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Discussiones Mathematicae Graph Theory 32(2) (2012)
191-204
DOI: https://doi.org/10.7151/dmgt.1599
The Vertex Monophonic Number of a Graph
A.P. Santhakumaran
Department of Mathematics | P.Titus
Department of Mathematics |
Abstract
For a connected graph G of order p ≥ 2 and a vertex x of G, a set S ⊆ V(G) is an x-monophonic set of G if each vertex v ∈ V(G) lies on an x −y monophonic path for some element y in S. The minimum cardinality of an x-monophonic set of G is defined as the x-monophonic number of G, denoted by mx(G). An x-monophonic set of cardinality mx(G) is called a mx-set of G. We determine bounds for it and characterize graphs which realize these bounds. A connected graph of order p with vertex monophonic numbers either p −1 or p −2 for every vertex is characterized. It is shown that for positive integers a, b and n ≥ 2 with 2 ≤ a ≤ b, there exists a connected graph G with radmG = a, diammG = b and mx(G) = n for some vertex x in G. Also, it is shown that for each triple m, n and p of integers with 1 ≤ n ≤ p −m −1 and m ≥ 3, there is a connected graph G of order p, monophonic diameter m and mx(G) = n for some vertex x of G.
Keywords: monophonic path, monophonic number, vertex monophonic number
2010 Mathematics Subject Classification: 05C12.
References
[1] | F. Buckley and F. Harary, Distance in Graphs (Addison-Wesley, Redwood City, CA, 1990). |
[2] | F. Buckley, F. Harary and L.U. Quintas, Extremal results on the geodetic number of a graph, Scientia A2 (1988) 17--26. |
[3] | G. Chartrand, F. Harary and P. Zhang, On the geodetic number of a graph, Networks 39 (2002) 1--6, doi: 10.1002/net.10007. |
[4] | G. Chartrand, G.L. Johns and P. Zhang, The detour number of a graph, Utilitas Mathematica 64 (2003) 97--113. |
[5] | G. Chartrand, G.L. Johns and P. Zhang, On the detour number and geodetic number of a graph, Ars Combinatoria 72 (2004) 3--15. |
[6] | F. Harary, Graph Theory (Addison-Wesley, 1969). |
[7] | F. Harary, E. Loukakis and C. Tsouros, The geodetic number of a graph, Math. Comput. Modeling 17(11) (1993) 87--95, doi: 10.1016/0895-7177(93)90259-2. |
[8] | A.P. Santhakumaran and P. Titus, Vertex geodomination in graphs, Bulletin of Kerala Mathematics Association, 2(2) (2005) 45--57. |
[9] | A.P. Santhakumaran and P. Titus, On the vertex geodomination number of a graph, Ars Combinatoria, to appear. |
[10] | A.P. Santhakumaran, P. Titus, The vertex detour number of a graph, AKCE International J. Graphs. Combin. 4(1) (2007) 99--112. |
[11] | A.P. Santhakumaran and P. Titus, Monophonic distance in graphs, Discrete Mathematics, Algorithms and Applications, to appear. |
Received 10 June 2010
Revised 11 February 2011
Accepted 14 February 2011
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