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\textit{弱可通行路径}，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{\it 弱可通行凸集}，若对于任意两个非相邻顶点$x,y \in S$，在$x$与$y$之间的任何弱可通行路径中的所有顶点也属于$S$。{\em 弱可通行凸性}是基于弱可通行凸集定义的图凸性空间。众多研究致力于确定配备凸性空间的图是否为{\em 凸几何}。凸集$S$中的元素$x$称为\emph{极端顶点}，若集合$S\backslash\{x\}$仍是凸集。图凸性空间被称为凸几何，若其满足Minkowski-Krein-Milman性质，即每个凸集都是其极端顶点的凸包。已知弦图、Ptolemaic图、弱极化图以及区间图可分别通过单音凸性、测地凸性、$m^3$凸性和可通行凸性刻画为凸几何。其他重要的图类也可通过此方式刻画。本文证明：图是弱可通行凸性下的凸几何当且仅当它是适当区间图。此外，本文还研究了某些经典图不变量在弱可通行凸性下的性质。