Boston, 1969.

[3] T. W. Haynes, S. T. Hedetniemi and P. J. Slater, “Fundamentals of Domination in Graphs,” Marcel Dekker, Inc., New York, 1998.

[4] V. R. Kulli and B. Janakiram, “The Minimal Dominating Graph,” Graph Theory Notes of New York, Vol. 28, Academy of Sciences, New York, 1995, pp. 12-15.

[5] V. R. Kulli, B. Janakiram and K. M. Niranjan, “The Common Minimal Domi-nating Graph,” Indian Journal of Pure and Applied Mathematics, Vol. 27, No. 2, 1996, pp. 193196.

[6] V. R. Kulli, B. Janaki-ram and K. M. Niranjan, “The Vertex Minimal Dominating Graph,” Acta Ciencia Indica, Vol. 28, 2002, pp. 435-440.

[7] V. R. Kulli, B. Janakiram and K. M. Niranjan, “The Dominating Graph,” Graph Theory Notes of New York, Vol. 46, New York Academy of Sciences, New York, 2004, pp. 5-8.

[8] H. B. Walikar, B. D. Acharya and E. Sampathkumar, “Recent Developments in the Theory of Domination in Graphs,” MRI Lecture Notes in Mathematices, Vol. 1, Mehta Research Institute, Alahabad, 1979.

[1] K. D. Dharmalingam, “Studies in Graph Theorey-Equitable Domination and Bottleneck Domination,” Ph.D. Thesis, Madurai Kamaraj University, Madurai, 2006.

[2] F. Harary, “Graph Theory,” Addison-Wesley, Boston, 1969.

[3] T. W. Haynes, S. T. Hedetniemi and P. J. Slater, “Fundamentals of Domination in Graphs,” Marcel Dekker, Inc., New York, 1998.

[4] V. R. Kulli and B. Janakiram, “The Minimal Dominating Graph,” Graph Theory Notes of New York, Vol. 28, Academy of Sciences, New York, 1995, pp. 12-15.

[5] V. R. Kulli, B. Janakiram and K. M. Niranjan, “The Common Minimal Domi-nating Graph,” Indian Journal of Pure and Applied Mathematics, Vol. 27, No. 2, 1996, pp. 193196.

[6] V. R. Kulli, B. Janaki-ram and K. M. Niranjan, “The Vertex Minimal Dominating Graph,” Acta Ciencia Indica, Vol. 28, 2002, pp. 435-440.

[7] V. R. Kulli, B. Janakiram and K. M. Niranjan, “The Dominating Graph,” Graph Theory Notes of New York, Vol. 46, New York Academy of Sciences, New York, 2004, pp. 5-8.

[8] H. B. Walikar, B. D. Acharya and E. Sampathkumar, “Recent Developments in the Theory of Domination in Graphs,” MRI Lecture Notes in Mathematices, Vol. 1, Mehta Research Institute, Alahabad, 1979.