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 n-cycle 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.
|Number of pages||12|
|Journal||Journal of Graph Theory|
|Publication status||Published - Nov 2000|
- Diameter of graphs
- Maximum degree
ASJC Scopus subject areas
- Geometry and Topology