Identifying codes and searching with balls in graphs

TL;DR

This paper investigates the minimum number of ball-based queries needed to identify an unknown vertex in various graph classes, considering both adaptive and non-adaptive strategies, with bounds established for hypercubes, random graphs, and bounded degree graphs.

Contribution

It provides new bounds on the number of ball queries required for vertex identification in different graph models, addressing both adaptive and non-adaptive scenarios.

Findings

Bounds established for hypercubes

Results for Erdős-Rényi random graphs

Analysis for graphs with bounded maximum degree

Abstract

Given a graph $G$ and a positive integer $R$ we address the following combinatorial search theoretic problem: What is the minimum number of queries of the form "does an unknown vertex $v \in V(G)$ belong to the ball of radius $r$ around $u$?" with $u \in V(G)$ and $r\le R$ that is needed to determine $v$. We consider both the adaptive case when the $j$th query might depend on the answers to the previous queries and the non-adaptive case when all queries must be made at once. We obtain bounds on the minimum number of queries for hypercubes, the Erd\H os-R\'enyi random graphs and graphs of bounded maximum degree .