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.
Downloads
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).
Downloads
Published
Issue
Section
License
-
Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.
