Efficiently parallelizable problems on a class of decomposable graphs

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 T_d(|V|, |E|) and P_d(|V|, |E|) denote the time complexity and processor complexity required to construct a parse tree representation T_G for a decomposable graph G = (V, E) on a PRAM model M_d. We define a general problem-solving paradigm to solve a wide class of subgraph optimization problems on decomposable graphs in O(T_d(|V|, |E|) + log |V(T_G)|) time using O(P _d(|V|, |E|) + |V(T_G)|)/log|V(T_G)|)) processors on M_d. 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).