Article in volume
Authors:
Title:
On the vertex position number of graphs
PDFSource:
Discussiones Mathematicae Graph Theory 44(3) (2024) 1169-1188
Received: 2022-09-14 , Revised: 2023-02-23 , Accepted: 2023-02-23 , Available online: 2023-03-19 , https://doi.org/10.7151/dmgt.2491
Abstract:
In this paper we generalise the notion of visibility from a point in an integer
lattice to the setting of graph theory. For a vertex $x$ of a graph $G$, we say
that a set $S \subseteq V(G)$ is an $x$-position set if for any $y \in S$
the shortest $x,y$-paths in $G$ contain no point of $S\setminus \{ y\}$.
We investigate the largest and smallest orders of maximum $x$-position sets in
graphs, determining these numbers for common classes of graphs and giving bounds
in terms of the girth, vertex degrees, diameter and radius. Finally we discuss
the complexity of finding maximum vertex position sets in graphs.
Keywords:
geodesic, vertex position set, vertex position number, general position number
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