The Gamma-Theta Conjecture holds for planar graphs

TL;DR

This paper proves the Gamma-Theta Conjecture for planar graphs, establishing a key equality among domination, eternal domination, and clique covering numbers for this class.

Contribution

It extends the validity of the Gamma-Theta Conjecture to all planar graphs, a significant class in graph theory.

Findings

The Gamma-Theta Conjecture holds for planar graphs.

The proof generalizes previous results for specific graph classes.

Supports the conjecture's applicability to broader graph classes.

Abstract

The Gamma-Theta Conjecture states that if the domination number of a graph is equal to its eternal domination number, then it is also equal to its clique covering number. This conjecture is known to be true for several graph classes, such as outerplanar graphs, subcubic graphs and $C_k$-free graphs, where $k \in \{3,4\}$. In this paper, we prove the Conjecture for the class of planar graphs.