Discrete Mathematics 272 (2003) 119 – 126
www.elsevier.com/locate/disc
Radius-edge-invariant and
diameter-edge-invariant graphs
H.B. Walikar
a
, Fred Buckley
b
, M.K. Itagi
a
a
Department of Mathematics, K.R.C.P.G. Centre, Belgaum 590001, India
b
Department of Mathematics, Baruch College (CUNY), New York,
NY 10010, USA
Received 23 May 2001; received in revised form 30 July 2001; accepted 30 November 2001
Dedicated to Frank Harary on the occasion of his eightieth birthday
Abstract
The eccentricity e(v)ofv is the distance to a farthest vertex from v. The radius r(G)is
the minimum eccentricity amongthe vertices of G and the diameter d(G) is the maximum
eccentricity. For graph G − e obtained by deletingedge e in G, we have r(G − e) ¿ r(G)
and d(G − e) ¿ d(G). If for all e in G, r(G − e)=r(G), then G is radius-edge-invariant.
Similarly, if for all e in G, d(G − e)=d(G), then G is diameter-edge-invariant. In this paper,
we study radius-edge-invariant and diameter-edge-invariant graphs and obtain characterizations
of radius-edge-invariant graphs and diameter-edge-invariant graphs of diameter two.
c
2003 Elsevier B.V. All rights reserved.
MSC: 05C15
Keywords: Diameter; Radius; Edge deletion
1. Introduction
Let G be a connected graph with vertex set V (G) and edge set E(G). The distance
d(u; v) between vertices u and v is the length of a shortest path joining u and v. The
eccentricity e(v)ofv is the distance to a farthest vertex from v. The radius r(G) and
diameter d(G) are the minimum and maximum eccentricities, respectively. The center
C(G) and periphery P(G) of graph G consist of the sets of vertices of minimum and
maximum eccentricity, respectively. Vertices within C(G) are called central vertices,
E-mail addresses: walikarhb@usa.net (H.B. Walikar), fred buckley@baruch.cuny.edu (F. Buckley),
medhi
huligol@yahoo.com (M.K. Itagi).
0012-365X/03/$ - see front matter
c
2003 Elsevier B.V. All rights reserved.
doi:10.1016/S0012-365X(03)00189-4