Wide-diameter and minimum length of fan

Let k be a positive integer and let G be a graph with V(G)| ≥k +1. For two distinct vertices x,y ∈ V(G), the k-wide-distance between x and y is the minimum I such that there exist k vertex-disjoint (x,y)-paths whose lengths are at most l. The k-wide-diameter d k (G) of G is the maximum value of the k-wide-distance between two distinct vertices of G. For x 0 ∈ V(G) and k distinct vertices x 1 ,x 2 ,…,x k ∈ V(G) - {x 0 }, we define f k (x 0 , {x 1 ,x 2 ,…,x k }) to be the minimum l such that there exist k vertex-disjoint paths P 1 ,P 2 ,…,P k , where P i is an (x 0 ,x i )-path of length at most l. We define f k (G) to be the maximum value of f k (x 0 , {x 1 ,x 2 ,…,x k }) over every x 0 ∈ V(G) and k distinct vertices x 1 ,x 2 ,…,x k ∈ V(G) - {x 0 }. We study relationships between d k (G) and f k (G). Among other results, we show that if G is a k-connected graph, k ≥ 2, then d k (G) - 1 ≤ f k (G) ≤ max {d k (G),(k - 1)d k (G) - 4k + 7}.