A walk $u_0u_1 \ldots u_{k-1}u_k$ is a \textit{weakly toll walk} if $u_0u_i \in E(G)$ implies $u_i = u_1$ and $u_ju_k\in E(G)$ implies $u_j=u_{k-1}$. A set $S$ of vertices of $G$ is {\it weakly toll convex} if for any two non-adjacent vertices $x,y \in S$ any vertex in a weakly toll walk between $x$ and $y$ is also in $S$. The {\em weakly toll convexity} is the graph convexity space defined over weakly toll convex sets. Many studies are devoted to determine if a graph equipped with a convexity space is a {\em convex geometry}. An \emph{extreme vertex} is an element $x$ of a convex set $S$ such that the set $S\backslash\{x\}$ is also convex. A graph convexity space is said to be a convex geometry if it satisfies the Minkowski-Krein-Milman property, which states that every convex set is the convex hull of its extreme vertices. It is known that chordal, Ptolemaic, weakly polarizable, and interval graphs can be characterized as convex geometries with respect to the monophonic, geodesic, $m^3$, and toll convexities, respectively. Other important classes of graphs can also be characterized in this way. In this paper, we prove that a graph is a convex geometry with respect to the weakly toll convexity if and only if it is a proper interval graph. Furthermore, some well-known graph invariants are studied with respect to the weakly toll convexity.

翻译：一条路径 $u_0u_1 \ldots u_{k-1}u_k$ 称为**弱容差路径**，若 $u_0u_i \in E(G)$ 蕴含 $u_i = u_1$，且 $u_ju_k\in E(G)$ 蕴含 $u_j=u_{k-1}$。图 $G$ 的顶点子集 $S$ 称为**弱容差凸集**，若对任意两个非邻接顶点 $x,y \in S$，在 $x$ 与 $y$ 之间的任意弱容差路径上的所有顶点也属于 $S$。**弱容差凸性**是基于弱容差凸集定义的图凸空间。许多研究致力于判定配备凸空间的图是否为**凸几何**。**极值顶点**是凸集 $S$ 中的元素 $x$，使得集合 $S\backslash\{x\}$ 仍为凸集。若图凸空间满足 Minkowski-Krein-Milman 性质（即每个凸集都是其极值顶点的凸包），则称该空间为凸几何。已知弦图、托勒密图、弱极化图与区间图可分别通过单音凸性、测地凸性、$m^3$ 凸性及容差凸性刻画为凸几何。其他重要图类也可通过类似方式刻画。本文证明，图关于弱容差凸性为凸几何当且仅当它是真区间图。此外，本文还研究了弱容差凸性下的若干经典图不变量。