Reconstructing Bounded Treelength Graphs with Linearithmic Shortest Path Distance Queries

Abstract

We consider the following graph reconstruction problem: given an unweighted connected graph $G=(V,E)$ with visible vertex set $V$ and an oracle which takes two vertices $u,v\in V$ and returns the shortest path distance between $u$ and $v$, how many queries are needed to reconstruct $E$? Specifically, we consider bounded degree $\hat{I}”$ and bounded treelength $\mathrm{tl}$ connected graphs and show that reconstruction can be done in $O_{\hat{I}”,\mathrm{tl}}(n\log n)$ queries with a deterministic algorithm. This result improves over the best known algorithm (deterministic or randomized) for this graph class by a $\log n$ factor and matches the known lower bound for the class of graphs with bounded chordality, which is a subclass of bounded treelength graphs.

9 pages, 2 figures