On Iiro Honkala's contributions to identifying codes

TL;DR

This paper presents a survey of Iiro Honkala’s contributions to the study of identifying codes with respect to several aspects: complexity of computing an identifying code, combinatorics in binary Hamming spaces, infinite grids, relationships between identifying codes and usual parameters in graphs, structural properties of graphs admitting identifying codes, and number of optimal identifying codes.

Abstract

A set $C$ of vertices in a graph $G=(V,E)$ is an identifying code if it is dominating and any two vertices of $V$ are dominated by distinct sets of codewords. This paper presents a survey of Iiro Honkala's contributions to the study of identifying codes with respect to several aspects: complexity of computing an identifying code, combinatorics in binary Hamming spaces, infinite grids, relationships between identifying codes and usual parameters in graphs, structural properties of graphs admitting identifying codes, and number of optimal identifying codes.

Figures (15)

• Figure 1: Different graphs and codes. Black vertices represent codewords.
• Figure 2: The subgraph induced by a clause $C_i = x_j \vee x_h \vee x_k$ for the transformation from 3-SAT to $1$-IdC.
• Figure 3: Partial representations of the four grids: (a) the square grid $G_S$; (b) the triangular grid $G_T$: black vertices are codewords (cf. Theorem \ref{['6triangleMark6']}); (c) the king grid $G_K$; (d) the hexagonal grid $G_H$ (with two possible representations: as a honeycomb or as a brick wall).
• Figure 4: A periodic $5$-IdC in the square grid $G_S$, of density $2/25$; codewords are in black.
• Figure 5: A periodic $1$-IdC in the square grid $G_S$, of density $7/20$; codewords are in black.

Theorems & Definitions (58)

• Definition 1
• Definition 2
• Example 1.1
• Theorem 1.2: Auger_2008
• Theorem 1.3
• Theorem 2.1
• Lemma 3.1
• Theorem 3.2: honkalaCodesIdentifyingSets2001*
• Theorem 3.3: honk02c*
• Theorem 3.4: karp98a