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The game saturation number of a graph. (English) Zbl 1365.05195
Summary: 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; $$\mathrm{sat}_g(\mathcal{F};H)$$ denotes the length under optimal play (when Max starts).
Let $$\mathcal{O}$$ denote the family of odd cycles and $$\mathcal{T}_n$$ the family of $$n$$-vertex trees, and write $$F$$ for $$\mathcal{F}$$ when $$\mathcal{F}=\{F\}$$. Our results include $$\mathrm{sat}_g(\mathcal{O};K_n)=\lfloor\frac n2\rfloor\lceil\frac n2\rceil,\;\mathrm{sat}_g(\mathcal{I}_n;K_n)=\tbinom{n-2}{2}+1$$ for $$n\geq6,\;\mathrm{sat}_g(K_{1,3};K_n)=2\lfloor\frac n2\rfloor$$ for $$n\geq8$$, and $$\mathrm{sat}_g(P_4;K_n)\in\{\lfloor\frac{4n}5\rfloor,\lceil\frac{4n}5\rceil\}$$ for $$n\geq5$$. We also determine $$\mathrm{sat}_g(P_4;K_{m,n})$$; with $$m\geq n$$, 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 $$\mathrm{sat}_g(C_4;K_{n,n})\geq\frac1{21}n^{13/12}-O(n^{35/36})$$. The results are very similar when Min plays first, except for the $$P_4$$-saturation game on $$K_{m,n}$$.

##### MSC:
 05C57 Games on graphs (graph-theoretic aspects) 91A43 Games involving graphs
##### Keywords:
saturation number; graph game; forbidden subgraph; spanning tree
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