Bounds on the fault-diameter of graphs

Let G be a ( k + 1 ) ‐connected or ( k + 1 ) ‐edge‐connected graph, where k ∈ ℕ . The k‐fault‐diameter and k‐edge‐fault‐diameter of G is the largest diameter of the subgraphs obtained from G by removing up to k vertices and edges, respectively. In this paper we give upper bounds on the k‐fault‐diameter and k‐edge‐fault‐diameter of graphs in terms of order. We show that the k‐fault‐diameter of a ( k + 1 ) ‐connected graph G of order n is bounded from above by n − k + 1 , and by approximately 4 k + 2 n if G is also triangle‐free. If G does not contain 4‐cycles then this bound can be improved further to approximately 5 n ( k − 1 ) 2 . We further show that the k‐edge‐fault‐diameter of a ( k + 1 ) ‐edge‐connected graph of order n is bounded by n – 1 if k = 1, by ⌊ 2 n − 1 3 ⌋ if k = 2, and by approximately 3 k + 2 n if k ≥ 3 , and give improved bounds for triangle‐free graphs. Some of the latter bounds strengthen, in some sense, bounds by Erdös, Pach, Pollack, and Tuza (J Combin Theory B 47 (1989) 73–79) on the diameter. All bounds presented are sharp or at least close to being optimal.