On a New Class of Codes for Identifying Vertices in Graphs

—We investigate a new class of codes for the optimal covering of vertices in an undirected graph Gsuch that any vertex in G can be uniquely identified by examining the vertices that cover it. We define a ball of radius t centered on a vertex v to be the set of vertices in G that are at distance at most t from v: The vertex v is then said to cover itself and every other vertex in the ball with center v: Our formal problem statement is as follows: Given an undirected graph G and an integer t  1, find a (minimal) set C of vertices such that every vertex in G belongs to a unique set of balls of radius t centered at the vertices in C: The set of vertices thus obtained constitutes a code for vertex identification. We first develop topology-independent bounds on the size of C: We then develop methods for constructing C for several specific topologies such as binary cubes, nonbinary cubes, and trees. We also describe the identification of sets of vertices using covering codes that un...

• Type: Journal
• Year: 1998
• Where: TIT
• Authors: Mark G. Karpovsky, Krishnendu Chakrabarty, Lev B. Levitin
• Keywords: Formal Problem Statement, TIT 1998, Undirected Graph, Undirected Graph Gsuch