Mitre C. Dourado ; Marisa Gutierrez ; Fábio Protti ; Silvia Tondato - Weakly toll convexity and proper interval graphs

dmtcs:9837 - Discrete Mathematics & Theoretical Computer Science, April 18, 2024, vol. 26:2 - https://doi.org/10.46298/dmtcs.9837
Weakly toll convexity and proper interval graphsArticle

Authors: Mitre C. Dourado ; Marisa Gutierrez ; Fábio Protti ; Silvia Tondato

    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.


    Volume: vol. 26:2
    Section: Graph Theory
    Published on: April 18, 2024
    Accepted on: March 27, 2024
    Submitted on: July 26, 2022
    Keywords: Mathematics - Combinatorics,Computer Science - Discrete Mathematics,05C75

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