Combinatorial properties of a general domination problem with parity constraints

We consider various properties of a general parity domination problem: given a graph G, one is looking for a subset S of the vertex set such that the open/closed neighborhood of each vertex contains an even/odd number of vertices in S (it is prescribed individually for each vertex which of these applies). We define the parameter s(G) to be the number of solvable instances out of 4 possibilities and study the properties of this parameter. Upper and lower bounds for general graphs and trees are given as well as a remarkable recurrence formula for rooted trees and—more generally—an algorithm for graphs of bounded tree-width. Furthermore, we give explicit formulas in several special cases and investigate random graphs.