On monophonic position sets in graphs
Abstract
The general position problem in graph theory asks for the largest set $S$ of vertices of a graph $G$ such that no shortest path of $G$ contains more than two vertices of $S$. In this paper we consider a variant of the general position problem called the \emph{monophonic position problem}, obtained by replacing `shortest path' by `induced path'. We prove some basic properties and bounds for the monophonic position number of a graph and determine the monophonic position number of some graph families, including unicyclic graphs, complements of bipartite graphs and split graphs. We show that the monophonic position number of trianglefree graphs is bounded above by the independence number. We present realisation results for the general position number, monophonic position number and monophonic hull number. Finally we discuss the complexity of the monophonic position problem.
 Publication:

arXiv eprints
 Pub Date:
 December 2020
 arXiv:
 arXiv:2012.10330
 Bibcode:
 2020arXiv201210330T
 Keywords:

 Mathematics  Combinatorics;
 05C12 05C69
 EPrint:
 Revised version. New author added together with new results on the complexity of the monophonic position problem. A small gap in the realisation theorem for mp vs. gp number has been fixed