Isolation game on graphs
Bo\v{s}tjan Bre\v{s}ar, Tanja Dravec, Daniel P. Johnston, Kirsti, Kuenzel, and Douglas F. Rall

TL;DR
This paper introduces a game-theoretic approach to graph isolation problems, analyzing the properties and bounds of the game isolation number for various graph families, especially paths and forests.
Contribution
It defines the game $$-isolation number, proves the Continuation Principle, and establishes bounds and exact values for paths and forests, advancing understanding of graph isolation games.
Findings
The difference in game $$-isolation numbers when starting player varies is at most 1.
For any graph, the game $$-isolation number with $=igraceket{K_2}$ is at most half the vertices.
Exact value of the game isolation number for paths $P_n$ when $n mod 5 eq 0$.
Abstract
Given a graph and a family of graphs , an -isolating set, as introduced by Caro and Hansberg, is any set such that contains no member of as a subgraph. In this paper, we introduce a game in which two players with opposite goals are together building an -isolating set in . Following the domination games, Dominator (Staller) wants that the resulting -isolating set obtained at the end of the game, is as small (as big) as possible, which leads to the graph invariant called the game -isolation number, denoted . We prove that the Continuation Principle holds in the -isolation game, and that the difference between the game -isolation numbers when either Dominator or Staller starts the game is at most . Considering two arbitrary families of graphs and $\cal…
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Taxonomy
TopicsGame Theory and Applications · Aquatic and Environmental Studies
