Distance Closed Domatic Number of Graphs

V. Sangeetha; T.N. Janakiraman

doi:https://doi.org/10.14445/22315373/IJMTT-V50P536

International Journal of Mathematics Trends and Technology

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Volume 50 | Number 4 | Year 2017 | Article Id. IJMTT-V50P536 | DOI : https://doi.org/10.14445/22315373/IJMTT-V50P536

Distance Closed Domatic Number of Graphs

V. Sangeetha, T.N. Janakiraman

Abstract

In a graph G = (V, E), a set S V(G) is said to be a distance closed set if for each vertex u S and for each w V – S, there exists at least one vertex v S such that d (u, v) = dG(u, w). A dominating set S is said to be a Distance Closed Dominating (D.C.D) set if S is distance closed.The cardinality of a minimum distance closed dominating set of G is called the distance closed domination number of G and is denoted by γdcl(G). The definition and the extensive study of the distance closed dominating sets in graphs are studied in [6].In this paper,the distance closed domatic number of some special classes of graphs are studied.Also, a general algorithm to find the structure of graphs with a given domatic number is proposed.

Keywords

Distance, eccentricity, radius, diameter, degree, paths, cycles, trees, self-centred graphs, complete graphs, complete bipartite graphs, regular graphs, distance closed dominating set, distance closed domination number, distance closed domatic number.

References

1. Bondy, J.A. and U.S.R. Murty(1976) Graph Theory with Applications. American Elsevier, New york.
2. Buckley, F. and Harary – Distance in graphs, Addison – Wesley, Redwood City, CA (1990).
3. Cockayne, E.J. and S.T. Hedetniemi (1977) Towards a theory of domination in Graphs. Networks,7,247 – 261.
4. Haynes, T.W., S.T. Hedetniemi and P.J. Slater (1998) Domination in Graphs: Advanced Topics. Marcel Dekker, New York.
5. Janakiraman, T.N. (1991) On Some eccentricity properties of the graph. Ph.D thesis, Madras University.
6. Janakiraman, T.N., P.J.A. Alphonse and V. Sangeetha (2010) Distance closed domination in graph. International Journal of Engineering Science Advanced Computing and Bio-Technology,1, 109 – 117.
7. Ore, O. (1962) Theory of Graphs. Amer. Soc. Colloq. Publ.,38, Amer. Math. Soc., Providence, RI.
8. Zelinka, B. (1984) Semidomatic numbers of directed graphs. Math. Slovaca.,34, 371 – 374.

Citation :

V. Sangeetha, T.N. Janakiraman, "Distance Closed Domatic Number of Graphs," International Journal of Mathematics Trends and Technology (IJMTT), vol. 50, no. 4, pp. 222-227, 2017. Crossref, https://doi.org/10.14445/22315373/IJMTT-V50P536