Discussiones Mathematicae Graph Theory 32(2) (2012) 191-204
doi: 10.7151/dmgt.1599

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The Vertex Monophonic Number of a Graph

A.P. Santhakumaran

Department of Mathematics
St.Xavier's College (Autonomous)
Palayamkottai - 627 002, India

P.Titus

Department of Mathematics
Anna University Tirunelveli
Tirunelveli - 627 007, India

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.

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Received 10 June 2010
Revised 11 February 2011
Accepted 14 February 2011