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Home > Journals > Active Journals > TAG > Vol. 11 > Iss. 1 (2024)

 

Abstract

In this paper we introduce the notion of $t_3$ \textit{convexity}, a natural restriction of triangle convexity. A \textit{triangle path} is a path allowing just short chords. A triangle path $P$ between two non-adjacent vertices in a graph $G$ is called $t_3$ \textit{path} if the first vertex of $P$ is among vertices from $P$ adjacent only to the second vertex of $P$, and the last vertex of $P$ is among vertices from $P$ adjacent only to the second-last vertex of $P$. A set $S \subseteq V(G)$ is $t_3$ \textit{convex} if for any two non-adjacent vertices $x, y \in S$ any vertex in a $t_3$ path between $x$ and $y$ is also in $S$. $t_3$ convexity consists of all $t_3$ convex subsets of $G$. We characterize $t_3$ convex graphs (graphs that are convex geometry with respect to $t_3$ convexity). We show that a graph $G$ is $t_3$ convex geometry if and only if $G$ is a chordal graph contains no $S_3$(sun) and co-rising sun as induced subgraph.

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Creative Commons Attribution 4.0 License
This work is licensed under a Creative Commons Attribution 4.0 License.

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