Odd and Even Dominating Sets with Open Neighborhoods

A subset D of the vertex set V of a graph is called an open oddd dominating set if each vertex in V is adjacent to an odd number of vertices in D (adjacency is irreflexive). In this paper we solve the existence and enumeration problems for odd open dominating sets (and analogously defined even open dominating sets) in the m × n grid graph and prove some structural results for those that do exist. We use a combination of combinatorial and linear algebraic methods, with particular reliance on the sequence of Fibonacci polynomials