  In this paper we study a maximization version of the classical Edge
Dominating Set (EDS) problem, namely, the Upper EDS problem, in the realm of
Parameterized Complexity. In this problem, given an undirected graph $G$, a
positive integer $k$, the question is to check whether $G$ has a minimal edge
dominating set of size at least $k$. We obtain the following results for Upper
EDS. We prove that Upper EDS admits a kernel with at most $4k^2-2$ vertices. We
also design a fixed-parameter tractable (FPT) algorithm for Upper EDS running
in time $2^{\mathcal{O}(k)} \cdot n^{\mathcal{O}(1)}$.
