Consider the following generalization of the classic binary search problem: A searcher is required to find a hidden target vertex x in a graph G. To do so, they iteratively perform queries to an oracle, each about a chosen vertex v. After each such call, the oracle responds whether the target was found and if not, the searcher receives as a reply the connected component in \(G-v\) which contains x. Additionally, each vertex v may have a different query cost c(v) and a different weight w(v). The goal is to find the optimal querying strategy which minimizes the weighted average-case cost required to find x. The problem is NP-hard even for uniform weights and query costs. Inspired by the progress on the edge query variant of the problem [SODA ’17], we establish a connection between searching and vertex separation. By doing so, we provide an \(O(\sqrt{\log n})\) -approximation algorithm for general graphs and a \((4+\epsilon )\) -approximation algorithm for the case when the input is a tree.