Algebraic connectivity of layered path graphs under node deletion

TL;DR

This paper investigates how node deletion affects the algebraic connectivity of layered path graphs, revealing that maintaining at least one upper-layer neighbor prevents connectivity deterioration, with applications in mobile robot formation control.

Contribution

It introduces layered path graphs and their subgraph cones to analyze the impact of node removal on algebraic connectivity, highlighting the importance of upper-layer neighbors.

Findings

Presence of at least one upper-layer neighbor preserves algebraic connectivity.

Node deletion without upper-layer neighbors can significantly reduce connectivity.

The study provides insights for robust formation control in mobile robotics.

Abstract

This paper studies the relation between node deletion and algebraic connectivity for graphs with a hierarchical structure represented by layers. To capture this structure, the concepts of layered path graph and its (sub)graph cone are introduced. The problem is motivated by a mobile robot formation control guided by a leader. In particular, we consider a scenario in which robots may leave the network resulting in the removal of the nodes and the associated edges. We show that the existence of at least one neighbor in the upper layer is crucial for the algebraic connectivity not to deteriorate by node deletion.