  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.
