# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

# IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

Rejection Rate (2018-2019): c. 84%

# Discussiones Mathematicae Graph Theory

## ON EDGE DETOUR GRAPHS

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

## Abstract

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.

## References

 [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.

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