20.500.12556/DiRROS-27965 Isolation game on graphs Given a graph $G$ and a family of graphs $\cal F$, an $\cal F$-isolating set, as introduced by Caro and Hansberg, is any set $S\subset V(G)$ such that $G - N[S]$ contains no member of $\cal F$ as a subgraph. In this paper, we introduce a game in which two players with opposite goals are together building an $\cal F$-isolating set in $G$. Following the domination games, Dominator (Staller) wants that the resulting $\cal F$-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 $\cal F$-isolation number, denoted $\iota_{\rm g}(G,\cal F)$. We prove that the Continuation Principle holds in the $\cal F$-isolation game, and that the difference between the game $\cal F$-isolation numbers when either Dominator or Staller starts the game is at most $1$. Considering two arbitrary families of graphs $\cal F$ and $\cal F'$, we find relations between them that ensure $\iota_{\rm g}(G,{\mathcal{F}}') \leq \iota_{\rm g}(G,{\mathcal{F}})$ for any graph $G$. A special focus is given on the isolation game, which takes place when ${\cal F}=\{K_2\}$. We prove that $\iota_{\rm g}(G,\{K_2\})\le |V(G)|/2$ for any graph $G$, and conjecture that $\lceil 3|V(G)|/7\rceil$ is the actual (sharp) upper bound. We prove that the isolation game on a forest when Dominator has the first move never lasts longer than the one in which Staller starts the game. Finally, we prove good lower and upper bounds on the game isolation numbers of paths $P_n$, which lead to the exact values $\iota_{\rm g}(P_n,\{K_2\})=\left\lfloor\frac{2n+2}{5}\right\rfloor$ when $n \equiv i \pmod 5$ and $i \in \{1,2,3\}$. isolation number graph games domination games continuation principle forest izolacijsko število igre na grafih dominacijske igre nadaljevalni princip gozd true false true Angleški jezik Slovenski jezik Neznano 2026-03-03 15:00:18 2026-03-03 15:00:19 2026-03-04 03:44:47 0000-00-00 00:00:00 2026 0 0 13 str. iss. 2, article no. 30 Vol. 42 Apr. 2026 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 2026-04-01 519.17 0911-0119 10.1007/s00373-026-03030-y 270312963 s00373-026-03030-y.pdf s00373-026-03030-y.pdf 1 23E1043F7BD1E0BF0D9C00E499EC664E e483a8c81ee7e97ed26aab01923b580900e9f59c4c666c6a7da86e5d7cf3661f d44b53f7-1709-11f1-8c73-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=41208 https://link.springer.com/article/10.1007/s00373-026-03030-y 1 53f4a5c5-1709-11f1-8c73-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=41207 Inštitut za matematiko, fiziko in mehaniko 0 0 0