Near-Linear Query Complexity for Graph Inference

Given a mental picture Ĝ = (V, Ê) of some unknown graph G = (V,E), how can we verify that our mental image is correct, when the unknown graph is only accessible by querying pairs of vertices and getting their shortest path distance in G? For graphs of bounded degree, we give an algorithm that uses a number of queries that is at most O(log n) times the minimum number achievable. For graphs that additionally have bounded treewidth, we give an algorithm that uses Õ(n) queries. When there is no a priori mental image and the goal is to reconstruct an unknown graph with few distance queries, we give an Õ(n) randomized algorithm for chordal graphs of bounded degree.