Let G be a connected graph. For two vertices u and v in G, a u-v geodesic is any shortest path joining u and v. The closed geodetic interval IG[u, v] consists of all vertices of G lying on any u-v geodesic. For S ⊆ V (G), S is a geodetic set in G if ∪u,v∈S IG[u, v] = V (G). Vertices u and v of G are neighbors if u and v are adjacent. The closed neighborhood NG[v] of vertex v consists of v and all neighbors of v. For S ⊆ V (G), S is a dominating set in G if ∪u∈S NG[u] = V (G). A geodetic dominating set in G is any geodetic set in G which is at the same time a dominating set in G. A geodetic dominating set in G is a minimal geodetic dominating set if it does not have a proper subset which is itself a geodetic dominating set in G. The maximum cardinality of a minimal geodetic dom- inating set in G is the upper geodetic domination number of G. This paper initiates the study of minimal geodetic dominating sets and upper geodetic domination numbers of connected graphs.
@article{bwmeta1.element.doi-10_7151_dmgt_1803,
author = {Hearty M. Nuenay and Ferdinand P. Jamil},
title = {On Minimal Geodetic Domination in Graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {35},
year = {2015},
pages = {403-418},
zbl = {1317.05041},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1803}
}
Hearty M. Nuenay; Ferdinand P. Jamil. On Minimal Geodetic Domination in Graphs. Discussiones Mathematicae Graph Theory, Tome 35 (2015) pp. 403-418. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1803/
[1] S. Canoy Jr. and G. Cagaanan, On the geodesic and hull numbers of graphs, Congr. Numer. 161 (2003) 97-104. | Zbl 1053.05038
[2] S. Canoy Jr. and G. Cagaanan, On the geodetic covers and geodetic bases of the composition G[Km], Ars Combin. 79 (2006) 33-45.
[3] G. Chartrand, F. Harary and P. Zhang, Geodetics sets in graphs, Discuss. Math. Graph Theory 20 (2000) 129-138. doi:10.7151/dmgt.1112[Crossref]
[4] G. Chartrand, F. Harary and P. Zhang, On the geodetic number of a graph, Networks 39 (2002) 1-6. doi:10.1002/net.10007[Crossref] | Zbl 0987.05047
[5] I. Aniversario, F. Jamil and S. Canoy Jr., The closed geodetic numbers of graphs, Util. Math. 74 (2007) 3-18. | Zbl 1176.05040
[6] S. Canoy Jr., G. Cagaanan and S. Gervacio, Convexity, geodetic and hull numbers of the join of graphs, Util. Math. 71 (2007) 143-159. | Zbl 1109.05040
[7] S. Canoy Jr. and I.J. Garces, Convex sets under some graph operations, Graphs Combin. 18 (2002) 787-793. doi:0.1007/s003730200065 | Zbl 1009.05054
[8] H. Escuardo, R. Gera, A. Hansberg, N. Jafari Rad and L. Volkmann, Geodetic domination in graphs, J. Combin. Math. Combin. Comput. 77 (2011) 89-101. | Zbl 1238.05072
[9] T. Haynes, S.T. Hedetniemi and P. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc. New York, 1998). | Zbl 0890.05002
[10] T. Haynes, S.M. Hedetniemi, S.T. Hedetniemi and M. Henning, Domination in graphs applied to electrical power networks, SIAM J. Discrete Math. 15 (2000) 519-529. doi:10.1137/S0895480100375831[Crossref]
[11] F. Buckley and F. Harary, Distance in Graphs (Redwood City, CA: Addison-Wesley, 1990). | Zbl 0688.05017
[12] M. Lema´nska, Weakly convex and convex domination numbers, Opuscula Math. 24 (2004)) 181-188.
[13] J. Bondy and G. Fan, A sufficient condition for dominating cycles, Discrete Math. 67 (1987) 205-208. doi:10.1016/0012-365X(87)90029-X[Crossref]
[14] E. Cockayne and S.T. Hedetniemi, Towards a theory of domination in graphs, Net- works 7 (1977) 247-261. doi:10.1002/net.3230070305[Crossref]
[15] H. Walikar, B. Acharya and E. Samathkumar, Recent Developments in the Theory of Domination in Graphs (Allahabad, 1979).
[16] T.L. Tacbobo, F.P.Jamil and S. Canoy Jr., Monophonic and geodetic domination in the join, corona and composition of graphs, Ars Combin. 112 (2013) 13-32. | Zbl 1313.05287