A divide-and-conquer approach for reconstruction of {C≥5}-free graphs via betweenness queries

Abstract

We study the query complexity of reconstructing {C≥5}-free graphs with respect to the betweenness oracle. In particular, we show that hidden {C≥5}-free graphs can be reconstructed by using O(Δ14⋅log2⁡n+Δ9⋅nlog2⁡n) betweenness queries in expectation, where Δ denotes the maximum degree of the given graph and n denotes the number of vertices. In addition, we propose two improved randomized algorithms for two subclasses of {C≥5}-free graphs, namely the distance-hereditary graphs and the chordal graphs. For the former class, our algorithm uses O(Δ10⋅log2⁡n+Δ5⋅nlog2⁡n) betweenness queries in expectation, and for the latter class, our algorithm uses O(Δ2⋅nlog2⁡n) betweenness queries in expectation.