Decreasing the diameter of bounded degree graphs

  • Authors:

  • Noga Alon;András Gyárfás;Miklós Ruszinkó

  • Affiliations:

  • Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv, Israel;Computer and Automation Research Institute of the Hungarian Academy of Sciences, Budapest, P.O. Box 63, Hungary-1518;Computer and Automation Research Institute of the Hungarian Academy of Sciences, Budapest, P.O. Box 63, Hungary-1518

  • Venue:
  • Journal of Graph Theory
  • Year:
  • 2000

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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 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. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 161–172, 2000 1991 Mathematics Subject classification: 05C12.