Weighted Triangle-free 2-matching Problem with Edge-disjoint Forbidden Triangles

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 set of triangles in find maximum containing no triangle . When all this known as triangle-free problem, which long-standing open problem. A main contribution paper to give first polynomial-time algorithm for under assumption that edge-disjoint triangles. In our algorithm, key ingredient extended formulation representing solution is, we introduce new variables represent convex hull feasible solutions projection another polytope higher dimensional space. Although has exponentially many inequalities, show separation can be solved polynomial time, leads