# On the Leaders' Graphical Characterization for Controllability of Path Related Graphs

## TL;DR

This paper introduces the concept of minimal perfect critical vertex sets for leader placement in undirected graphs, providing conditions for special cases and a complete solution for path graphs, advancing controllability analysis.

## Contribution

It defines minimal perfect critical vertex sets using Laplacian eigenvectors and offers a complete algorithm for leader placement in path graphs.

## Key findings

- Complete solution for leader placement in path graphs
- Proof that minimal perfect critical 3 vertex sets do not exist
- Necessary and sufficient conditions for special minimal perfect critical vertex sets

## Abstract

The problem of leaders location plays an important role in the controllability of undirected graphs.The concept of minimal perfect critical vertex set is introduced by drawing support from the eigenvector of Laplace matrix. Using the notion of minimal perfect critical vertex set, the problem of finding the minimum number of controllable leader vertices is transformed into the problem of finding all minimal perfect critical vertex sets. Some necessary and sufficient conditions for special minimal perfect critical vertex sets are provided, such as minimal perfect critical 2 vertex set, and minimal perfect critical vertex set of path or path related graphs. And further, the leaders location problem for path graphs is solved completely by the algorithm provided in this paper. An interesting result that there never exist a minimal perfect critical 3 vertex set is proved, too.