# On Minimal Geodetic Domination in Graphs

Hearty M. Nuenay; Ferdinand P. Jamil

Discussiones Mathematicae Graph Theory (2015)

- Volume: 35, Issue: 3, page 403-418
- ISSN: 2083-5892

## Access Full Article

top## Abstract

top## How to cite

topHearty M. Nuenay, and Ferdinand P. Jamil. "On Minimal Geodetic Domination in Graphs." Discussiones Mathematicae Graph Theory 35.3 (2015): 403-418. <http://eudml.org/doc/271213>.

@article{HeartyM2015,

abstract = {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.},

author = {Hearty M. Nuenay, Ferdinand P. Jamil},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {minimal geodetic dominating set; upper geodetic domination number},

language = {eng},

number = {3},

pages = {403-418},

title = {On Minimal Geodetic Domination in Graphs},

url = {http://eudml.org/doc/271213},

volume = {35},

year = {2015},

}

TY - JOUR

AU - Hearty M. Nuenay

AU - Ferdinand P. Jamil

TI - On Minimal Geodetic Domination in Graphs

JO - Discussiones Mathematicae Graph Theory

PY - 2015

VL - 35

IS - 3

SP - 403

EP - 418

AB - 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.

LA - eng

KW - minimal geodetic dominating set; upper geodetic domination number

UR - http://eudml.org/doc/271213

ER -

## References

top- [1] S. Canoy Jr. and G. Cagaanan, On the geodesic and hull numbers of graphs, Congr. Numer. 161 (2003) 97-104. Zbl1053.05038
- [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] Zbl0987.05047
- [5] I. Aniversario, F. Jamil and S. Canoy Jr., The closed geodetic numbers of graphs, Util. Math. 74 (2007) 3-18. Zbl1176.05040
- [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. Zbl1109.05040
- [7] S. Canoy Jr. and I.J. Garces, Convex sets under some graph operations, Graphs Combin. 18 (2002) 787-793. doi:0.1007/s003730200065 Zbl1009.05054
- [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. Zbl1238.05072
- [9] T. Haynes, S.T. Hedetniemi and P. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc. New York, 1998). Zbl0890.05002
- [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). Zbl0688.05017
- [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. Zbl1313.05287

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.