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：一个行走$u_0u_1 \ldots u_{k-1}u_k$称为\textit{弱拟费行走}，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 弱拟费凸集}，如果对任意两个非相邻顶点$x,y \in S$，在$x$与$y$之间的任意弱拟费行走中的所有顶点也属于$S$。{\em 弱拟费凸性}是基于弱拟费凸集定义的图凸性空间。大量研究致力于判定配备凸性空间的图是否为{\em 凸几何}。凸集$S$的一个元素$x$称为\textit{极值顶点}，若集合$S\backslash\{x\}$也凸。图凸性空间被称为凸几何，当其满足Minkowski-Krein-Milman性质（即每个凸集是其极值顶点的凸包）。已知弦图、托勒密图、弱极化图及区间图可分别通过单音凸性、测地凸性、$m^3$凸性及拟费凸性刻画为凸几何。其他重要图类也可通过类似方式刻画。本文证明，图关于弱拟费凸性为凸几何当且仅当该图是恰当区间图。此外，我们还研究了若干经典图不变量在弱拟费凸性下的性质。