Predominating a vertex in the connected domination game

TL;DR

This paper studies the connected domination game, establishing bounds on the number of moves needed when a vertex is pre-dominated, and introduces a new principle to replace the classical one.

Contribution

It introduces the Connected Game Continuation Principle and provides sharp bounds on game length after pre-dominating a vertex, advancing understanding of the connected domination game.

Findings

Established bounds for $ ext{γ}_{ ext{cg}}(G|x)$ relative to $ ext{γ}_{ ext{cg}}(G)$.

Proved the bounds are sharp with specific graph examples.

Characterized graphs where $ ext{γ}_{ ext{cg}}'(G|x) = ext{infinity}$.

Abstract

The connected domination game is played just as the domination game, with an additional requirement that at each stage of the game the vertices played induce a connected subgraph. The number of moves in a D-game (an S-game, resp.) on a graph $G$ when both players play optimally is denoted by $\gamma_{\rm cg}(G)$ ($\gamma_{\rm cg}'(G)$, resp.). Connected Game Continuation Principle is established as a substitute for the classical Continuation Principle which does not hold for the connected domination game. Let $G|x$ denote the graph $G$ together with a declaration that the vertex $x$ is already dominated. The first main result asserts that if $G$ is a graph with $\gamma_{\rm cg}(G) \geq 3$ and $x \in V(G)$, then $\gamma_{\rm cg}(G|x) \leq 2 \gamma_{\rm cg}(G) - 3$ and the bound is sharp. The second main theorem states that if $G$ is a graph with $n(G) \geq 2$ and $x \in V(G)$, then $\gamma_{\rm cg}(G|x) \geq \left \lceil \frac12 \gamma_{\rm cg}(G) \right \rceil$ and the bound is sharp. Graphs $G$ and their vertices $x$ for which $\gamma_{\rm cg}'(G|x) = \infty$ holds are also characterized.