Connected edge monophonic number of a graph

• Santhakumaran, A. P. ^[1] ; Titus, P. ^[2] ; Balakrishnan, P. ^[2]
  1. [1] Hindustan University

     Hindustan University

     India

     
  2. [2] University College of Engineering Nagercoil.
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 32, Nº. 3, 2013, págs. 215-234
• Idioma: inglés
• DOI: 10.4067/S0716-09172013000300002
• Enlaces
  • Texto completo
• Resumen

  • For a connected graph G of order n,a set S of vertices is called an edge monophonic set of G if every edge of G lies on a monophonic path joining some pair of vertices in S, and the edge monophonic number me(G) is the minimum cardinality of an edge monophonic set. An edge monophonic set S of G is a connected edge mono-phonic set if the subgraph induced by S is connected, and the connected edge monophonic number mce(G) is the minimum cardinality of a connected edge monophonic set of G. Graphs of order n with connected edge monophonic number 2, 3 or n are characterized. It is proved that there is no non-complete graph G of order n > 3 with me(G) = 3 and mce(G)= 3. It is shown that for integers k,l and n with 4 < k < l < n, there exists a connected graph G of order n such that me(G) = k and mce(G) = l.Also, for integers j,k and l with 4 < j < k < l, there exists a connected graph G such that me(G)= j,mce(G)= k and gce(G) = l,where gce(G) is the connected edge geodetic number ofa graph G.

• Referencias bibliográficas
  • Citas [1] F. Buckley and F. Harary, Distance in Graphs, Addison-Wesley, Redwood City, CA, (1990).
  • [2] G. Chartrand, F. Harary and P. Zhang, On the geodetic number of a graph, Networks, 39(1), pp. 1-6, (2002).
  • [3] G. Chartrand, F. Harary , H. C. Swart and P. Zhang, Geodomination in graphs, Bulletin of the ICA, 31, pp. 51-59, (2001).
  • [4] F. Harary, Graph Theory, Addision-Wesely (1969).
  • [5] F. Harary, E. Loukakis and C. Tsouros, The geodetic number of a graph, Mathl. Comput. Modeling, 17(11), pp. 89-95, (1993).
  • [6] R. Muntean and P. Zhang, On geodomonation in graphs, Congr. Numer., 143, pp. 161-174, (2000).
  • [7] A. P. Santhakumaran P. Titus and P. Balakrishnan, Edge monophonic number of a graph, communicated.
  • [8] A. P. Santhakumaran and S. V. Ullas Chandran, On the edge geodetic number and k-edge geodetic number of a graph, Inter. J. Math. Combin.,...
  • [9] A. P. Santhakumaran and S. V. Ullas Chandran, The edge geodetic number and cartesian product of a graph, Discussiones Mathematicae Graph...