Fan-type theorem for path-connectivity

A graphG is said to bek-path-connected if every pair of distinct vertices inG are joined by a path of length at leastk. We prove that if max{deg _G ^x , deg _G ^y }k for every pair of verticesx,y withd _G (x,y)=2 in a 2-connected graphG, whered _G (x,y) is the distance betweenx andy inG, thenG isk-path-connected.