A Note on Edge-Domsaturation Number of a Graph

Affiliation(s)

Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli- 627 012, India.

Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli- 627 012, India.

ABSTRACT

The edge-domsaturation number ds'(G) of a graph G = (V, E) is the least positive integer k such that every edge of G lies in an edge dominating set of cardinality k. In this paper, we characterize unicyclic graphs G with ds'(G) = q – Δ'(G) + 1 and investigate well-edge dominated graphs. We further define γ'–-critical, γ'+-critical, ds'–-critical, ds'+-critical edges and study some of their properties.

The edge-domsaturation number ds'(G) of a graph G = (V, E) is the least positive integer k such that every edge of G lies in an edge dominating set of cardinality k. In this paper, we characterize unicyclic graphs G with ds'(G) = q – Δ'(G) + 1 and investigate well-edge dominated graphs. We further define γ'–-critical, γ'+-critical, ds'–-critical, ds'+-critical edges and study some of their properties.

Cite this paper

D. Nidha and M. Kala, "A Note on Edge-Domsaturation Number of a Graph,"*Open Journal of Discrete Mathematics*, Vol. 2 No. 3, 2012, pp. 109-113. doi: 10.4236/ojdm.2012.23021.

D. Nidha and M. Kala, "A Note on Edge-Domsaturation Number of a Graph,"

References

[1] F. Harary, “Graph Theory,” Addison-Wesley Publishing Company, Boston, 1969.

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

[3] B. D. Acharya, “The Strong Domination Number of a Graph and Related Concepts,” Journal of Mathematical Physics, Vol. 14, No. 5, 1980, pp. 471-475.

[4] S. Arumugam and R. Kala, “Domsaturation Number of a Graph,” Indian Journal of Pure and Applied Mathematics, Vol. 33, No. 11, 2002, pp. 1671-1676.

[5] S. Arumugam and S. Velammal, “Edge Domination in Graphs,” Taiwanese Journal of Mathematics, Vol. 2, No. 2, 1998, pp. 173-179.

[6] A. Finbow, B. L. Hartnell and R. Nowakowski, “Well Dominated Graphs: A Collection of Covered Ones,” Ars Combinatoria, Vol. 25, No. A, 1988, pp. 5-10.

[1] F. Harary, “Graph Theory,” Addison-Wesley Publishing Company, Boston, 1969.

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

[3] B. D. Acharya, “The Strong Domination Number of a Graph and Related Concepts,” Journal of Mathematical Physics, Vol. 14, No. 5, 1980, pp. 471-475.

[4] S. Arumugam and R. Kala, “Domsaturation Number of a Graph,” Indian Journal of Pure and Applied Mathematics, Vol. 33, No. 11, 2002, pp. 1671-1676.

[5] S. Arumugam and S. Velammal, “Edge Domination in Graphs,” Taiwanese Journal of Mathematics, Vol. 2, No. 2, 1998, pp. 173-179.

[6] A. Finbow, B. L. Hartnell and R. Nowakowski, “Well Dominated Graphs: A Collection of Covered Ones,” Ars Combinatoria, Vol. 25, No. A, 1988, pp. 5-10.