In the literature, there are quite a few sequential and parallel algorithms to solve problems on decomposable graphs utilizing distinct techniques. 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 graph 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(P d(|V|, |E|) + |V(TG)|)/log|V(TG)|)) processors on Md. We also demonstrate the following examples fitting into our paradigm: (1) The maximum independent set problem on trees, (2) The maximum matching problem on series-parallel graphs, and (3) The efficient domination problem on series-parallel graphs. Our results improve the previously best known results of (1) and (2).
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