Parameterized Complexity of Odd Domination and its Generalization

In the $\text{textsc}OddDomination$ problem, given a graph $G$ and a positive integer $k$, the task is to determine whether there exists a vertex subset $D$ of $G$ such that the closed neighborhood of each vertex in $G$ contains an odd number of vertices from $D$. In this paper, we investigate the computational complexity of the problem. When parameterized by the solution size $k$, we establish W[1]-hardness on some restricted graphs and a sharp boundary between fixed-parameter tractability and W[1]-hardness with respect to the girth of the input graph. Then, we address the problem when parameterized by several structural graph parameters. Furthermore, we investigate the parameterized complexity of $\text{textsc}ParityDomination$, which is a generalization of $\text{textsc}OddDomination$.