The Game Saturation Number of a Graph

Authors

James M. Carraher
William B. Kinnersley, University of Rhode IslandFollow
Benjamin Reiniger
Douglas B. West

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_n) = 2[T_n/2] for n ≥ 8, and sat_g(P₄; K_n) ∈ {[4n/f] , [4n/5]} for n ≥ 5. We also determine sat_g(P₄; K_m;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 sat_g(C4; K_n,n) ≥ 1/21n^13/12 – O(n^35/36). The results are very similar when Min plays first, except for the P₄-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