The Game Saturation Number of a Graph

Given a family ${\mathcal F}$ and a host graph $H$, a graph $G\subseteq H$ is ${\mathcal F}$-saturated relative to $H$ if no subgraph of $G$ lies in ${\mathcal F}$ but adding any edge from $E(H)-E(G)$ to $G$ creates such a subgraph. In the ${\mathcal F}$-saturation game on $H$, players Max and Min alternately add edges of $H$ to $G$, avoiding subgraphs in ${\mathcal F}$, until $G$ becomes ${\mathcal F}$-saturated relative to $H$. They aim to maximize or minimize the length of the game, respectively; $\textrm{sat}_g({\mathcal F};H)$ denotes the length under optimal play (when Max starts). Let ${\mathcal O}$ denote the family of all odd cycles and ${\mathcal T}$ the family of $n$-vertex trees, and write $F$ for ${\mathcal F}$ when ${\mathcal F}=\{F\}$. Our results include $\textrm{sat}_g({\mathcal O};K_{2k})=k^2$, $\textrm{sat}_g({\mathcal T};K_n)=\binom{n-2}{2}+1$ for $n\ge6$, $\textrm{sat}_g(K_{1,3};K_n)=2\lfloor n/2 \rfloor$ for $n\ge8$, $\textrm{sat}_g(K_{1,r+1};K_n)=\frac{rn}{2}-\frac{r^2}{8}+O(1)$, and $|\textrm{sat}_g(P_4;K_n)-(4n-1)/5|\le 1$. We also determine $\textrm{sat}_g(P_4;K_{m,n})$; with $m\ge n$, it is $n$ when $n$ is even, $m$ when $n$ is odd and $m$ is even, and $m+\lfloor n/2 \rfloor$ when $mn$ is odd. Finally, we prove the lower bound $\textrm{sat}_g(C_4;K_{n,n})\ge\frac{1}{10.4}n^{13/12}-O(n^{35/36})$. The results are very similar when Min plays first, except for the $P_4$-saturation game on $K_{m,n}$.

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