Randomized algorithms and upper bounds for multiple domination in graphs and networks

We consider four different types of multiple domination and provide new improved upper bounds for the kk- and kk-tuple domination numbers. They generalize two classical bounds for the domination number and are better than a number of known upper bounds for these two multiple domination parameters. Also, we explicitly present and systematize randomized algorithms for finding multiple dominating sets, whose expected orders satisfy new and recent upper bounds. The algorithms for kk- and kk-tuple dominating sets are of linear time in terms of the number of edges of the input graph, and they can be implemented as local distributed algorithms. Note that the corresponding multiple domination problems are known to be NP-complete.

► Four different types of multiple domination in graphs/networks are considered. ► New improved upper bounds for the kk- and kk-tuple domination numbers are provided. ► Efficient randomized algorithms for finding small-size multiple dominating sets are devised. ► The algorithms can be implemented in parallel or as local distributed algorithms.