Realizability of Fault Tolerant Graphs∗

A connected graph G is optimal-κ if the connectivity κ(G) = δ(G), where δ(G) is the minimum degree of G. It is super-κ if every minimum vertex cut isolates a vertex. An optimal-κ graph G is moptimal-κ if for any vertex set S ⊆ V (G) with |S| ≤ m, G−S is still optimal-κ. The maximum integer of such m, denoted by Oκ(G), is the vertex fault tolerance of G with respect to the property of optimal-κ. The concept of vertex fault tolerance with respect to the property of super-κ, denoted by Sκ(G), is defined in a similar way. In a previous paper, we have proved that min{κ1(G)−δ(G), δ(G)−1} ≤ Oκ(G) ≤ δ(G)−1 and min{κ1(G)− δ(G)− 1, δ(G)− 1} ≤ Sκ(G) ≤ δ(G)− 1. We also have Sκ(G) ≤ Oκ(G) ≤ δ(G)− 1. In this paper, we study the realizability problems concerning with the above three bounds. By construction, we proved that for any non-negative integers a, b, c with a ≤ b ≤ c, (i) there exists a graph G such that κ1(G)− δ(G) = a, Oκ(G) = b, and δ(G)− 1 = c; (ii) there exists a graph G with κ1(G)− δ(G)− 1 = a, Sκ(G) = b, and δ(G)−1 = c; (iii) there exists a graph G such that Sκ(G) = a, Oκ(G) = b and δ(G)−1 = c.