  We introduce the cycle intersection graph of a graph, an adaptation of the
cycle graph of a graph, and use the structure of these graphs to prove an upper
bound for the decycling number of all even graphs. This bound is shown to be
significantly better when an even graph admits a cycle decomposition in which
any two cycles intersect in at most one vertex. Links between the cycle rank of
the cycle intersection graph of an even graph and the decycling number of the
even graph itself are found. The problem of choosing an ideal cycle
decomposition is addressed and is presented as an optimization problem over the
space of cycle decompositions of even graphs.
