# Products of Geodesic Graphs and the Geodetic Number of Products

Jake A. Soloff; Rommy A. Márquez; Louis M. Friedler

Discussiones Mathematicae Graph Theory (2015)

- Volume: 35, Issue: 1, page 35-42
- ISSN: 2083-5892

## Access Full Article

top## Abstract

top## How to cite

topJake A. Soloff, Rommy A. Márquez, and Louis M. Friedler. "Products of Geodesic Graphs and the Geodetic Number of Products." Discussiones Mathematicae Graph Theory 35.1 (2015): 35-42. <http://eudml.org/doc/271222>.

@article{JakeA2015,

abstract = {Given a connected graph and a vertex x ∈ V (G), the geodesic graph Px(G) has the same vertex set as G with edges uv iff either v is on an x − u geodesic path or u is on an x − v geodesic path. A characterization is given of those graphs all of whose geodesic graphs are complete bipartite. It is also shown that the geodetic number of the Cartesian product of Km,n with itself, where m, n ≥ 4, is equal to the minimum of m, n and eight.},

author = {Jake A. Soloff, Rommy A. Márquez, Louis M. Friedler},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {geodesic graph; geodetic number; Cartesian products},

language = {eng},

number = {1},

pages = {35-42},

title = {Products of Geodesic Graphs and the Geodetic Number of Products},

url = {http://eudml.org/doc/271222},

volume = {35},

year = {2015},

}

TY - JOUR

AU - Jake A. Soloff

AU - Rommy A. Márquez

AU - Louis M. Friedler

TI - Products of Geodesic Graphs and the Geodetic Number of Products

JO - Discussiones Mathematicae Graph Theory

PY - 2015

VL - 35

IS - 1

SP - 35

EP - 42

AB - Given a connected graph and a vertex x ∈ V (G), the geodesic graph Px(G) has the same vertex set as G with edges uv iff either v is on an x − u geodesic path or u is on an x − v geodesic path. A characterization is given of those graphs all of whose geodesic graphs are complete bipartite. It is also shown that the geodetic number of the Cartesian product of Km,n with itself, where m, n ≥ 4, is equal to the minimum of m, n and eight.

LA - eng

KW - geodesic graph; geodetic number; Cartesian products

UR - http://eudml.org/doc/271222

ER -

## References

top- [1] A.P. Santhakumaran and P. Titus, Geodesic graphs, Ars Combin. 99 (2011) 75-82. Zbl1265.05204
- [2] W. Imrich and S. Klavžar, Product Graphs: Structure and Recognition (Wiley, New York, 2000).
- [3] B. Brešar, M. Kovše and A. Tepeh Horvat, Geodetic sets in graphs in: M. Dehmer (Eds.), Structural Analysis of Complex Networks, Springer Science+Business Media, LLC, New York (2011) 197-218. doi:10.1007/978-0-8176-4789-6 8[Crossref] Zbl1221.05107
- [4] B. Brešar, S. Klavžar and A. Tepeh Horvat, On the geodetic number and related metric sets in Cartesian product graphs, Discrete Math. 308 (2008) 5555-5561. doi:10.1016/j.disc.2007.10.007[WoS][Crossref] Zbl1200.05060
- [5] F. Harary, E. Loukakis and C. Tsouros, The geodetic number of a graph, Math. Comput. Modelling 17 (1993) 89-95. doi:10.1016/0895-7177(93)90259-2[Crossref][WoS] Zbl0825.68490
- [6] 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
- [7] J. Cáceres, C. Hernando, M. Mora, I.M. Pelayo and M.L. Puertas, On the geodetic and the hull numbers in strong product graphs, Comput. Math. Appl. 60 (2010) 3020-3031. doi:10.1016/j.camwa.2010.10.001[WoS][Crossref] Zbl1207.05043
- [8] T. Jiang, I. Pelayo and D. Pritikin, Geodesic convexity and Cartesian products in graphs, manuscript (2004).
- [9] Y. Ye, C. Lu and Q. Liu, The geodetic numbers of Cartesian products of graphs, Math. Appl. 20 (2007) 158-163. Zbl1125.05058

## NotesEmbed ?

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