A hidden graph is a graph whose edge set is hidden. A distance oracle of a graph G is a black-box that receives two vertices of G and outputs the distance between the two vertices. Given a hidden graph, the reconstruction problem aims to identify the edges of the hidden graph by accessing a distance oracle, and the verification problem aims to check whether the hidden graph is equal to another given graph (not hidden).

If the hidden graph G is a connected chordal graph, a Las Vegas reconstruction algorithm using $O({\mathrm{\Delta }}^3{2}^{\mathrm{\Delta }}\cdot n(2^{\mathrm{\Delta }}+{\log }^{2}n)\log n)$ distance queries is known, where Δ is the maximum degree of G and n is the number of vertices of G. Improving upon this result, we present a reconstruction algorithm using only $O({\mathrm{\Delta }}^{2}n{\log }^{2}n)$ distance queries. As a byproduct, we obtain a deterministic algorithm for the verification of chordal graphs with $O({\mathrm{\Delta }}^{2}n\log n)$ distance queries. Additionally, we derive a deterministic algorithm of reconstructing connected interval graphs using only $O(\mathrm{\Delta }n)$ distance queries, and prove that reconstructing or verifying a connected interval graph needs $\mathrm{\Omega }(\mathrm{\Delta }n)$ distance queries, which implies that this algorithm is the best possible in terms of the number of distance queries needed.

隐藏图是其边集被隐藏的图。图G的距离预言是一个黑盒，它接收G的两个顶点并输出两个顶点之间的距离。给定一个隐藏图，重建问题旨在通过访问距离预言来识别隐藏图的边缘，而验证问题旨在检查该隐藏图是否等于另一个给定图（未隐藏）。

如果隐藏图G是连通弦图，则使用$Ø（{\mathrm{\Delta }}^3{2}^{\mathrm{\Delta }}\cdot ñ（2^{\mathrm{\Delta }}+{\mathrm{日志}}^2ñ）\mathrm{日志}ñ）$距离查询是已知的，其中Δ是最大程度ģ和Ñ是顶点的数量ģ。为了改善这个结果，我们提出了一种仅使用$Ø（{\mathrm{\Delta }}^2ñ{\mathrm{日志}}^2ñ）$距离查询。作为副产品，我们获得了确定性算法，用于验证弦图$Ø（{\mathrm{\Delta }}^2ñ\mathrm{日志}ñ）$距离查询。此外，我们推导了一种确定性算法，该算法仅使用$Ø（\mathrm{\Delta }ñ）$ 距离查询，并证明需要重建或验证连接的间隔图 $\mathrm{\Omega }（\mathrm{\Delta }ñ）$ 距离查询，这意味着就所需的距离查询而言，该算法是最好的。