Graph complexity measures and monotonicity

The structural complexity of instances of computational problems has been an important research area in theoretical computer science in the last decades. Its success manifested itself in tree-width—a complexity measure for undirected graphs. Many practically relevant problems that are difficult in general, can be efficiently solved on graphs of bounded tree-width. For directed graphs, there are several similar measures, among others entanglement, directed tree-width, DAG-width and Kelly-width, that have shown their advantages as well as disadvantages. In this thesis, we work on complexity measures for directed graphs that can be described in terms of graph searching games. A graph searching game is played on a given graph by a team of cops and a robber. The cops try to capture the robber whose intent is to escape. Precise rules define a game and a complexity measure on graphs. The complexity of a graph is usually the minimal number of cops needed to capture the robber. Hereby, an important issue is monotonicity, a property of winning cop strategies. It implies the existence of suitable decompositions of the given graphs which allow for efficient algorithms for difficult computational problems. We discuss monotonicity issues of graph searching games, discuss problems appearing when monotonicity is introduced, and develop methods for solving those problems. In the first part, we consider the game defining entanglement. While, in general, there are no monotone winning strategies for the cops in the usual sense, we prove a weak variant of monotonicity for two cops. This implies the existence of corresponding graph decompositions. The second part is dedicated to the relationship between structural complexity and imperfect information. The latter is a powerful tool to extend expressiveness of games. The most involved result in that part is that parity games with bounded imperfect information can be efficiently solved on classes of directed graphs of bounded DAG-width. Its proof heavily relies on a notion of games with multiple robbers. We develop the technique of omitting cop placements while transforming strategies. The crucial question about monotonicity is whether monotonicity costs, the number of additional cop needed to capture the robber monotonically, can be bounded by a constant. We analyse the only known example by Kreutzer and Ordyniak which shows that the monotonicity costs for DAG-width can be positive. We define a notion of weak monotonicity for DAG-width and investigate the problem of boundedness of weak monotonicity costs. Furthermore, we define a structural decomposition of graphs that corresponds to a winning cop strategy in a game with weak monotonicity. The last part gives an overview of known results concerning boundedness of values of different measures on the same graph classes.