On the General Position Number of Mycielskian Graphs

On the General Position Number of Mycielskian Graphs

The general position problem for graphs was inspired by the no-three-in-line problem from discrete geometry. A set \(S\) of vertices of a graph \(G\) is a \emph{general position set} if no shortest path in \(G\) contains three or more vertices of \(S\). The \emph{general position number} of \(G\) is...

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Veröffentlicht in:arXiv.org 2024-03
Hauptverfasser: Elias John Thomas, Chandran, Ullas, Tuite, James, Gabriele Di Stefano
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Apexes
Graph theory
Graphs
Lower bounds
Shortest-path problems
Trees (mathematics)
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Chandran, Ullas
Tuite, James
Gabriele Di Stefano
description The general position problem for graphs was inspired by the no-three-in-line problem from discrete geometry. A set \(S\) of vertices of a graph \(G\) is a \emph{general position set} if no shortest path in \(G\) contains three or more vertices of \(S\). The \emph{general position number} of \(G\) is the number of vertices in a largest general position set. In this paper we investigate the general position numbers of the Mycielskian of graphs. We give tight upper and lower bounds on the general position number of the Mycielskian of a graph \(G\) and investigate the structure of the graphs meeting these bounds. We determine this number exactly for common classes of graphs, including cubic graphs and a wide range of trees.
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subjects Apexes
Graph theory
Graphs
Lower bounds
Shortest-path problems
Trees (mathematics)
title On the General Position Number of Mycielskian Graphs
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