Upper double monophonic number of a graph.
ResumenA set S of a connected graph G of order n is called a double monophonic set of G if for every pair of vertices x, y in G there exist vertices u, v in S such that x, y lie on a u − v monophonic path. The double monophonic number dm(G) of G is the minimum cardinality of a double monophonic set. A double monophonic set S in a connected graph G is called a minimal double monophonic set if no proper subset of S is a double monophonic set of G. The upper double monophonic number of G is the maximum cardinality of a minimal double monophonic set of G, and is denoted by dm⁺(G). Some general properties satisfied by upper double monophonic sets are discussed. It is proved that for a connected graph G of order n, dm(G) = n if and only if dm⁺(G) = n. It is also proved that dm(G) = n − 1 if and only if dm⁺ (G) = n − 1 for a non-complete graph G of order n with a full degree vertex. For any positive integers 2 ≤ a ≤ b, there exists a connected graph G with dm(G) = a and dm⁺(G) = b.
F. Buckley and F. Harary, Distance in Graphs, Addison Wesley, Redwood city, CA, (1990).
G. Chartrand, F. Harary and P. Zhang, On the geodetic number of a graph, Networks 39, pp. 1-6, (2002).
F. Harary, Graph Theory, Addision Wesley, U.S.A.,(1969).
F. Harary, E. Loukakis and C. Tsouros, The geodetic number of a graph, Math. Comput. Modeling 17, pp. 89 - 95, (1993).
A. P. Santhakumaran and T. Jebaraj, The upper double geodetic number of a graph, Malaysian Journal of Science 30 (3): 225- 229, (2011).
A. P. Santhakumaran and T. Jebaraj, The double geodetic number of a graph, Discuss. Math. Graph Theory, 32, pp. 109-119, (2012).
A. P. Santhakumaran and T. Venkata Raghu, The double monophonic number of a graph, International Journal of Computational and Applied Mathematics, 11 (1), pp. 21-26, (2016).