In this paper, we sketch characteristics of those problems which can be systematically solved on decomposable graphs. Trees, series-parallel graphs, outerplanar graphs, and bandwidth-k graphs all belong to decomposable graphs. Let Td(|V|,|E|) and Pd(|V|,|E|) denote the time complexity and processor complexity required to construct a parse tree representation TG for a decomposable G = (V, E) on a PRAM model Md. We define a general problem-solving paradigm to solve a wide class of subgraph optimization problems on decomposable graphs in O(Td(|V|,|E|) + log|V(TG)|) time using O(Pd(|V|,|E|) + |V(T G)|/log|V(TG)|) processors on Md. By using our paradigm, we show the following parallel complexities: (a) The maximum independent set problem on trees can be solved in O(log|V|) time using O(|V|/log|V|) processors on an EREW PRAM, (b) The maximum matching problem on series-parallel, graphs can be solved in O(log|E|) time using O(|E|/log|E|) processors on an EREW PRAM, (c) The efficient domination problem on series-parallel graphs can be solved in O(log|E|) time using O(|E|/log|E|) processors on an EREW PRAM.