Discussiones Mathematicae Graph Theory 30(1) (2010) 155-174
doi: 10.7151/dmgt.1484

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A.P. Santhakumaran and S. Athisayanathan

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


For two vertices u and v in a graph G = (V,E), the detour distance D(u, v) is the length of a longest u-v path in G. A u-v path of length D(u, v) is called a u-v detour. A set S ⊆V is called an edge detour set if every edge in G lies on a detour joining a pair of vertices of S. The edge detour number dn1(G) of G is the minimum order of its edge detour sets and any edge detour set of order dn1(G) is an edge detour basis of G. A connected graph G is called an edge detour graph if it has an edge detour set. It is proved that for any non-trivial tree T of order p and detour diameter D, dn1(T) ≤ p-D+1 and dn1(T) = p-D+1 if and only if T is a caterpillar. We show that for each triple D, k, p of integers with 3 ≤ k ≤ p-D+1 and D ≥ 4, there is an edge detour graph G of order p with detour diameter D and dn1(G) = k. We also show that for any three positive integers R, D, k with k ≥ 3 and R < D ≤ 2R, there is an edge detour graph G with detour radius R, detour diameter D and dn1(G) = k. Edge detour graphs G with detour diameter D ≤ 4 are characterized when dn1(G) = p-2 or dn1(G) = p-1.

Keywords: detour, edge detour set, edge detour basis, edge detour number.

2010 Mathematics Subject Classification: 05C12.


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Received 25 January 2009
Revised 28 May 2009
Accepted 28 May 2009