Wide-diameter and minimum length of fan

AbstractLet 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 l such that there exist k vertex-disjoint (x,y)-paths whose lengths are at most l. The k-wide-diameter dk(G) of G is the maximum value of the k-wide-distance between two distinct vertices of G. For x0∈V(G) and k distinct vertices x1,x2,…,xk∈V(G)−{x0}, we define fk(x0,{x1,x2,…,xk}) to be the minimum l such that there exist k vertex-disjoint paths P1,P2,…,Pk, where Pi is an (x0,xi)-path of length at most l. We define fk(G) to be the maximum value of fk(x0,{x1,x2,…,xk}) over every x0∈V(G) and k distinct vertices x1,x2,…,xk∈V(G)−{x0}. We study relationships between dk(G) and fk(G). Among other results, we show that if G is a k-connected graph, k⩾2, then dk(G)−1⩽fk(G)⩽max{dk(G),(k−1)dk(G)−4k+7}