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2010 | 30 | 1 | 155-174
Tytuł artykułu

On edge detour graphs

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Kategorie tematyczne
Wydawca
Rocznik
Tom
30
Numer
1
Strony
155-174
Opis fizyczny
Daty
wydano
2010
otrzymano
2009-01-25
poprawiono
2009-05-28
zaakceptowano
2009-05-28
Twórcy
  • Research Department of Mathematics, St. Xavier's College (Autonomous), Palayamkottai - 627 002, India
  • Research Department of Mathematics, St. Xavier's College (Autonomous), Palayamkottai - 627 002, India
Bibliografia
  • [1] F. Buckley and F. Harary, Distance in Graphs (Addison-Wesley, Reading MA, 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, G.L. Johns, and P. Zang, Detour number of a graph, Util. Math. 64 (2003) 97-113.
  • [4] G. Chartrand and P. Zang, Distance in graphs-taking the long view, AKCE J. Graphs. Combin. 1 (2004) 1-13.
  • [5] G. Chartrand and P. Zang, Introduction to Graph Theory (Tata McGraw-Hill, New Delhi, 2006).
  • [6] A.P. Santhakumaran and S. Athisayanathan, Weak edge detour number of a graph, Ars Combin., to appear.
  • [7] A.P. Santhakumaran and S. Athisayanathan, Edge detour graphs, J. Combin. Math. Combin. Comput. 69 (2009) 191-204.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1484
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