    Graphs with unique zero forcing sets and Grundy dominating sets

The concept of zero forcing was introduced in the context of linear algebra, and was further studied by both graph theorists and linear algebraists. It is based on the process of activating vertices of a graph G starting from a set of vertices that are already active, and applying the rule that an active vertex with exactly one non-active neighbor forces that neighbor to become active. A set S⊂ V(G) is called a zero forcing set of G if initially only vertices of S are active and the described process enforces all vertices of G to become active. The size of a minimum zero forcing set in G is called the zero forcing number of G. While a minimum zero forcing set can only be unique in edgeless graphs, we consider the weaker uniqueness condition, notably that for every two minimum zero forcing sets in a graph G there is an automorphism that maps one to the other. We characterize the class of trees that enjoy this condition by using properties of minimum path covers of trees. In addition, we investigate both variations of uniqueness for several concepts of Grundy domination, which first appeared in the context of domination games, yet they are also closely related to zero forcing. For each of the four variations of Grundy domination we characterize the graphs that have only one Grundy dominating set of the given type, and characterize those forests that enjoy the weaker (isomorphism based) condition of uniqueness. The latter characterizations lead to efficient algorithms for recognizing the corresponding classes of forests.

Authors

09/01/2020

Reconfiguration graphs of zero forcing sets

This paper begins the study of reconfiguration of zero forcing sets, and...
05/07/2018

Domination Cover Number of Graphs

A set D ⊆ V for the graph G=(V, E) is called a dominating set if any ver...
09/06/2019

On the vertices belonging to all, some, none minimum dominating set

We characterize the vertices belonging to all minimum dominating sets, t...
09/25/2017

On the error of a priori sampling: zero forcing sets and propagation time

Zero forcing is an iterative process on a graph used to bound the maximu...
11/17/2021

On the coalition number of graphs

Let G be a graph with vertex set V. Two disjoint sets V_1, V_2 ⊆ V form ...
10/14/2017

On complexity of mutlidistance graph recognition in R^1

Let A be a set of positive numbers. A graph G is called an A-embeddable ...
09/17/2019

Rotational Uniqueness Conditions Under Oblique Factor Correlation Metric

In an addendum to his seminal 1969 article Jöreskog stated two sets of c...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.