We present a new parallel tree contraction scheme which takes O(log n) contraction phases to reduce a tree to its root, and implement this scheme in O(log n log log n) time using O(n/log log n) processors on an arbitrary CRCW PRAM. We then show a data structure to represent a connected distance-hereditary graph G in the form of a rooted tree. Applying our tree contraction scheme on the above data structure together with graph theoretical properties, we solve the problems of finding a minimum connected γ-dominating set and finding a minimum γ-dominating clique on G in O(log n log log n) time using O((n + m)/log log n) processors on an arbitrary CRCW PRAM, where n and m are the number of vertices and edges in G, respectively.
Hsieh, S. Y., Ho, C. W., Hsu, T. S., Ko, M. T., & Chen, G. H. (1998). A new simple parallel tree contraction scheme and its application on distance-hereditary graphs. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 1457 LNCS, pp. 298–309). Springer Verlag. https://doi.org/10.1007/bfb0018548