Total effective resistance (Kirchhoff index) is a fundamental graph metric with applications in network robustness, consensus algorithms, and graph neural networks. We study Total Effective Resistance Minimization (TERM): Given a graph G and a budget k, find a subgraph H with at most k edges to minimize the sum of pairwise effective resistances. While efficient approximation algorithms exist for spanning trees ( \(k=n-1\) ) and large budgets ( \(k \gg n\) ), the challenging small-budget regime ( \(k-n=o(n)\) ) lacks non-trivial approximations. We bridge this gap with an \(O(\sqrt{n})\) -approximation algorithm for all \(k \ge n-1\) , achieved through a greedy edge deletion framework that combines solutions for extreme budget regimes. Further, we study a natural Single-Source TERM variant (STERM), which minimizes the sum of effective resistances from a source vertex. Surprisingly, while TERM is NP-hard for spanning trees, STERM admits polynomial-time algorithms in this regime. We establish new connections: an \(\alpha \) -approximation for STERM yields a \(2\alpha \) -approximation for TERM. In addition, we make progress toward resolving STERM’s complexity status by proving APX-hardness via a reduction from 3-Dimensional Matching.