Criticality for Maker-Breaker domination games with predomination

TL;DR

The paper investigates Maker-Breaker domination games with predomination, introducing MBD critical predominated graphs where Staller’s win on $(G,D)$ is robust to any augmentation of $D$. It develops tree-based tools to characterize Staller-win predominated trees and provides a complete linear-time recognition algorithm for MBD-critical trees. Extending to cacti, it defines $F$-cacti via double-odd replacements and proves that a broad class of atomic MBD-critical graphs arises from these constructions, with matching-based strategies underpinning Dominator’s side. The work also formulates a broader hypergraph perspective, linking MBD-critical graphs to Maker-Breaker critical hypergraphs and outlining future directions for Dominator-criticality and richer graph families.

Abstract

A predominated graph is a pair $(G,D)$, where $G$ is a graph and the vertices in $D\subseteq V(G)$ are considered already dominated. Maker-Breaker domination game critical (MBD critical) predominated graphs are introduced as the predominated graphs $(G,D)$ on which Staller wins the game, but Dominator wins on $(G, D \cup \{v\})$ for every vertex $v \in V(G) \setminus D$. Tools are developed for handling the Maker-Breaker domination game on trees which lead to a characterization of Staller-win predominated trees. MBD critical predominated trees are characterized and an algorithm is designed which verifies in linear time whether a given predominated tree is MBD critical. A large class of MBD critical predominated cacti is presented and Maker-Breaker critical hypergraphs constructed.

Figures (6)

• Figure 1: A tree $T$, the tree $S(T)$, and $G$ with a substructure $S(T)$.
• Figure 2: The partition of the vertices of the tree $T$ into black, white and gray vertices, along with the four substructures of $T$. The colorings on substructures are compatible on their intersections, by Proposition \ref{['prop:coloring-substructures']}.
• Figure 3: Two MBD critical predominated trees (above) and an atomic MBD critical predominated tree. The substructures $F$ and $F'$ as in Theorem \ref{['thm:critical-trees']} (i) are also marked. The predominated vertices are marked with a square around them.
• Figure 4: Two atomic MBD critical cactus graphs $H_1$ and $H_2$, both obtained from the atomic MBD critical tree depicted at the bottom of Fig. \ref{['fig:atomic-critical-2x']}.
• Figure 5: Matchings $M(H_1,x)$ and $M(H_2,y)$, as defined in the proof of Theorem \ref{['thm:cC']}.

Theorems & Definitions (34)

• Definition 2.1
• Proposition 2.2
• proof
• Corollary 2.3
• Proposition 3.2
• proof
• Definition 3.3
• Definition 4.1
• Definition 4.2
• Theorem 4.3: bujtas-2023