A weighted graph $φG$ encodes a finite metric space $D_{φG}$. When is $D$ totally decomposable? When does it embed in $\ell_1$ space? When does its representing matrix have $\leq 1$ positive eigenvalue? We give useful lemmata and prove that these questions can be answered without examining $φ$ if and only if $G$ has no $K_{2,3}$ minor. We also prove results toward the following conjecture. $D_{φG}$ has $\leq n$ positive eigenvalues for all $φ$, if and only if $G$ has no $K_{2,3,...,3}$ minor, with $n$ threes.