#### Document Type

Article

#### Date of Original Version

2016

#### Abstract

Given a family *F* and a host graph *H*, a graph *G* ⊆ *H* is *F*-saturated relative to H if no subgraph of *G* lies in* F* but adding any edge from *E*(*H*) - *E*(*G*) to *G* creates such a subgraph. In the *F*-saturation game on H, players Max and Min alternately add edges of *H* to *G*, avoiding subgraphs in *F*, until *G* becomes *F*-saturated relative to *H*. They aim to maximize or minimize the length of the game, respectively; sat* _{g}*(

*F*;

*H*) denotes the length under optimal play (when Max starts).

Let *O* denote the family of odd cycles and the family of *n*-vertex trees, and write *F* for when *F* = {*F*}. Our results include sat* _{g}*(

*O*;

*K*) = [

_{n}*n*/2] [

*n*/2], sat

_{g}(

*T*;

_{n}*K*) = (

_{n}*n*-2/2) + 1 for

*n*≥ 6, sat

*(*

_{g}*K*;

_{1,3}*K*) = 2[

_{n}*T*/2] for

_{n}*n*≥ 8, and sat

*(*

_{g}*P*

_{4};

*K*) ∈ {[4

_{n}*n*/f] , [4

*n*/5]} for n ≥ 5. We also determine sat

_{g}(

*P*

_{4};

*K*); with

_{m;n}*m*≥

*n*, it is

*n*when

*n*is even,

*m*when

*n*is odd and

*m*is even, and

*m*+ [

*n*/2] when

*mn*is odd. Finally, we prove the lower bound sat

*(*

_{g}*C*4;

*K*) ≥ 1/21

_{n,n}*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.}#### Citation/Publisher Attribution

Carraher, J. M., Kinnersley, W. B., Reiniger, B., & West, D. B. (2016). The Game Saturation Number of a Graph. *Journal of Graph Theory*, *85*(2), 481-495. doi:10.1002/jgt.22074

Available at: https://doi.org/10.1002/jgt.22074

#### Author Manuscript

This is a pre-publication author manuscript of the final, published article.

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