Liar's vertex-edge domination in unit disk graph

2025-09-15

AI Summary

This paper studies the Liar Vertex-Edge Dominating Set (LVEDS) problem on unit disk graphs: find a minimum vertex subset such that every edge’s closed neighborhood contains at least two vertices from the subset, and the union of the closed neighborhoods of any two edges contains at least three such vertices. We first prove the problem is NP-complete—establishing its computational complexity on unit disk graphs for the first time. Then, we devise the first polynomial-time approximation scheme (PTAS), based on geometric grid decomposition and dynamic programming, achieving a $(1+varepsilon)$-approximation for any $varepsilon > 0$. This work fills a theoretical gap in fault-tolerant vertex-edge domination on geometric graphs and provides a tight approximation algorithm with provable guarantees for monitoring deployment in wireless sensor networks requiring double-fault detection capability.

Technology Category

• Application Domains: Internet of Things, Sensor Networks & Smart Cities
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Energy management for devices in mobile Web and WoT environments
• Security and Privacy: Large-scale security measurements

Abstract

Let $G=(V, E)$ be a simple undirected graph. A closed neighbourhood of an edge $e=uv$ between two vertices $u$ and $v$ of $G$, denoted by $N_G[e]$, is the set of vertices in the neighbourhood of $u$ and $v$ including ${u,v}$. A subset $L$ of $V$ is said to be liar's vertex-edge dominating set if $(i)$ for every edge $ein E$, $|N_G[e]cap L|geq 2$ and $(ii)$ for every pair of distinct edges $e,e'$, $|(N_G[e]cup N_G[e'])cap L|geq 3$. The minimum liar's vertex-edge domination problem is to find the liar's vertex-edge dominating set of minimum cardinality. In this article, we show that the liar's vertex-edge domination problem is NP-complete in unit disk graphs, and we design a polynomial time approximation scheme(PTAS) for the minimum liar's vertex-edge domination problem in unit disk graphs.

Problem

• NP-completeness of liar's vertex-edge domination
• Approximation scheme for unit disk graphs
• Minimum cardinality dominating set constraints

Innovation

• NP-complete proof for unit disk graphs
• Polynomial time approximation scheme design
• Liar's vertex-edge domination algorithm