$δ$ -双曲图的偏心地形
Eccentricity terrain of $δ$-hyperbolic graphs
摘要：
一个图$G=(V,E)$是$δ$ -双曲的，如果对于任何四个顶点$u,v,w,x$，三个距离和$d(u,v)+d(w,x)$，$d(u,w)+d(v,x)$和$d(u,x)+d(v,w)$中较大的两个最多相差$2δ\geq 0$。最近的工作表明，许多现实世界的图有小双曲$δ$。本文描述了$δ$ -双曲图的偏心地形.偏心函数$e_G(v)=\max\{d(v,u) : u \in V\}$将$G$的顶点集划分为偏心层$C_{k}(G) = \{v \in V : e(v)=rad(G)+k\}$、$k \in \mathbb{N}$，其中$rad(G)=\min\{e_G(v): v\in V\}$是$G$的半径。本文研究了沿最短路径的顶点偏心率层，识别出丘陵、平原、山谷、阶地和高原等地形特征。引入了$β$ -伪凸性的概念，它蕴涵了Gromov的$ε$ -拟凸性，并说明了$δ$ -双曲图中伪凸集的丰富性。特别地，它表明所有集合$C_{\leq k}(G)=\{v\in V : e_G(v) \leq rad(G) + k\}$、$k\in \mathbb{N}$都是$(2δ-1)$ -伪凸的。此外，得到了顶点偏心率的几个界，这些界给出了有效地逼近所有偏心率的几种方法。一个$O(δ|E|)$时间偏心率近似$\hat{e}(v)$，所有$v\in V$，提出了使用两个相互远离的顶点的距离，并满足$e_G(v)-2δ\leq \hat{e}(v) \leq {e_G}(v)$。证明了存在两个偏心率逼近生成树$T$，一个在$O(δ|E|)$时间可构造，另一个在$O(|E|)$时间可构造，它们分别满足${e}_G(v) \leq e_T(v) \leq {e}_G(v)+4δ+1$和${e}_G(v) \leq e_T(v) \leq {e}_G(v)+6δ$.因此，树的偏心地形给出了一个很好的近似（直到$δ$ -双曲线图的偏心地形的附加误差$O(δ))$）。
英文摘要：
A graph $G=(V,E)$ is $δ$-hyperbolic if for any four vertices $u,v,w,x$, the two larger of the three distance sums $d(u,v)+d(w,x)$, $d(u,w)+d(v,x)$, and $d(u,x)+d(v,w)$ differ by at most $2δ\geq 0$. Recent work shows that many real-world graphs have small hyperbolicity $δ$. This paper describes the eccentricity terrain of a $δ$-hyperbolic graph. The eccentricity function $e_G(v)=\max\{d(v,u) : u \in V\}$ partitions the vertex set of $G$ into eccentricity layers $C_{k}(G) = \{v \in V : e(v)=rad(G)+k\}$, $k \in \mathbb{N}$, where $rad(G)=\min\{e_G(v): v\in V\}$ is the radius of $G$. The paper studies the eccentricity layers of vertices along shortest paths, identifying such terrain features as hills, plains, valleys, terraces, and plateaus. It introduces the notion of $β$-pseudoconvexity, which implies Gromov's $ε$-quasiconvexity, and illustrates the abundance of pseudoconvex sets in $δ$-hyperbolic graphs. In particular, it shows that all sets $C_{\leq k}(G)=\{v\in V : e_G(v) \leq rad(G) + k\}$, $k\in \mathbb{N}$, are $(2δ-1)$-pseudoconvex. Additionally, several bounds on the eccentricity of a vertex are obtained which yield a few approaches to efficiently approximating all eccentricities. An $O(δ|E|)$ time eccentricity approximation $\hat{e}(v)$, for all $v\in V$, is presented that uses distances to two mutually distant vertices and satisfies $e_G(v)-2δ\leq \hat{e}(v) \leq {e_G}(v)$. It also shows existence of two eccentricity approximating spanning trees $T$, one constructible in $O(δ|E|)$ time and the other in $O(|E|)$ time, which satisfy ${e}_G(v) \leq e_T(v) \leq {e}_G(v)+4δ+1$ and ${e}_G(v) \leq e_T(v) \leq {e}_G(v)+6δ$, respectively. Thus, the eccentricity terrain of a tree gives a good approximation (up-to an additive error $O(δ))$ of the eccentricity terrain of a $δ$-hyperbolic graph.
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