  Problems of the following kind have been the focus of much recent research in
the realm of parameterized complexity: Given an input graph (digraph) on $n$
vertices and a positive integer parameter $k$, find if there exist $k$ edges
(arcs) whose deletion results in a graph that satisfies some specified parity
constraints. In particular, when the objective is to obtain a connected graph
in which all the vertices have even degrees---where the resulting graph is
\emph{Eulerian}---the problem is called Undirected Eulerian Edge Deletion. The
corresponding problem in digraphs where the resulting graph should be strongly
connected and every vertex should have the same in-degree as its out-degree is
called Directed Eulerian Edge Deletion. Cygan et al. [\emph{Algorithmica,
2014}] showed that these problems are fixed parameter tractable (FPT), and gave
algorithms with the running time $2^{O(k \log k)}n^{O(1)}$. They also asked, as
an open problem, whether there exist FPT algorithms which solve these problems
in time $2^{O(k)}n^{O(1)}$. In this paper we answer their question in the
affirmative: using the technique of computing \emph{representative families of
co-graphic matroids} we design algorithms which solve these problems in time
$2^{O(k)}n^{O(1)}$. The crucial insight we bring to these problems is to view
the solution as an independent set of a co-graphic matroid. We believe that
this view-point/approach will be useful in other problems where one of the
constraints that need to be satisfied is that of connectivity.
