This paper introduces a new class of games called symmetric complementing games. These games are interesting since their related complexity classes include many well-known graph problems: Finding minimum spanning forests; k-connectivity and k-blocks; and recognition of chordal graphs, comparability graphs, interval graphs, split graphs, permutation graphs, and constant valence planar graphs. For these problems probabilistic sequential algorithms requiring simultaneously logarithmic space and polynomial time are given. Furthermore, probabilistic parallelism algorithms requiring simultaneously logarithmic time and a polynomial number of processors are also given.