# The vertex monophonic number of a graph

Discussiones Mathematicae Graph Theory (2012)

- Volume: 32, Issue: 2, page 191-204
- ISSN: 2083-5892

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topA.P. Santhakumaran, and P. Titus. "The vertex monophonic number of a graph." Discussiones Mathematicae Graph Theory 32.2 (2012): 191-204. <http://eudml.org/doc/270812>.

@article{A2012,

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.},

author = {A.P. Santhakumaran, P. Titus},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {monophonic path; monophonic number; vertex monophonic number},

language = {eng},

number = {2},

pages = {191-204},

title = {The vertex monophonic number of a graph},

url = {http://eudml.org/doc/270812},

volume = {32},

year = {2012},

}

TY - JOUR

AU - A.P. Santhakumaran

AU - P. Titus

TI - The vertex monophonic number of a graph

JO - Discussiones Mathematicae Graph Theory

PY - 2012

VL - 32

IS - 2

SP - 191

EP - 204

AB - 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.

LA - eng

KW - monophonic path; monophonic number; vertex monophonic number

UR - http://eudml.org/doc/270812

ER -

## References

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