Edge Detour Monophonic Number of a Graph
DOI:
https://doi.org/10.4067/S071609172013000200007Keywords:
Monophonic number, Edge monophonic number, Detour monophonic number, Edge detour monophonic number.Abstract
For a connected graph G of order at least two, an edge detour monophonic set of G is a set S of vertices such that every edge of G lies on a detour monophonic path joining some pair of vertices in S. The edge detour monophonic number of G is the minimum cardinality of its edge detour monophonic sets and is denoted by edm(G) .We determine bounds for it and characterize graphs which realize these bounds. Also, certain general properties satisfied by an edge detour monophonic set are studied. It is shown that for positive integers a, b and c with 2 < a < b < c, there exists a connected graph G such that m(G) = a, m!(G) = b and edm(G) = c,where m(G) is the monophonic number and m! (G) is the edge monophonic number of G. Also, for any integers a and b with 2 < a < b, there exists a connected graph G such that dm(G) = a and edm(G)= b,where dm(G) is the detour monophonic number of a graph G.References
[1] F. Buckley and F. Harary, Distance in Graphs, AddisonWesley, Redwood City, CA, (1990).
[2] G. Chartrand, F. Harary and P. Zhang, On the geodetic number of a graph, Networks, 39 (1), pp. 16, (2002).
[3] W. Hale, Frequency Assignment; Theory and Applications, Proc. IEEE, 68, pp. 14971514, (1980).
[4] F. Harary, Graph Theory, AddisonWesley, (1969).
[5] F. Harary, E. Loukakis, and C. Tsouros, The Geodetic Number of a Graph, Math. Comput. Modeling 17 (11), pp. 8795, (1993).
[6] T. Mansour and M. Schork, Wiener, hyperWiener detour and hyper detour indices of bridge and chain graphs, J. Math. Chem., 47, pp. 7298, (2010).
[7] A.P. Santhakumaran, P. Titus and P. Balakrishnan, Some Realisation Results on Edge Monophonic Number of a Graph, communicated.
[8] A.P. Santhakumaran, P. Titus and K. Ganesamoorthy, On the Monophonic Number of a Graph, communicated.
[9] P. Titus, K. Ganesamoorthy and P. Balakrishnan, The Detour Monophonic Number of a Graph, J. Combin. Math. Combin. Comput., 84, pp. 179188, (2013).
[10] P. Titus and K. Ganesamoorthy, On the Detour Monophonic Number of a Graph, Ars Combinatoria, to appear.
[2] G. Chartrand, F. Harary and P. Zhang, On the geodetic number of a graph, Networks, 39 (1), pp. 16, (2002).
[3] W. Hale, Frequency Assignment; Theory and Applications, Proc. IEEE, 68, pp. 14971514, (1980).
[4] F. Harary, Graph Theory, AddisonWesley, (1969).
[5] F. Harary, E. Loukakis, and C. Tsouros, The Geodetic Number of a Graph, Math. Comput. Modeling 17 (11), pp. 8795, (1993).
[6] T. Mansour and M. Schork, Wiener, hyperWiener detour and hyper detour indices of bridge and chain graphs, J. Math. Chem., 47, pp. 7298, (2010).
[7] A.P. Santhakumaran, P. Titus and P. Balakrishnan, Some Realisation Results on Edge Monophonic Number of a Graph, communicated.
[8] A.P. Santhakumaran, P. Titus and K. Ganesamoorthy, On the Monophonic Number of a Graph, communicated.
[9] P. Titus, K. Ganesamoorthy and P. Balakrishnan, The Detour Monophonic Number of a Graph, J. Combin. Math. Combin. Comput., 84, pp. 179188, (2013).
[10] P. Titus and K. Ganesamoorthy, On the Detour Monophonic Number of a Graph, Ars Combinatoria, to appear.
Published
20130624
How to Cite
[1]
A. P. Santhakumaran, P. Titus, K. Ganesamoorthy, and P. Balakrishnan, “Edge Detour Monophonic Number of a Graph”, Proyecciones (Antofagasta, On line), vol. 32, no. 2, pp. 183198, Jun. 2013.
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Artículos

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