With the aim of extending the coverage and improving the performance of impulse radio ultra-wideband (UWB) systems, this paper focuses on developing a novel single differential encoded decode and forward (DF) non-cooperative relaying scheme (NCR). To favor simple receiver structures, differential noncoherent detection is employed which enables effective energy capture without any channel estimation. Putting emphasis on the general case of multi-hop relaying, we illustrate an original algorithm for the joint power allocation and path selection (JPAPS), minimizing an approximate expression of the overall bit error rate (BER). In particular, after deriving a closed-form power allocation strategy, the optimal path selection is reduced to a shortest path problem on a connected graph, which can be solved without any topology information with complexity O(N-3), N being the number of available relays of the network. An approximate scheme is also presented, which reduces the complexity to O(N-2) while showing a negligible performance loss, and for benchmarking purposes, an exhaustive-search based multi-hop DF cooperative strategy is derived. Simulation results for various network setups corroborate the effectiveness of the proposed low-complexity JPAPS algorithm, which favorably compares to existing AF and DF relaying methods.