Domination problems on trees and their homogeneous extensions

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 solving the r–dominating set problem on these graphs.