Discussiones Mathematicae Graph Theory 31(3) (2011) 461-473
doi: 10.7151/dmgt.1558

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THE CONNECTED FORCING CONNECTED VERTEX DETOUR NUMBER OF A GRAPH

A.P. Santhakumaran

Research Department of Mathematics
St. Xavier's College (Autonomous)
Palayamkottai - 627 002, India
e-mail: apskumar1953@yahoo.co.in

P. Titus

Department of Mathematics
Anna University Tirunelveli
Tirunelveli - 627 007, India
e-mail: titusvino@yahoo.com

Abstract

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 cfcdx(Sx), is the cardinality of a minimum connected x-forcing subset for Sx. The connected forcing connected x-detour number of G is cfcdx(G) = mincfcdx(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.

Keywords: vertex detour number, connected vertex detour number, upper connected vertex detour number, forcing connected vertex detour number, connected forcing connected vertex detour number.

2010 Mathematics Subject Classification: 05C12.

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Received 1 September 2009
Revised 10 May 2010
Accepted 12 May 2010