Minimum Power Dominating Sets of Random Cubic Graphs

We present two heuristics for finding a small power dominating set of cubic graphs. We analyze the performance of these heuristics on random cubic graphs using differential equations. In this way, we prove that the proportion of vertices in a minimum power dominating set of a random cubic graph is asymptotically almost surely at most 0.067801. We also provide a corresponding lower bound of 1/29.7 ≈ 0.03367 using known results on bisection width.