Self-Stabilizing Algorithms for {k}-Domination

In the self-stabilizing algorithmic paradigm for distributed computing each node has only a local view of the system, yet in a nite amount of time the system converges to a global state, satisfying some desired property. A function f : V (G) ! f0; 1; 2; : : : ; kg is a fkgdominating function if j2N[i]f(j) k for all i 2 V (G). In this paper we present self-stabilizing algorithms for nding a minimal fkg-dominating function in an arbitrary graph. Our rst algorithm covers the general case, where k is arbitrary. This algorithm requires an exponential number of moves, however we believe that its scheme is interesting on its own, because it can insure that when a node moves, its neighbors hold correct values in their variables. For the case that k = 2 we propose a linear time self-stabilizing algorithm.