Odd Domination in Direct and Strong Products of Graphs

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.