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## Discussiones Mathematicae Graph Theory

2011 | 31 | 3 | 461-473
Tytuł artykułu

### The connected forcing connected vertex detour number of a graph

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
For any vertex x in a connected graph G of order p ≥ 2, a set S of vertices of V is an x-detour set of G if each vertex v in G lies on an x-y detour for some element y in S. A connected x-detour set of G is an x-detour set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected x-detour set of G is the connected x-detour number of G and is denoted by cdₓ(G). For a minimum connected x-detour set Sₓ of G, a subset T ⊆ Sₓ is called a connected x-forcing subset for Sₓ if the induced subgraph G[T] is connected and Sₓ is the unique minimum connected x-detour set containing T. A connected x-forcing subset for Sₓ of minimum cardinality is a minimum connected x-forcing subset of Sₓ. The connected forcing connected x-detour number of Sₓ, denoted by $cf_{cdx}(Sₓ)$, is the cardinality of a minimum connected x-forcing subset for Sₓ. The connected forcing connected x-detour number of G is $cf_{cdx}(G) = mincf_{cdx}(Sₓ)$, where the minimum is taken over all minimum connected x-detour sets Sₓ in G. Certain general properties satisfied by connected x-forcing sets are studied. The connected forcing connected vertex detour numbers of some standard graphs are determined. It is shown that for positive integers a, b, c and d with 2 ≤ a < b ≤ c ≤ d, there exists a connected graph G such that the forcing connected x-detour number is a, connected forcing connected x-detour number is b, connected x-detour number is c and upper connected x-detour number is d, where x is a vertex of G.
Słowa kluczowe
EN
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
461-473
Opis fizyczny
Daty
wydano
2011
otrzymano
2009-09-01
poprawiono
2010-05-10
zaakceptowano
2010-05-12
Twórcy
autor
• Research 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] G. Chartrand, H. Escuadro and P. Zang, Detour Distance in Graphs, J. Combin. Math. Combin. Comput. 53 (2005) 75-94.
• [3] G. Chartrand, F. Harary and P. Zang, 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. Zang, The Detour Number of a Graph, Utilitas Mathematica 64 (2003) 97-113.
• [5] G. Chartrand, G.L. Johns and P. Zang, 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 (1993) 87-95, doi: 10.1016/0895-7177(93)90259-2.
• [8] T. Mansour and M. Schork, Wiener, hyper-Wiener detour and hyper-detour indices of bridge and chain graphs, J. Math. Chem. 47 (2010) 72-98, doi: 10.1007/s10910-009-9531-7.
• [9] A.P. Santhakumaran and P. Titus, Vertex Geodomination in Graphs, Bulletin of Kerala Mathematics Association 2 (2005) 45-57.
• [10] A.P. Santhakumaran and P. Titus, On the Vertex Geodomination Number of a Graph, Ars Combinatoria, to appear.
• [11] A.P. Santhakumaran and P. Titus, The Vertex Detour Number of a Graph, AKCE International Journal of Graphs and Combinatorics 4 (2007) 99-112.
• [12] A.P. Santhakumaran and P. Titus, The Connected Vertex Geodomination Number of a Graph, Journal of Prime Research in Mathematics 5 (2009) 101-114.
• [13] A.P. Santhakumaran and P. Titus, The Connected Vertex Detour Number of a Graph, communicated.
• [14] A.P. Santhakumaran and P. Titus, The Upper Connected Vertex Detour Number of a Graph, communicated.
Typ dokumentu
Bibliografia
Identyfikatory