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：设$u_0u_1 \ldots u_{k-1}u_k$是一条路径，若$u_0u_i \in E(G)$蕴含$u_i = u_1$，且$u_ju_k\in E(G)$蕴含$u_j=u_{k-1}$，则称其为\textit{弱toll路径}。图$G$的顶点子集$S$称为{\it 弱toll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toll路径中的顶点均属于$S$。{\em 弱toll凸性}定义为基于弱toll凸集的图凸性空间。许多研究致力于判定具有凸性空间的图是否为{\em 凸几何}。\emph{极顶点}是指凸集$S$中使得$S\backslash\{x\}$仍为凸集的元素$x$。若图凸性空间满足Minkowski-Krein-Milman性质（即每个凸集均为其极顶点的凸包），则称其为凸几何。已知弦图、托勒密图、弱极化图及区间图可分别由单音凸性、测地凸性、$m^3$凸性和toll凸性刻画为凸几何。其他重要图类亦可由此方式刻画。本文证明：图关于弱toll凸性为凸几何当且仅当它是真区间图。此外，本文还研究了若干经典图不变量在弱toll凸性下的性质。