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 $d(u,v)+d(w,x)$, $d(u,w)+d(v,x)$, $d(u,x)+d(v,w)$ differ by at most $2\delta \ge 0$. 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 vertices of G into eccentricity layers $C_k(G)=\{v\in V:e_G(v)=rad(G)+k\}$, $k\in \mathbb{N}$, 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. It shows that all sets $C_{\le k}(G)=\{v\in V:e_G(v)\le rad(G)+k\}$, $k\in \mathbb{N}$, are $(2\delta -1)$-pseudoconvex. Several bounds on the eccentricity of a vertex are obtained which yield a few approaches to efficiently approximating all eccentricities.

图 $G=（V，Ë）$如果是任意四个顶点，则为δ双曲型$ü，v，w，X$，三个距离总和中的两个较大 $d（ü，v）+d（w，X）$， $d（ü，w）+d（v，X）$， $d（ü，X）+d（v，w）$ 最多相差 $2\delta \ge 0$。本文描述了δ双曲线图的偏心地形。偏心功能${Ë}_G（v）=\mathrm{最高}\{d（v，ü）：ü\in V\}$将G的顶点划分为偏心层$C_{ķ}（G）=\{v\in V：{Ë}_G（v）=\mathrm{[R}\mathrm{一种}d（G）+ķ\}$， $ķ\in \mathbb{ñ}$，在哪里 $\mathrm{[R}\mathrm{一种}d（G）=\mathrm{分}\{{Ë}_G（v）：v\in V\}$是G的半径。本文研究了沿最短路径的顶点的偏心层，确定了丘陵，平原，山谷，阶地和高原等地形特征。它介绍了β-伪凸性的概念，它表示Gromov的ϵ-拟凸性，并说明了δ-双曲图中伪凸集的数量。它显示所有集合$C_{\le ķ}（G）=\{v\in V：{Ë}_G（v）\le \mathrm{[R}\mathrm{一种}d（G）+ķ\}$， $ķ\in \mathbb{ñ}$， 是 $（2\delta -\mathrm{1个}）$-伪凸。获得顶点偏心的几个边界，这产生了一些有效地逼近所有偏心的方法。