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## Discussiones Mathematicae Graph Theory

2012 | 32 | 2 | 321-330
Tytuł artykułu

### The vertex detour hull number of a graph

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Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
For vertices x and y in a connected graph G, the detour distance D(x,y) is the length of a longest x - y path in G. An x - y path of length D(x,y) is an x - y detour. The closed detour interval I_D[x,y] consists of x,y, and all vertices lying on some x -y detour of G; while for S ⊆ V(G), $I_D[S] = ⋃_{x,y ∈ S} I_D[x,y]$. A set S of vertices is a detour convex set if $I_D[S] = S$. The detour convex hull $[S]_D$ is the smallest detour convex set containing S. The detour hull number dh(G) is the minimum cardinality among subsets S of V(G) with $[S]_D = V(G)$. Let x be any vertex in a connected graph G. For a vertex y in G, denoted by $I_D[y]^x$, the set of all vertices distinct from x that lie on some x - y detour of G; while for S ⊆ V(G), $I_D[S]^x = ⋃_{y ∈ S} I_D[y]^x$. For x ∉ S, S is an x-detour convex set if $I_D[S]^x = S$. The x-detour convex hull of S, $[S]^x_D$ is the smallest x-detour convex set containing S. A set S is an x-detour hull set if $[S]^x_D = V(G) -{x}$ and the minimum cardinality of x-detour hull sets is the x-detour hull number dhₓ(G) of G. For x ∉ S, S is an x-detour set of G if $I_D[S]^x = V(G) - {x}$ and the minimum cardinality of x-detour sets is the x-detour number dₓ(G) of G. Certain general properties of the x-detour hull number of a graph are studied. It is shown that for each pair of positive integers a,b with 2 ≤ a ≤ b+1, there exist a connected graph G and a vertex x such that dh(G) = a and dhₓ(G) = b. It is proved that every two integers a and b with 1 ≤ a ≤ b, are realizable as the x-detour hull number and the x-detour number respectively. Also, it is shown that for integers a,b and n with 1 ≤ a ≤ n -b and b ≥ 3, there exist a connected graph G of order n and a vertex x such that dhₓ(G) = a and the detour eccentricity of x, $e_D(x) = b$. We determine bounds for dhₓ(G) and characterize graphs G which realize these bounds.
Słowa kluczowe
EN
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
321-330
Opis fizyczny
Daty
wydano
2012
otrzymano
2010-08-31
poprawiono
2011-05-26
zaakceptowano
2011-06-06
Twórcy
autor
• Department of Mathematics, St. Xavier's College (Autonomous), Palayamkottai - 627 002, India
autor
• Department of Mathematics, Amrita Vishwa Vidyapeetham University, Amritapuri Campus, Clappana, Kollam - 690 525, India
Bibliografia
• [1] F. Buckley and F. Harary, Distance in Graphs (Addison-Wesley, Reading MA, 1990).
• [2] G. Chartrand, H. Escuadro and P. Zhang, Detour distance in graphs, J. Combin. Math. Combin. Comput. 53 (2005) 75-94.
• [3] G. Chartrand, G.L. Johns and P. Zhang, Detour number of a graph, Util. Math. 64 (2003) 97-113.
• [4] G. Chartrand, G.L. Johns and P. Zhang, On the detour number and geodetic number of a graph, Ars Combin. 72 (2004) 3-15.
• [5] G. Chartrand, L. Nebesky and P. Zhang, A survey of Hamilton colorings of graphs, preprint.
• [6] G. Chartrand and P. Zhang, Introduction to Graph Theory (Tata McGraw- Hill Edition, New Delhi, 2006).
• [7] W. Hale, Frequency Assignment, in: Theory and Applications, Proc. IEEE 68 (1980) 1497-1514, doi: 10.1109/PROC.1980.11899.
• [8] A.P. Santhakumaran and S. Athisayanathan, Connected detour number of a graph, J. Combin. Math. Combin. Comput. 69 (2009) 205-218.
• [9] A.P. Santhakumaran and P. Titus, The vertex detour number of a graph, AKCE J. Graphs. Combin. 4 (2007) 99-112.
• [10] A.P. Santhakumaran and S.V. Ullas Chandran, The detour hull number of a graph, communicated.
Typ dokumentu
Bibliografia
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