The vertex monophonic number of a graph

A.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 -