Expected Linear time Minimum Spanning Tree algorithm
Nothing in particular happens if there is an isolated vertex. The algorithm is correct even if $H$ consists of no edges, or if it consists of all edges. The algorithm proceeds by finding a minimum spanning forest on $H$, and using it to prune the contracted graph. IF $H$ consists of no edges, then nothing is pruned, but the algorithm remains correct - this only affects the running time.

Note also that the algorithm is for the more general minimum spanning forest problem. Hence there is absolutely no problem if the graph is not connected. The algorithm will find a minimum spanning tree in each connected component.