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 xymonophonic set of G if each vertex v Є V(G) lies on an xu monophonic path or a yu monophonic path for some element u in S. The minimum cardinality of an xy monophonic set of G is defined as the xymonophonic number of G, denoted by m_{xy} (G). An xymonophonic set of cardinality m_{xy} (G) is called a m_{xy} 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 m_{xy} (G) = n for some edge xy in G.
References
F. Buckley and F. Harary, Distance in Graphs, AddisonWesley, Redwood City, CA, (1990).
F. Harary, Graph Theory, AddisonWesley, Reading Mass, (1969).
A. P. Santhakumaran and P. Titus, Monophonic distance in graphs, Discrete Mathematics, Algorithms and Applications, Vol. 3, No. 2, pp. 159169, (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. 189202, (2012).
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