The authors introduce the generalized saturation game, where two players, Max and Mini, alternately claim edges of the complete graph Kn such that the graph of the claimed edges remains F-free throughout the game. The H-score of the game is the number of copies of H in the final graph, and the players aim to maximize or minimize this score, respectively.
The authors first study path-saturation games with star scores. They show that for paths Ps with s ≥ 7 and stars Sℓ with ℓ ≥ 3, the H-score is Θ(nℓ-1). They also determine the exact H-scores for smaller path lengths.
Next, the authors consider S4-saturation games, where they analyze the number of P3's and P5's in the final graph. They provide a complete characterization of the P3-score for n ≤ 12 and give upper and lower bounds for larger n.
The authors then study cycle-free saturation games, where the final graph is a forest. They obtain tight bounds on the Sk-score and P4-score in this setting.
Finally, they address P5-saturation games, determining the number of triangles and paths of length 3 in the final graph.
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