 
Summary: Decreasing the Diameter of Bounded Degree Graphs
Noga Alon
Andr´as Gy´arf´as
Mikl´os Ruszink´o §
February 22, 2002
To the memory of Paul Erdos
Abstract
Let fd(G) denote the minimum number of edges that have to be added to a graph G to transform
it into a graph of diameter at most d. We prove that for any graph G with maximum degree D
and n > n0(D) vertices, f2(G) = n  D  1 and f3(G) n  O(D3
). For d 4, fd(G) depends
strongly on the actual structure of G, not only on the maximum degree of G. We prove that the
maximum of fd(G) over all connected graphs on n vertices is n/ d/2  O(1). As a byproduct, we
show that for the ncycle Cn, fd(Cn) = n/(2 d/2  1)  O(1) for every d and n, improving earlier
estimates of Chung and Garey in certain ranges.
1 Preliminaries and results
Extremal problems concerning the diameter of graphs have been initiated by Erdos, R´enyi and S´os in [4]
and [5]. Problems concerning the change of diameter if edges are added or deleted have been initiated
by Chung and Garey in [2], followed by a survey of Chung [1] which contains further references, e.g.
the paper by Schoone, Bodlaender and Leeuwen [6]. A related problem, decreasing the diameter of a
