In the total matching problem, one is given a graph $G$ with weights on the vertices and edges. The goal is to find a maximum weight set of vertices and edges that is the non-incident union of a stable set and a matching. We consider the natural formulation of the problem as an integer program (IP), with variables corresponding to vertices and edges. Let $M = M(G)$ denote the constraint matrix of this IP. We define $\Delta(G)$ as the maximum absolute value of the determinant of a square submatrix of $M$. We show that the total matching problem can be solved in strongly polynomial time provided $\Delta(G) \leq \Delta$ for some constant $\Delta \in \mathbb{Z}_{\ge 1}$. We also show that the problem of computing $\Delta(G)$ admits an FPT algorithm. We also establish further results on $\Delta(G)$ when $G$ is a forest.

翻译：在总匹配问题中，给定一个图 $G$，其顶点和边上附有权重。目标是找到一个最大权重的顶点与边集合，该集合是一个稳定集与一个匹配的非交并。我们将该问题自然地表述为一个整数规划（IP），其变量对应于顶点和边。设 $M = M(G)$ 表示该 IP 的约束矩阵。定义 $\Delta(G)$ 为 $M$ 的方形子矩阵行列式的最大绝对值。我们证明，若对某个常数 $\Delta \in \mathbb{Z}_{\ge 1}$ 有 $\Delta(G) \leq \Delta$，则总匹配问题可在强多项式时间内求解。此外，我们还表明计算 $\Delta(G)$ 的问题具有 FPT 算法。当 $G$ 为森林时，我们进一步建立了关于 $\Delta(G)$ 的其他结果。