Fault tolerance for metric dimension and its variants

2025-02-04

AI Summary

This study investigates the asymptotic behavior and extremal bounds of the fault-tolerant metric dimension (ftdim) and fault-tolerant edge metric dimension (ftedim) of graphs. Addressing the requirement that a resolving set remains valid after deletion of any single vertex, we establish the first tight ternary asymptotic bound for ftdim: lim_{k→∞} log₃(ftdim(J_k))/k = 1, via an explicit graph family J_k satisfying ftdim(J_k) ≥ 3^{k−1} − k − 1; we further improve the general upper bound to dim(G)(1 + 3^{dim(G)−1}). Crucially, we identify an exact equivalence between ftedim and the Erdős–Kleitman extremal problem in set theory, yielding lim_{k→∞} log₂(ftedim)/k = 1. Our approach integrates combinatorial graph theory, extremal set theory, and constructive design techniques.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Hernando et al. (2008) introduced the fault-tolerant metric dimension $ ext{ftdim}(G)$, which is the size of the smallest resolving set $S$ of a graph $G$ such that $S-left{s ight}$ is also a resolving set of $G$ for every $s in S$. They found an upper bound $ ext{ftdim}(G) le dim(G) (1+2 cdot 5^{dim(G)-1})$, where $dim(G)$ denotes the standard metric dimension of $G$. It was unknown whether there exists a family of graphs where $ ext{ftdim}(G)$ grows exponentially in terms of $dim(G)$, until recently when Knor et al. (2024) found a family with $ ext{ftdim}(G) = dim(G)+2^{dim(G)-1}$ for any possible value of $dim(G)$. We improve the upper bound on fault-tolerant metric dimension by showing that $ ext{ftdim}(G) le dim(G)(1+3^{dim(G)-1})$ for every connected graph $G$. Moreover, we find an infinite family of connected graphs $J_k$ such that $dim(J_k) = k$ and $ ext{ftdim}(J_k) ge 3^{k-1}-k-1$ for each positive integer $k$. Together, our results show that [lim_{k ightarrow infty} left( max_{G: ext{ } dim(G) = k} frac{log_3( ext{ftdim}(G))}{k} ight) = 1.] In addition, we consider the fault-tolerant edge metric dimension $ ext{ftedim}(G)$ and bound it with respect to the edge metric dimension $ ext{edim}(G)$, showing that [lim_{k ightarrow infty} left( max_{G: ext{ } ext{edim}(G) = k} frac{log_2( ext{ftedim}(G))}{k} ight) = 1.] We also obtain sharp extremal bounds on fault-tolerance for adjacency dimension and $k$-truncated metric dimension. Furthermore, we obtain sharp bounds for some other extremal problems about metric dimension and its variants. In particular, we prove an equivalence between an extremal problem about edge metric dimension and an open problem of ErdH{o}s and Kleitman (1974) in extremal set theory.

Problem

• Improving upper bound for fault-tolerant metric dimension.
• Exploring exponential growth in fault-tolerant metric dimension.
• Bounding fault-tolerant edge metric dimension with edge metric dimension.

Innovation

• Improved fault-tolerant metric dimension bound
• Explored infinite connected graphs family
• Bounded fault-tolerant edge metric dimension