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：一条行走 $u_0u_1 \ldots u_{k-1}u_k$ 称为\textit{弱环行走}，如果 $u_0u_i \in E(G)$ 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{\it 弱环凸集}，如果对于任意两个非相邻顶点 $x,y \in S$，$x$ 与 $y$ 之间的任意弱环行走中的所有顶点也属于 $S$。{\em 弱环凸性}是基于弱环凸集定义的图凸性空间。许多研究致力于判定配备凸性空间的图是否为{\em 凸几何}。凸集 $S$ 中的一个元素 $x$ 称为\emph{极值点}，如果集合 $S\backslash\{x\}$ 仍是凸集。图凸性空间被称为凸几何，当它满足 Minkowski–Krein–Milman 性质，即每个凸集都是其极值点的凸包。已知弦图、Ptolemaic 图、弱极化图及区间图可分别通过单声凸性、测地凸性、$m^3$ 凸性与环凸性刻画为凸几何。其他重要的图类也可通过类似方式刻画。本文证明，图关于弱环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环凸性下的性质。