  The weighted $\mathcal{T}$-free $2$-matching problem is the following
problem: given an undirected graph $G$, a weight function on its edge set, and
a set $\mathcal{T}$ of triangles in $G$, find a maximum weight $2$-matching
containing no triangle in $\mathcal{T}$. When $\mathcal{T}$ is the set of all
triangles in $G$, this problem is known as the weighted triangle-free
$2$-matching problem, which is a long-standing open problem. A main
contribution of this paper is to give a first polynomial-time algorithm for the
weighted $\mathcal{T}$-free $2$-matching problem under the assumption that
$\mathcal{T}$ is a set of edge-disjoint triangles. In our algorithm, a key
ingredient is to give an extended formulation representing the solution set,
that is, we introduce new variables and represent the convex hull of the
feasible solutions as a projection of another polytope in a higher dimensional
space. Although our extended formulation has exponentially many inequalities,
we show that the separation problem can be solved in polynomial time, which
leads to a polynomial-time algorithm for the weighted $\mathcal{T}$-free
$2$-matching problem.
