Comment on "Edge Geodetic Covers in Graphs"
DOI:
https://doi.org/10.4067/S0716-09172015000400003Keywords:
Geodetic cover, Geodetic number, Edge geodetic cover, Edge geodetic number.Abstract
In this paper we show by counter example that one of the main results in the paper "Edge Geodetic Covers in Graphsby Mariano and Canoy (International Mathematical Forum, 4, 2009, no. 46, 2301 - 2310) does not hold. Further, we partially characterize connected graphs G of order n for which its edge geodetic number ge(G) = n — 1.References
[1] M. Atici, On the edge geodetic number of a graph. International Journal of Computer Mathematics, 80, pp. 853-861, (2003).
[2] F. Buckley and F. Harary, Distance in Graphs, Addison-Wesley, Redwood City, CA, (1990).
[3] F. Buckley, F. Harary, and L. Quintas, Extremal results on the geodetic number of a graph. SCIENTIA Series A: Mathematical Sciences, 2, pp. 17-26, (1988).
[4] S. Canoy Jr., S. Gervacio, and G. Cagaanan, Convexity, geodetic, and hull numbers of the join of graphs, Utilitas Mathematica, 71, pp. 143- 159, (2006).
[5] R. E. Mariano and S. R. Canoy, Jr., Edge Geodetic Covers in Graphs, International Mathematical Forum, 4, No. 46, pp. 2301-2310, (2009).
[6] G. Chartrand, F. Harary, and P. Zhang, Geodetic sets in graph. Discuss. Math. Graph Theory, 20, pp. 129-138, (2000).
[7] G. Chartrand, F. Harary and P. Zhang, On the Geodetic Number of a Graph, Networks, 39 (1), pp. 1-6, (2002).
[8] G. Chartrand, F. Harary , H. C. Swart and P. Zhang, Geodomination in Graphs, Bulletin of the ICA, 31, pp. 51-59, (2001).
[9] F. Harary, Graph Theory, Addision-Wesely, (1969).
[10] F. Harary, E. Loukakis, C. TSouros, The geodetic number of a graph, Mathl. Comput.Modeling,17 (11), pp. 89-95, (1993).
[11] R. Muntean and P. Zhang, On Geodomonation in Graphs, Congr. Numer., 143, pp. 161-174, (2000).
[12] A. P. Santhakumaran, P. Titus and J. John, The Upper Connected Geodetic Number and Forcing Connected Geodetic Number of a Graph, Discrete Applied Mathematics,157, pp. pp.1571-1580, (2009).
[13] A. P.Santhakumaran, P. Titus and J. John, On the Connected Geodetic Number of a Graph, J. Comb. Math. Comb.Compu., 69, pp. 219- 229, (2009).
[14] A. P. Santhakumaran and J. John, Edge Geodetic Number of a Graph, Journal of Discrete Mathematical Sciences & Cryptography, 10(3), pp. 415-432, (2007).
[15] A. P. Santhakumaran and J. John, Connected Edge Geodetic Number of a Graph, SCIENTIA Series A: Mathematical Sciences, 17, pp. 67- 82, (2009).
[16] A. P.Santhakumaran and J. John, The upper edge geodetic number and the forcing edge geodetic number of a Graph, OPUSCULA MATHEMATICA, 29 (4), pp. 427-441, (2009).
[17] A. P.Santhakumaran and S. V. Ullas Chandran, On the edge geodetic number and k-edge geodetic number of a graph, Int. J. of Math. Combinatorics, 3, pp. 85-93, (2008).
[18] A. P.Santhakumaran and S. V. Ullas Chandran, The k-edge geodetic number of a graph, Utilitas Mathematica, 88, pp. 119-137, (2012).
[2] F. Buckley and F. Harary, Distance in Graphs, Addison-Wesley, Redwood City, CA, (1990).
[3] F. Buckley, F. Harary, and L. Quintas, Extremal results on the geodetic number of a graph. SCIENTIA Series A: Mathematical Sciences, 2, pp. 17-26, (1988).
[4] S. Canoy Jr., S. Gervacio, and G. Cagaanan, Convexity, geodetic, and hull numbers of the join of graphs, Utilitas Mathematica, 71, pp. 143- 159, (2006).
[5] R. E. Mariano and S. R. Canoy, Jr., Edge Geodetic Covers in Graphs, International Mathematical Forum, 4, No. 46, pp. 2301-2310, (2009).
[6] G. Chartrand, F. Harary, and P. Zhang, Geodetic sets in graph. Discuss. Math. Graph Theory, 20, pp. 129-138, (2000).
[7] G. Chartrand, F. Harary and P. Zhang, On the Geodetic Number of a Graph, Networks, 39 (1), pp. 1-6, (2002).
[8] G. Chartrand, F. Harary , H. C. Swart and P. Zhang, Geodomination in Graphs, Bulletin of the ICA, 31, pp. 51-59, (2001).
[9] F. Harary, Graph Theory, Addision-Wesely, (1969).
[10] F. Harary, E. Loukakis, C. TSouros, The geodetic number of a graph, Mathl. Comput.Modeling,17 (11), pp. 89-95, (1993).
[11] R. Muntean and P. Zhang, On Geodomonation in Graphs, Congr. Numer., 143, pp. 161-174, (2000).
[12] A. P. Santhakumaran, P. Titus and J. John, The Upper Connected Geodetic Number and Forcing Connected Geodetic Number of a Graph, Discrete Applied Mathematics,157, pp. pp.1571-1580, (2009).
[13] A. P.Santhakumaran, P. Titus and J. John, On the Connected Geodetic Number of a Graph, J. Comb. Math. Comb.Compu., 69, pp. 219- 229, (2009).
[14] A. P. Santhakumaran and J. John, Edge Geodetic Number of a Graph, Journal of Discrete Mathematical Sciences & Cryptography, 10(3), pp. 415-432, (2007).
[15] A. P. Santhakumaran and J. John, Connected Edge Geodetic Number of a Graph, SCIENTIA Series A: Mathematical Sciences, 17, pp. 67- 82, (2009).
[16] A. P.Santhakumaran and J. John, The upper edge geodetic number and the forcing edge geodetic number of a Graph, OPUSCULA MATHEMATICA, 29 (4), pp. 427-441, (2009).
[17] A. P.Santhakumaran and S. V. Ullas Chandran, On the edge geodetic number and k-edge geodetic number of a graph, Int. J. of Math. Combinatorics, 3, pp. 85-93, (2008).
[18] A. P.Santhakumaran and S. V. Ullas Chandran, The k-edge geodetic number of a graph, Utilitas Mathematica, 88, pp. 119-137, (2012).
How to Cite
[1]
A. P. Santhakumaran and S. V. Ullas Chandran, “Comment on "Edge Geodetic Covers in Graphs"”, Proyecciones (Antofagasta, On line), vol. 34, no. 4, pp. 343-350, 1.
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