  Given a graph $G$ rooted at a vertex $r$ and weight functions, $\gamma ,
\tau: E(G) \to \mathbb{R}$, the generalized cable-trench problem (CTP) is to
find a single spanning tree that simultaneously minimizes the sum of the total
edge cost with respect to $\tau$ and the single-source shortest paths cost with
respect to $\gamma$. Although this problem is provably $NP$-complete in the
general case, we examine certain tractable instances involving various graph
constructions of trees and cycles, along with quantities associated to edges
and vertices that arise out of these constructions. We show that given a path
decomposition oracle, for graphs in which all cycles are edge disjoint, there
exists a fast method to determine the cable-trench tree. Further, we examine
properties of graphs which contribute to the general intractability of the CTP
and present some open questions in this direction.
