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.

Nordhaus and Gaddum proved in 1956 that the sum of the chromatic number χ of a graph G and its complement is at most | G | + 1 . The Nordhaus–Gaddum graphs are the class of graphs satisfying this inequality with equality, and are well-understood. In this paper we consider a hereditary generalization: graphs G for which all induced subgraphs H of G satisfy χ ( H ) + χ ( H ¯ ) ≥ | H | . We characterize the forbidden induced subgraphs of this class and find its intersection with a number of common classes, including line graphs. We also discuss χ -boundedness and algorithmic results.