Abstract

. A graph is a homogeneous extension of a tree iff the reduction of all homogeneous sets (sometimes called modules) to single vertices gives a tree. We show that these graphs can be recognized in linear sequential and polylogarithmic parallel time using modular decomposition. As an application of some results on homogeneous sets we present a linear time algorithm computing the vertex sets of the connected components of the complement of an arbitrary graph. Moreover we present efficient parallel algorithms solving the problems r--dominating set, r-- dominating clique and connected r--dominating set (and thus the Steiner tree problem) on trees by reducing these problems to algebraic tree computations. Using these algorithms we can compute minimum r--dominating cliques and minimum connected r--dominating sets in homogeneous extensions of trees in linear sequential and logarithmic parallel time using a linear number of processors. Finally, we give a more involved sequential algorithm solv...

Citations