Majorization and the minimum number of dominating sets

A recent Wagner result, which characterizes graphs without isolates and with a minimum number of dominating sets, is based on an earlier characterization restricted to trees. A self-contained proof of both characterizations has been found. Each graph in question is proved to include a union of disjoint stars as a spanning subgraph which is induced by pendant edges of the graph. Majorization among numerical multisets which represent distribution of vertices among stars helps to see that the solution is equipartite. Continuization yields the optimal size of stars. Additional characterizations limited to graphs with upper-/lower-bounded domination number are obtained. A linear-time algorithm for partitioning a tree into minimal number of stars is presented. Concluding remarks indicate how majorization applied to disjoint unions of subdivided stars leads to a characterization of graphs with nontrivial minimum number of totally dominating sets. Amazingly, these graphs of order different from 38 make up a well-defined class of graphs with a maximum number of efficient dominating sets. (C) 2013 Elsevier B.V. All rights reserved.