Total isolation game in graphs

TL;DR

The paper studies the total isolation game on graphs, introducing the game total isolation number $\iota_{gt}(G)$ (Dominator-start) and $\iota'_{gt}(G)$ (Staller-start). It shows that the Continuation Principle does not hold for this game, evidenced by a counterexample on $P_5$, and develops a framework based on greedy strategies and Stage 1/Stage 2 analyses to bound the game size. It establishes a tight family of bounds: for connected $G$ of order $n$, $\iota_{gt}(G) < \frac{5}{6}n$; more refined bounds in terms of minimum degree $\delta$ and maximum degree $\Delta$ are $\iota_{gt}(G) \le \left(\frac{2\delta-1}{3\delta-2}\right)n - \frac{\Delta-2}{3\delta-2}$ and $\iota'_{gt}(G) \le \left(\frac{2\delta-1}{3\delta-2}\right)n - \frac{(\delta-1)(2\delta-3)}{3\delta-2}$, with corollaries for $\delta=2$ and for diameter-2 graphs ($\iota_{gt}(G) \le \tfrac{2}{3}n$). The results contribute to competitive graph theory by extending domination-game techniques to the total isolation setting and by providing near-optimal bounds for broad graph families, including a general conjecture for components of size at least 3.

Abstract

The total isolation game is played on a graph $G$ by two players who take turns playing a vertex such that if $S$ is the set of already played vertices, then a vertex can be selected only if it is adjacent to a vertex that belongs to a (nontrivial) component of the graph $G - N_G(S)$ of order at least $2$ or a vertex that is isolated in $G - N_G(S)$ and belongs to the set $S$, where $N_G(S)$ is the set of vertices adjacent to a vertex in $S$. Dominator wishes to finish the game with the minimum number of played vertices, while Staller has the opposite goal. The game total isolation number $ι_{\rm gt}(G)$ is the number of moves in the Dominator-start game where both players play optimally. We prove that if $G$ is a connected graph of order $n \ge 3$, then $ι_{\rm gt}(G) < \frac{5}{6}n$. Furthermore if $G$ has minimum degree at least $2$, then we prove that $ι_{\rm gt}(G) \le \frac{3}{4}n$. More generally, if $G$ is a connected graph of order $n \ge 3$ with minimum degree $δ$ where $δ\ge 2$, then we prove that $ι_{\rm gt}(G) \le \left( \frac{2δ-1}{3δ-2} \right) n$. Among other results it is proved that if $G$ is a graph of order $n$ with diameter $2$, then $ι_{\rm gt}(G) \le \frac{2}{3}n$.

Theorems & Definitions (12)

• Theorem 1
• Claim 1
• Claim 2
• Corollary 1
• Theorem 2
• Claim 3
• Corollary 2
• Corollary 3
• Theorem 3
• Theorem 4