Independent-Set Threshold Games and Geodetic Removal on Odd Cycles

In the geodetic removal game, players select vertices until the convex hull of the unselected vertices, taken along shortest paths in the original graph, ceases to be the whole graph. We prove that the second player wins on every odd cycle of order at least five, resolving a conjecture of Benesh, Ernst, Meyer, Salmon and Sieben. The proof identifies the terminal sets with maximum independent sets of an auxiliary cycle and uses a dynamic pairing strategy. More generally, consider the impartial game in which vertices are selected until their union contains an independent set of a prescribed size r, with r at least 2. On every path or cycle of order at least 2r, the second player can force termination on exactly move 2r - 2. We classify the remaining feasible path thresholds and prove the same exact-turn result for feasible thresholds on bipartite graphs with a given perfect matching. The strategies maintain paired selected vertices until a final move deliberately breaks the pairing. They admit constant-time responses after linear initialization.