Centroidal bases in graphs

TL;DR

This paper introduces the concept of centroidal bases in graphs, establishing bounds on their size, analyzing their properties in specific graph classes, and examining the computational complexity of finding such bases.

Contribution

It defines the centroidal dimension, relates it to existing graph parameters, characterizes graphs reaching bounds, and studies its computational complexity and approximation algorithms.

Findings

Bounds the centroidal dimension between rac{ ext{ln} n}{ ext{ln} ext{ln} n} and n-1.

Characterizes graphs that reach the upper bound.

Shows the problem is computationally hard and provides an approximation algorithm.

Abstract

We introduce the notion of a centroidal locating set of a graph $G$, that is, a set $L$ of vertices such that all vertices in $G$ are uniquely determined by their relative distances to the vertices of $L$. A centroidal locating set of $G$ of minimum size is called a centroidal basis, and its size is the centroidal dimension $CD(G)$. This notion, which is related to previous concepts, gives a new way of identifying the vertices of a graph. The centroidal dimension of a graph $G$ is lower- and upper-bounded by the metric dimension and twice the location-domination number of $G$, respectively. The latter two parameters are standard and well-studied notions in the field of graph identification. We show that for any graph $G$ with $n$ vertices and maximum degree at least~2, $(1+o(1))\frac{\ln n}{\ln\ln n}\leq CD(G) \leq n-1$. We discuss the tightness of these bounds and in particular, we characterize the set of graphs reaching the upper bound. We then show that for graphs in which every pair of vertices is connected via a bounded number of paths, $CD(G)=\Omega\left(\sqrt{|E(G)|}\right)$, the bound being tight for paths and cycles. We finally investigate the computational complexity of determining $CD(G)$ for an input graph $G$, showing that the problem is hard and cannot even be approximated efficiently up to a factor of $o(\log n)$. We also give an $O\left(\sqrt{n\ln n}\right)$-approximation algorithm.