We identify a graph without proper cycles, which is comatching with a cycle,The result is then extended to certain general families of graphs with cyclomatic number \(1\), formed by attaching trees to cycles.
Citation
Edward J.Farrell, Andrew A.Hunte. \(D\)-Graphs for Graphs with Cyclomatic Number \(1\)[J], Ars Combinatoria, Volume 129. 63-69. .