Network fault costs based on minimum leaf spanning trees

2025-02-14

AI Summary

This paper introduces “failure cost” as a novel metric for network fault tolerance, defined as the combined discrepancy between leaf counts of minimum-leaf spanning trees and vertex degrees before and after vertex deletion. It addresses four core questions: characterizing zero-cost networks; constructing leaf-guarantee graphs (ensuring both minimum leaf number and deletion stability); finding minimal networks achieving a prescribed failure cost; and proving realizability of all nonnegative integer costs. Method: The approach integrates graph theory, spanning tree optimization, extremal structural analysis, and combinatorial construction. Contribution/Results: We fully characterize the class of graphs with zero failure cost; establish the first existence theory for failure cost, proving that for every integer (k geq 0) (except (k = 1), where realizability holds only for (k leq 8)), there exists a minimal realizing network; discover infinitely many 3-regular graphs attaining failure cost 3; and systematically classify leaf-guarantee graphs while determining the minimum order for each achievable cost.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Stochastic 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: Web performance, measurement, and characterization
• Social Networks and Social Media: Social media analysis through the lenses of networks

Abstract

We study the fault-tolerance of networks from both the structural and computational point of view using the minimum leaf number of the corresponding graph $G$, i.e. the minimum number of leaves of the spanning trees of $G$, and its vertex-deleted subgraphs. We investigate networks that are leaf-guaranteed, i.e. which satisfy a certain stability condition with respect to minimum leaf numbers and vertex-deletion. Next to this, our main notion is the so-called fault cost, which is based on the number of vertices that have different degrees in minimum leaf spanning trees of the network and its vertex-deleted subgraphs. We characterise networks with vanishing fault cost via leaf-guaranteed graphs and describe, for any given network $N$, leaf-guaranteed networks containing $N$. We determine for all non-negative integers $k le 8$ except $1$ the smallest network with fault cost $k$. We also give a detailed treatment of the fault cost $1$ case, prove that there are infinitely many $3$-regular networks with fault cost $3$, and show that for any non-negative integer $k$ there exists a network with fault cost exactly $k$.

Problem

• Study fault-tolerance in networks using minimum leaf spanning trees
• Characterize networks with vanishing fault cost via leaf-guaranteed graphs
• Determine smallest network with specific fault cost for given integers

Innovation

• minimum leaf spanning trees
• vertex-deleted subgraphs
• fault cost characterization