Characterization of efficiently parallel solvable problems on distance-hereditary graphs

In this paper, we sketch common properties of a class of so-called subgraph optimization problems that can be systematically solved on distance-hereditary graphs. Based on the found properties, we then develop a general problem-solving paradigm that solves these problems efficiently in parallel. As a by-product, we also obtain new linear-time algorithms by a sequential simulation of our parallel algorithms. Let T_d(|V|, |E|) and P_d(|V|, |E|) denote the time and processor complexities, respectively, required to construct a decomposition tree of a distance-hereditary graph G = (V, E) on a PRAM model M_d. Based on the proposed paradigm, we show that the maximum independent set problem, the maximum clique problem, the vertex connectivity problem, the domination problem, and the independent domination problem can be sequentially solved in O(|V| + |E|) time, and solved in parallel in O(T_d(|V|, |E|) + log |V|) time using O(P_d(|V|, |E|) + |V|/log |V|) processors on M_d. By constructing a decomposition tree under a CREW PRAM, we also show that T_d(|V|, |E|) = O(log² |V|) and P_d(|V|, |E|) = O(|V| + |E|).