Learning causal relationships between variables is a fundamental task in causal inference and directed acyclic graphs (DAGs) are a popular choice to represent the causal relationships. As one can recover a causal graph only up to its Markov equivalence class from observations, interventions are often used for the recovery task. Interventions are costly in general and it is important to design algorithms that minimize the number of interventions performed. In this work, we study the problem of identifying the smallest set of interventions required to learn the causal relationships between a subset of edges (target edges). Under the assumptions of faithfulness, causal sufficiency, and ideal interventions, we study this problem in two settings: when the underlying ground truth causal graph is known (subset verification) and when it is unknown (subset search). For the subset verification problem, we provide an efficient algorithm to compute a minimum sized interventional set; we further extend these results to bounded size non-atomic interventions and node-dependent interventional costs. For the subset search problem, in the worst case, we show that no algorithm (even with adaptivity or randomization) can achieve an approximation ratio that is asymptotically better than the vertex cover of the target edges when compared with the subset verification number. This result is surprising as there exists a logarithmic approximation algorithm for the search problem when we wish to recover the whole causal graph. To obtain our results, we prove several interesting structural properties of interventional causal graphs that we believe have applications beyond the subset verification/search problems studied here.