The Game Saturation Number of a Graph

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; satg(F ;H) denotes the length under optimal play (when Max starts). Let O denote the family of odd cycles and Tn the family of n-vertex trees, and write F for F when F = {F}. Our results include satg(O;Kn) = ⌊ n 2 ⌋ ⌈ n 2 ⌉ , satg(Tn;Kn) = n−2 2 ) +1 for n ≥ 6, satg(K1,3;Kn) = 2 ⌊ n 2 ⌋ for n ≥ 8, and satg(P4;Kn) ∈ { ⌊ 4n 5 ⌋ , ⌈ 4n 5 ⌉ }. We also determine satg(P4;Km,n); with 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 satg(C4;Kn,n) ≥ 1 20.8n − O(n35/36). The results are very similar when Min plays first, except for the P4-saturation game on Km,n.