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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1599,
author = {A.P. Santhakumaran and P. Titus},
title = {The vertex monophonic number of a graph},
journal = {Discussiones Mathematicae Graph Theory},
volume = {32},
year = {2012},
pages = {191-204},
zbl = {1255.05071},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1599}
}
A.P. Santhakumaran; P. Titus. The vertex monophonic number of a graph. Discussiones Mathematicae Graph Theory, Tome 32 (2012) pp. 191-204. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1599/
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