ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory


Discussiones Mathematicae Graph Theory 32(2) (2012) 321-330
DOI: 10.7151/dmgt.1602

The Vertex Detour Hull Number of a Graph

A.P. Santhakumaran

Department of Mathematics
St. Xavier's College (Autonomous)
Palayamkottai - 627 002, India

S.V. Ullas Chandran

Department of Mathematics
Amrita Vishwa Vidyapeetham University
Amritapuri Campus, Clappana
Kollam - 690 525, India


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 ID[x,y] consists of x,y, and all vertices lying on some x −y detour of G; while for S ⊆ V(G), ID[S] = ∪x,y ∈ SID[x,y]. A set S of vertices is a detour convex set if ID[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 ID[y]x, the set of all vertices distinct from x that lie on some x −y detour of G; while for S ⊆ V(G), ID[S]x = ∪y ∈ SID[y]x. For x ∉ S, S is an x-detour convex set if ID[S]x = S. The x-detour convex hull of S, [S]xD is the smallest x-detour convex set containing S. A set S is an x-detour hull set if [S]xD = 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 ID[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, eD(x) = b. We determine bounds for dhx(G) and characterize graphs G which realize these bounds.

Keywords: detour, detour number, detour hull number, x-detour number, x-detour hull number

2010 Mathematics Subject Classification: 05C12.


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Received 31 August 2010
Revised 26 May 2011
Accepted 6 June 2011