Edge fixed monophonic number of a graph.
Keywords:
Monophonic path, Vertex monophonic number, Edge fixed monophonic numberAbstract
For an edge xy in a connected graph G of order p ≥ 3, a set SCV(G)is an xy-monophonic set of G if each vertex v Є V(G) lies on an x-u monophonic path or a y-u monophonic path for some element u in S. The minimum cardinality of an xy- monophonic set of G is defined as the xy-monophonic number of G, denoted by mxy (G). An xy-monophonic set of cardinality mxy (G) is called a mxy -set of G. We determine bounds for it and find the same for special classes of graphs. It is shown that for any three positive integers r, d and n ≥ 2 with 2 ≤ r ≤ d, there exists a connected graph G with monophonic radius r, monophonic diameter d and mxy (G) = n for some edge xy in G.
References
F. Buckley and F. Harary, Distance in Graphs, Addison-Wesley, Redwood City, CA, (1990).
F. Harary, Graph Theory, Addison-Wesley, Reading Mass, (1969).
A. P. Santhakumaran and P. Titus, Monophonic distance in graphs, Discrete Mathematics, Algorithms and Applications, Vol. 3, No. 2, pp. 159-169, (2011).
A. P. Santhakumaran and P. Titus, A note on ‘Monophonic distance in graphs’, Discrete Mathematics, Algorithms and Applications, Vol. 4, No. 2 (2012), DOI: 10.1142/S1793830912500188.
A. P. Santhakumaran and P. Titus, The vertex monophonic number of a graph, Discussiones Mathematicae Graph Theory, 32, pp. 189-202, (2012).
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