Short chains in circle and in square coverings

The problem of finding short paths avoiding obstacles in a packing was extensively studied. Similar question concerning coverings received less attention. Here we consider locally finite coverings of the plane (i) by discs of radius 1, (ii) by squares of area 1. Given two discs (squares resp.) centered at x and \(x'\) one can ask for the least number of discs (squares) needed to connect the given discs (squares resp.). Estimates of type \((C + o(1)) d(x,x')\) as the Euclidean distance \( d(x,x') \rightarrow \infty \) will be given. For discs we prove \(C= \frac{2}{\sqrt{3}} \thickapprox 1.15 \dots \), for squares we prove \(C = 2\sqrt{2}\). The latter constant turns out to be the best possible.