Weakly toll convexity and proper interval graphs
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 $x$...
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| Published in: | Discrete Mathematics and Theoretical Computer Science Vol. 26:2; no. Graph Theory; pp. 1 - 16 |
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| Main Authors: | , , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Nancy
DMTCS
01.08.2024
Discrete Mathematics & Theoretical Computer Science |
| Subjects: | |
| ISSN: | 1365-8050, 1462-7264, 1365-8050 |
| Online Access: | Get full text |
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| Summary: | 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. |
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| Bibliography: | SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14 |
| ISSN: | 1365-8050 1462-7264 1365-8050 |
| DOI: | 10.46298/dmtcs.9837 |