... subset of vertices C ⊆ V ,and an integer r ≥ 1; for any vertex v ∈ V , let Br(v) denote the ball of radius rcentered at v, i.e., the set of all vertices linked to v by a path of at most r edges.If ... si+2trivially r-cover the same verticesbecause r ≥ 3. Other constructions can be thought of. We give, without proof, one example which givessmaller graphs than the ones in the proof of Theorem ... by theconcatenation of the part of P∗going from v1to vkand the part of Pr+1going from vkto x. The length of P is greater than or equal to q + r + 1: there are q edges from v1tovq+1and...