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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1484,
author = {A.P. Santhakumaran and S. Athisayanathan},
title = {On edge detour graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {30},
year = {2010},
pages = {155-174},
zbl = {1214.05019},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1484}
}
A.P. Santhakumaran; S. Athisayanathan. On edge detour graphs. Discussiones Mathematicae Graph Theory, Tome 30 (2010) pp. 155-174. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1484/
[000] [1] F. Buckley and F. Harary, Distance in Graphs (Addison-Wesley, Reading MA, 1990). | Zbl 0688.05017
[001] [2] G. Chartrand, H. Escuadro and P. Zang, Detour distance in graphs, J. Combin. Math. Combin. Comput. 53 (2005) 75-94. | Zbl 1074.05029
[002] [3] G. Chartrand, G.L. Johns, and P. Zang, Detour number of a graph, Util. Math. 64 (2003) 97-113. | Zbl 1033.05032
[003] [4] G. Chartrand and P. Zang, Distance in graphs-taking the long view, AKCE J. Graphs. Combin. 1 (2004) 1-13. | Zbl 1062.05051
[004] [5] G. Chartrand and P. Zang, Introduction to Graph Theory (Tata McGraw-Hill, New Delhi, 2006).
[005] [6] A.P. Santhakumaran and S. Athisayanathan, Weak edge detour number of a graph, Ars Combin., to appear. | Zbl 1249.05102
[006] [7] A.P. Santhakumaran and S. Athisayanathan, Edge detour graphs, J. Combin. Math. Combin. Comput. 69 (2009) 191-204. | Zbl 1200.05071