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## Discussiones Mathematicae Graph Theory

2012 | 32 | 2 | 191-204
Tytuł artykułu

### The vertex monophonic number of a graph

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
EN
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
191-204
Opis fizyczny
Daty
wydano
2012
otrzymano
2010-06-10
poprawiono
2011-02-11
zaakceptowano
2011-02-14
Twórcy
autor
• Department of Mathematics, St.Xavier's College (Autonomous), Palayamkottai - 627 002, India
autor
• Department of Mathematics, Anna University, Tirunelveli, Tirunelveli - 627 007, India
Bibliografia
• [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.
Typ dokumentu
Bibliografia
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