May 2012
Discussiones Mathematicae: Graph Theory;2012, Vol. 32 Issue 2, p191
Academic Journal
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 ε 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.


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