# In a graph, is it always possible to construct a set of cycle bases, with each and every edge Is shared by at most 2 cycle bases?

Given a graph with a list of edges, is it possible to always construct a set of cycle bases for those edges, such that each and every edge is shared by at most 2 cycle bases?

The above question assumes that each and every edge must somehow belong to at least one cycle. IN other words, there is no vertex that is connected to one and only one edge.