从图的边界距离矩阵重构图
Reconstructing a graph from the distance matrix of its boundary
AI总结：
本文定义了图的边界距离矩阵，证明了块图和单圈图可由其边界距离矩阵唯一确定，并提出了重构算法，同时猜想该唯一确定性适用于所有连通图。
摘要：
对于连通图$G$的一个顶点$v$，若存在$G$的另一顶点$u$，使得$v$的任何邻居到$u$的距离都不超过$v$到$u$的距离，则称$v$为$G$的边界顶点。$G$的边界$\partial(G)$是其所有边界顶点的集合。图$G=([n],E)$的边界距离矩阵$\hat{D}_G$是阶数为$\kappa$的方阵，其中$\kappa$为$\partial(G)$的阶数，满足对于任意$i,j\in \partial(G)$，$[\hat{D}_G]_{ij}=d_G(i,j)$。给定阶数为$\kappa$的方阵$\hat{B}$，我们证明了$\hat{B}$成为树$T$的叶子集合（恰好是其边界）距离矩阵$\hat{D}_T$的条件。我们证明，若$G$是块图或单圈图，则$G$由其边界距离矩阵$\hat{D}_{G}$唯一确定；并猜想该结论对任意连通图$G$均成立，前提是$G$的阶数$n$和边界（即边界距离矩阵）均已预先指定。此外，我们给出了从边界距离矩阵重构1-块图（或单圈图）的算法，其最坏时间复杂度为$O(\kappa n)$（或$O(n^2)$）。
英文摘要：
A vertex $v$ of a connected graph $G$ is said to be a boundary vertex of $G$ if for some other vertex $u$ of $G$, no neighbor of $v$ is further away from $u$ than $v$. The boundary $\partial(G)$ of $G$ is the set of all of its boundary vertices. The boundary distance matrix $\hat{D}_G$ of a graph $G=([n],E)$ is the square matrix of order $κ$, being $κ$ the order of $\partial(G)$, such that for every $i,j\in \partial(G)$, $[\hat{D}_G]_{ij}=d_G(i,j)$. Given a square matrix $\hat{B}$ of order $κ$, we prove under which conditions $\hat{B}$ is the distance matrix $\hat{D}_T$ of the set of leaves of a tree $T$, which is precisely its boundary. We show that if $G$ is either a block graph or a unicyclic graph, then $G$ is uniquely determined by the boundary distance matrix $\hat{D}_{G}$ of $G$ and we also conjecture that this statement holds for every connected graph $G$, whenever both the order $n$ and the boundary (and thus also the boundary distance matrix) of $G$ are prefixed. Moreover, an algorithm for reconstructing a 1-block graph (resp., a unicyclic graph) from its boundary distance matrix is given, whose time complexity in the worst case is $O(κn)$ (resp., $O(n^2)$).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
