Weakly toll convexity and proper interval graphs

TL;DR

This paper introduces a weakly toll convexity on graphs via weakly toll walks and studies the resulting convex geometries. It proves a tight characterization: a graph is a convex geometry under $WT$-convexity if and only if it is a proper interval graph. It further analyzes invariants, showing that for a proper interval graph $G$, $wtn(G)=wth(G)=|Ext(G)|$, while for trees these invariants equal 2. These results connect a novel convexity with a classical graph class and motivate extensions to broader graph families and related combinatorial parameters such as Carathéodory, Radon, and Helly numbers.

Abstract

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.

Figures (6)

• Figure 1: Gimbel's graphs.
• Figure 2: Forbidden configurations for an end vertex in proper interval graphs.
• Figure 3: (a) Interval model of graph $G=B_8$ (see Figure 1); (b) clique tree $G_I$, associated with canonical representation $I=Q_1,Q_2,\ldots,Q_7$; (c) interval model of the graph $H=G[Q_3,Q_5]$; (d) clique tree $H_{J}$, associated with canonical representation $J=I[Q_3,Q_5]$.
• Figure 4: Interval graph with weakly toll number three.
• Figure 5: The graph $G-N[s_1]$ is not connected, but $G-N[s_2]$ is connected. Note that $G_2=G[N[q_2], N[s_2]]=G[\{q_2,d,e,s_2\}]$ and $Q'_2=\{d,e\}$. By (1) in Proposition \ref{['l1']}, $q_2,d,e,s_2$ is a weakly toll walk, and since $N[q_2]\cap Q'_2\subseteq N[s_1]$, it follows that $d \in N[s_1]$ and $s_1,d,q_2,d,e,s_2$ is a weakly toll walk which captures $d$, $e$ and $q_2$. In addition, $G_1=G[N[s_1],N[q_1]]=G[\{s_1,a,b,d,q_1\}]$. Since $G_1-N[q_1]$ is a connected graph, by (1) in Proposition \ref{['l1']}, $s_1,a,b,q_1$ is a weakly toll walk which captures $a$ and $b$.

Theorems & Definitions (30)

• Theorem 2.1
• Theorem 2.2
• Corollary 2.1
• Theorem 2.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof