Geodesic
The vertex monophonic number of a graph
Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 2, pp. 191-204

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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 
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Santhakumaran, A.; Titus, P. The vertex monophonic number of a graph. Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 2, pp. 191-204. http://geodesic.mathdoc.fr/item/DMGT_2012_32_2_a0/