20.500.12556/DiRROS-30029 Bounds on the game isolation number and exact values for paths and cycles The isolation game is played on a graph $G$ by two players who take turns playing a vertex such that if $X$ is the set of already played vertices, then a vertex can be selected only if it dominates a vertex from a nontrivial component of $G \setminus N_G[X]$, where $N_G[X]$ is the set of vertices in $X$ or adjacent to a vertex in $X$. Dominator wishes to finish the game with the minimum number of played vertices, while Staller has the opposite goal. The game isolation number $\iota_{\rm g}(G)$ is the number of moves in the Dominator-start game where both players play optimally. If Staller starts the game the invariant is denoted by $\iota_{\rm g}'(G)$. In this paper, $\iota_{\rm g}(C_n)$, $\iota_{\rm g}(P_n)$, $\iota_{\rm g}'(C_n)$, and $\iota_{\rm g}'(P_n)$ are determined for all $n$. It is proved that there are only two graphs that attain equality in the upper bound $\iota_{\rm g}(G) \le \frac{1}{2}|V(G)|$, and that there are precisely eleven graphs which attain equality in the upper bound $\iota_{\rm g}'(G) \le \frac{1}{2}|V(G)|$. For trees $T$ of order at least three it is proved that $\iota_{\rm g}(T) \le \frac{5}{11}|V(T)|$. A new infinite family of graphs $G$ is also constructed for which $\iota_{\rm g}(G) = \iota_{\rm g}'(G) = \frac{3}{7}|V(G)|$ holds. isolating set isolation game paths and cycles trees izolacijska množica izolacijska igra poti in cikli drevesa true false true Angleški jezik Slovenski jezik Neznano 2026-06-12 09:14:42 2026-06-12 09:14:42 2026-06-13 03:52:10 0000-00-00 00:00:00 2026 0 0 22 str. no. 2 article no. 33 Vol. 28 2026 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 519.17 1365-8050 10.46298/dmtcs.16132 281382915 2507.08503.pdf 2507.08503.pdf 1 2ACD69B2C51D44023D5F9D8BEE74C84A 0b05e6c9767f4dbf577a48ccbe28a78f44567829c76b4d850c4292a860598fcb 462f662e-662f-11f1-aebd-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=45281 https://dmtcs.episciences.org/18488 1 636d307a-662e-11f1-aebd-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=45280 Inštitut za matematiko, fiziko in mehaniko 0 0 0