Graphs with unique Grundy dominating sets

TL;DR

The paper investigates graphs with a unique or iso-unique Grundy-type dominating structure across four variants linked to domination games and zero forcing. It establishes that unique Grundy domination is impossible for nontrivial connected graphs, while iso-unique Grundy domination graphs are exactly the complete graphs; for zero forcing-related variants, iso-unique trees are characterized via path covers and can be recognized in $O(n^2)$ time. It also provides linear-time recognition for iso-unique Grundy total domination trees and shows that all forests are iso-unique L-Grundy graphs, with broader, unresolved questions for the remaining iso-unique classes. The results yield both sharp structural classifications and practical recognition algorithms, highlighting the rarity of uniqueness phenomena in Grundy domination across graphs and trees.

Abstract

Given a graph $G$ consider a procedure of building a dominating set $D$ in $G$ by adding vertices to $D$ one at a time in such a way that whenever vertex $x$ is added to $D$ there exists a vertex $y\in N_G[x]$ that becomes dominated only after $x$ is added to $D$. The maximum cardinality of a set $D$ obtained in the described way is called the Grundy domination number of $G$ and $D$ a Grundy dominating set. While a Grundy dominating set of a connected graph $G$ is not unique unless $G$ is the trivial graph, we consider a natural weaker uniqueness condition, notably that for every two Grundy dominating sets in a graph $G$ there is an automorphism that maps one to the other. We investigate both versions of uniqueness for several concepts of Grundy domination, which appeared in the context of domination games and 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.

Figures (3)

• Figure 1: An iso-unique Z-Grundy domination graph; vertices of a $\gamma_{gr}^Z$-set are shaded, while white vertices form a minimum zero forcing set.
• Figure 2: An iso-unique zero forcing tree; paths of a minimum path cover are circled by ellipses; vertices of a minimum zero forcing are shaded.
• Figure 3: Graph $H$ with two $\gamma_{gr}^{t}(H)$-sets marked by black vertices.

Theorems & Definitions (38)

• Proposition 1
• proof
• Corollary 2
• Corollary 3
• proof
• Theorem 4
• proof
• Theorem 5
• Proposition 6
• Lemma 7