We study low-density parity-check (LDPC) codes with iterative decoding on block fading (BF) Relay Channels. We use coded cooperation, a variant of decode-and-forward with a smaller outage probability than the latter. We derive a new code structure by extending the previously published full-diversity root-LDPC code, designed for the BF point-to-point channel, to exhibit a rate-compatibility property which is necessary for coded cooperation. Density evolution equations are determined in order to plot the word error rate performance for infinite length codes. We show that our code construction exhibits near-outage limit performance for all block lengths and for a range of coding rates up to 0.5, which is the highest possible coding rate that achieves full-diversity on the relay channel.