Odd Domination in Direct and Strong Products of Graphs

摘要

An odd open dominating set of a graph is a subset of the graph's vertices with the property that the open neighborhood of each vertex in the graph contains an odd number of vertices in the subset. An odd closed r-dominating set is a subset of the graph's vertices with the property that the closed r-ball centered at each vertex in the graph contains an odd number of vertices in the subset. We show that the n-fold direct product of simple graphs has an odd open dominating set if and only if each factor has an odd open dominating set. Secondly, we show that the n-fold strong product of simple graphs has an odd closed r-dominating set if and only if each factor has an odd closed r-dominating set.