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

Voir la notice de l'article provenant de la source Library of Science

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 
@article{DMGT_2012_32_2_a0,
     author = {Santhakumaran, A. and Titus, P.},
     title = {The vertex monophonic number of a graph},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {191--204},
     publisher = {mathdoc},
     volume = {32},
     number = {2},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2012_32_2_a0/}
}
TY  - JOUR
AU  - Santhakumaran, A.
AU  - Titus, P.
TI  - The vertex monophonic number of a graph
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2012
SP  - 191
EP  - 204
VL  - 32
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGT_2012_32_2_a0/
LA  - en
ID  - DMGT_2012_32_2_a0
ER  - 
%0 Journal Article
%A Santhakumaran, A.
%A Titus, P.
%T The vertex monophonic number of a graph
%J Discussiones Mathematicae. Graph Theory
%D 2012
%P 191-204
%V 32
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGT_2012_32_2_a0/
%G en
%F DMGT_2012_32_2_a0
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/

[1] F. Buckley and F. Harary, Distance in Graphs (Addison-Wesley, Redwood City, CA, 1990).

[2] F. Buckley, F. Harary and L.U. Quintas, Extremal results on the geodetic number of a graph, Scientia A2 (1988) 17-26.

[3] G. Chartrand, F. Harary and P. Zhang, On the geodetic number of a graph, Networks 39 (2002) 1-6, doi: 10.1002/net.10007.

[4] G. Chartrand, G.L. Johns and P. Zhang, The detour number of a graph, Utilitas Mathematica 64 (2003) 97-113.

[5] G. Chartrand, G.L. Johns and P. Zhang, On the detour number and geodetic number of a graph, Ars Combinatoria 72 (2004) 3-15.

[6] F. Harary, Graph Theory (Addison-Wesley, 1969).

[7] F. Harary, E. Loukakis and C. Tsouros, The geodetic number of a graph, Math. Comput. Modeling 17(11) (1993) 87-95, doi: 10.1016/0895-7177(93)90259-2.

[8] A.P. Santhakumaran and P. Titus, Vertex geodomination in graphs, Bulletin of Kerala Mathematics Association, 2(2) (2005) 45-57.

[9] A.P. Santhakumaran and P. Titus, On the vertex geodomination number of a graph, Ars Combinatoria, to appear.

[10] A.P. Santhakumaran, P. Titus, The vertex detour number of a graph, AKCE International J. Graphs. Combin. 4(1) (2007) 99-112.

[11] A.P. Santhakumaran and P. Titus, Monophonic distance in graphs, Discrete Mathematics, Algorithms and Applications, to appear.