We study a two-player game played on undirected graphs called Trail Trap , which is a variant of a game known as Partizan Edge Geography . One player starts by choosing any edge and moving a token from one endpoint to the other; the other player then chooses a different edge and does the same. Alternating turns, each player moves their token along an unused edge from its current vertex to an adjacent vertex, until one player cannot move and loses. We present an algorithm to determine which player has a winning strategy when the graph is a tree and partially characterize the trees on which a given player wins. Additionally, we show that it is NP-hard to determine if Player 2 has a winning strategy on Trail Trap from the starting position, even for connected bipartite planar graphs with maximum degree 4. We determine which player has a winning strategy for certain subclasses of complete bipartite graphs and grid graphs, and we propose several open problems for further study.

The localization game is a variant of the game of Cops and Robber in which the robber is invisible and moves between adjacent vertices, but the cops can probe any k vertices of the graph to obtain the distance between probed vertices and the robber. The localization number of a graph is the minimum k needed for the cops to be able to locate the robber in finite time. The localization capture time is the minimum number of rounds needed for the cops to locate the robber. The localization capture time conjecture claims that there exists a constant C such that the localization number of every connected graph on n vertices is at most Cn . While it is known that the conjecture holds for trees, in this paper we significantly improve the known upper bound for the localization capture time of trees. We also prove the conjecture for a subclass of outerplanar graphs and present a generalization of the localization game that appears useful for making further progress towards the conjecture.

The connected domination game is a variant of the domination game where the played vertices must form a connected subgraph at all stages of the game. In this paper we prove that deciding whether the game connected domination number is smaller than a given integer is PSPACE-complete using log-space reductions for both Dominator- and Staller-start connected domination game.