Simulated annealing construction of shortest (spanning/nonspanning
and closed/open) paths on generál connected graphs is discussed. A brief graphtheoretical analysis of the problem is given. A theorem has been proved that for connected graphs the shortest paths are semielementary, that is each edge on the path is visited at most twice in opposite directions. This observation considerably reduces the search space. Tasks may be further specified depending on whether the initial and terminal vertices are given or not. Similarly, in construction of shortest open paths a subtask is considered when the path must visit a prescribed subset of graph vertices. Illustrative calculations demonstrate that the proposed method results for incomplete graphs in the paths that are dosely related to optimal solutions.