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

06/08/2019
by   Li Dai, et al.
0

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.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
06/07/2022

Vertex-critical (P_3+ℓ P_1)-free and vertex-critical (gem, co-gem)-free graphs

A graph G is k-vertex-critical if χ(G)=k but χ(G-v)<k for all v∈ V(G) wh...
research
02/16/2021

(-k)-critical trees and k-minimal trees

In a graph G=(V,E), a module is a vertex subset M of V such that every v...
research
06/03/2023

Valid path-based graph vertex numbering

A labelling of a graph is an assignment of labels to its vertex or edge ...
research
12/19/2019

On existence of perfect bitrades in Hamming graphs

A pair (T_0,T_1) of disjoint sets of vertices of a graph G is called a p...
research
10/26/2021

Connected greedy colourings of perfect graphs and other classes: the good, the bad and the ugly

The Grundy number of a graph is the maximum number of colours used by th...
research
11/15/2017

On consistent vertex nomination schemes

Given a vertex of interest in a network G_1, the vertex nomination probl...
research
09/19/2019

A note on minimal art galleries

We will consider some extensions of the polygonal art gallery problem. I...

Please sign up or login with your details

Forgot password? Click here to reset