  We study the problem of determining whether a given graph~$G=(V,E)$ admits a
matching~$M$ whose removal destroys all odd cycles of~$G$ (or equivalently
whether~$G-M$ is bipartite). This problem is equivalent to determine
whether~$G$ admits a~$(2,1)$-coloring, which is a~$2$-coloring of~$V(G)$ such
that each color class induces a graph of maximum degree at most~$1$. We
determine a dichotomy related to the~{\sf NP}-completeness of this problem,
where we show that it is~{\sf NP}-complete even for $3$-colorable planar graphs
of maximum degree~$4$, while it is known that the problem can be solved in
polynomial time for graphs of maximum degree at most~$3$. In addition we
present polynomial-time algorithms for some graph classes, including graphs in
which every odd cycle is a triangle, graphs of small dominating sets,
and~$P_5$-free graphs. Additionally, we show that the problem is fixed
parameter tractable when parameterized by the clique-width, which implies
polynomial-time solution for many interesting graph classes, such as
distance-hereditary, outerplanar, and chordal graphs. Finally,
an~$O\left(2^{O\left(vc(G)\right)} \cdot n\right)$-time algorithm and a kernel
of at most~$2\cdot nd(G)$ vertices are presented, where~$vc(G)$ and~$nd(G)$ are
the vertex cover number and the neighborhood diversity of~$G$, respectively.
