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Progress on Self Identifying Codes

Devin Jean, Suk Seo

TL;DR

The paper studies self-identifying codes (SIC), a variant of identifying codes where a detector set $S$ satisfies $N_S[x]\neq\varnothing$ and $\bigcap_{v\in N_S[x]}N[v]=\{x\}$, enabling unique localization of any vertex. It proves SIC minimization is NP-complete via a 3-SAT reduction and analyzes SIC across key graph families, delivering exact and tight density bounds for hypercubes, cubic graphs, and several infinite grids using share-arguments and structural reasoning. It establishes fundamental existence criteria (no semi-closed-twins) and relates SIC to standard IC, while characterizing extremal cubic graphs and providing density benchmarks for infinite graphs with practical implications for sensor placement and fault detection in networks.

Abstract

The concept of an identifying code for a graph was introduced by Karpovsky, Chakrabarty, and Levitin in 1998 as the problem of covering the vertices of a graph such that we can uniquely identify any vertex in the graph by examining the vertices that cover it. An application of an identifying code would be to detect a faulty processor in a multiprocessor system. In 2020, a variation of identify code called "self-identifying code" was introduced by Junnila and Laihonen, which simplifies the task of locating the malfunctioning processor. In this paper, we continue to explore self-identifying codes. In particular, we prove the problem of determining the minimum cardinality of a self-identifying code for an arbitrary graph is NP-complete and we investigate minimum-sized self-identifying code in several classes of graphs, including cubic graphs and infinite grids.

Progress on Self Identifying Codes

TL;DR

The paper studies self-identifying codes (SIC), a variant of identifying codes where a detector set satisfies and , enabling unique localization of any vertex. It proves SIC minimization is NP-complete via a 3-SAT reduction and analyzes SIC across key graph families, delivering exact and tight density bounds for hypercubes, cubic graphs, and several infinite grids using share-arguments and structural reasoning. It establishes fundamental existence criteria (no semi-closed-twins) and relates SIC to standard IC, while characterizing extremal cubic graphs and providing density benchmarks for infinite graphs with practical implications for sensor placement and fault detection in networks.

Abstract

The concept of an identifying code for a graph was introduced by Karpovsky, Chakrabarty, and Levitin in 1998 as the problem of covering the vertices of a graph such that we can uniquely identify any vertex in the graph by examining the vertices that cover it. An application of an identifying code would be to detect a faulty processor in a multiprocessor system. In 2020, a variation of identify code called "self-identifying code" was introduced by Junnila and Laihonen, which simplifies the task of locating the malfunctioning processor. In this paper, we continue to explore self-identifying codes. In particular, we prove the problem of determining the minimum cardinality of a self-identifying code for an arbitrary graph is NP-complete and we investigate minimum-sized self-identifying code in several classes of graphs, including cubic graphs and infinite grids.

Paper Structure

This paper contains 9 sections, 21 theorems, 2 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Properties and Existence Criteria
  3. NP-completeness of Self Identifying Codes
  4. SIC for Special Classes of Graphs
  5. Hypercubes
  6. SIC in cubic graphs
  7. Upper Bound on SIC(G)
  8. Lower Bound on SIC(G)
  9. SIC for Infinite Grids

Key Result

Theorem 1.1

A set $S \subseteq V(G)$ is an SIC if and only if $N_S[u] - N_S[v] \neq \varnothing$ for all distinct $u,v \in V(G)$.

Figures (10)

  • Figure 1: Example of locating an intruder in an IC
  • Figure 2: Examples of optimal IC (a), RED:IC, (b), DET:IC (c), SIC (d), and ERR:IC (e) solutions on the Petersen graph
  • Figure 3: Variable and clause graphs
  • Figure 4: Construction of $G$ from $(x_1 \lor \overline{x_2} \lor x_3) \land (\overline{x_1} \lor x_2 \lor x_5) \land (\overline{x_1} \lor \overline{x_2} \lor x_4) \land (\overline{x_3} \lor \overline{x_4} \lor \overline{x_5})$with $N = 4$, $M = 3$, $K = 109$
  • Figure 5: Optimal SIC for $Q_n$ with $n \le 5$
  • ...and 5 more figures

Theorems & Definitions (40)

  • Definition 1: junn20o
  • Theorem 1.1: junn20o
  • Definition 2: fouc16a
  • Definition 3
  • Corollary 2.1
  • Corollary 2.2
  • Theorem 2.1: jean24b
  • proof
  • Corollary 2.3
  • Theorem 2.2
  • ...and 30 more