Infinite Order Domination in Graphs

The previously studied notions of smart and foolproof finite order domination of a simple graph \( G = (V, E) \) are generalised in the sense that safe configurations in \( G \) are not merely sought after \( k \geq 1 \) moves, but in the limiting cases where \( k \to \infty \). Some general properties of these generalised domination parameters are established, after which the parameter values are found for certain simple graph structures (such as paths, cycles, multipartite graphs, and products of complete graphs, cycles, and paths).