arXiv subjects

Samuel G. Corregidor

Publications and source records attributed to Samuel G. Corregidor.

3 recordsLinked to original sources

arXiv2022

Vertex separators, chordality and virtually free groups

In this paper we consider some results obtained for graphs using minimal vertex separators and generalized chordality and translate them to the context of Geometric Group Theory. Using these new tools, we are able to give two new characterizations for a group to be virtually free. Furthermore, we prove that the Baumslag-Solitar group $BS(1,n)$ is $k$-chordal for some $k$ if and only if $|n|<3$ and we give an application of generalized chordality to the study of the word problem.

math.GR↗

arXiv2020

Finite metric and $k$-metric bases on ultrametric spaces

Given a metric space $(X,d)$, a set $S\subseteq X$ is called a $k$-\emph{metric generator} for $X$ if any pair of different points of $X$ is distinguished by at least $k$ elements of $S$. A $k$-\emph{metric basis} is a $k$-metric generator of the minimum cardinality in $X$. We prove that ultrametric spaces do not have finite $k$-metric bases for $k>2$. We also characterize when the metric and 2-metric bases of an ultrametric space are finite and, when they are finite, we characterize them. Finally, we prove that an ultrametric space can be easily recovered knowing only the metric basis and the coordinates of the points in it.

math.GN↗

arXiv2019

A note on $k$-metric dimensional graphs

Given a graph $G = (V,E)$, a set $S \subset V$ is called a $k$-\emph{metric generator} for $G$ if any pair of different vertices of $G$ is distinguished by at least $k$ elements of $S$. A graph is $k$-\emph{metric dimensional} if $k$ is the largest integer such that there exists a $k$-metric generator for $G$. This paper studies some bounds on the number $k$ for which a graph is $k$-metric dimensional.

math.CO↗