Total isolation game in graphs

TL;DR

This paper introduces and analyzes the total isolation game on graphs, establishing upper bounds on the game total isolation number based on graph properties like order, minimum degree, and diameter.

Contribution

It provides new bounds for the game total isolation number for various classes of graphs, including connected graphs with minimum degree constraints and diameter conditions.

Findings

For connected graphs of order n ≥ 3, the total isolation game number is less than 5/6 of n.

Graphs with minimum degree at least 2 have a game number at most 3/4 of n.

Graphs with diameter 2 have a game number at most 2/3 of n.

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 $\iota_{\rm gt}(G)$ 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 $\iota_{\rm gt}(G) < \frac{5}{6}n$. Furthermore if $G$ has minimum degree at least $2$, then we prove that $\iota_{\rm gt}(G) \le \frac{3}{4}n$. More generally, if $G$ is a connected graph of order $n \ge 3$ with minimum degree $\delta$ where $\delta \ge 2$, then we prove that $\iota_{\rm gt}(G) \le \left( \frac{2\delta-1}{3\delta-2} \right) n$. Among other results it is proved that if $G$ is a graph of order $n$ with diameter $2$, then $\iota_{\rm gt}(G) \le \frac{2}{3}n$.