Cops, Robber and Medianwidth Parameters

In previous work, we introduced median decompositions of graphs, a generalisation of tree decompositions where a graph can be modelled after any median graph, along with a hierarchy of i-medianwidth parameters (mwi)i≥1 starting from treewidth and converging to the clique number. We introduce another graph parameter based on the concept of median decompositions, to be called i-latticewidth and denoted by lwi, for which we restrict the modelling median graph of a decomposition to be isometrically embeddable into the Cartesian product of i paths. The sequence (lwi)i≥1 gives rise to a hierarchy of parameters starting from pathwidth and converging to the clique number. We characterise the i-latticewidth of a graph in terms of maximal intersections of bags of i path decompositions of the graph. We study a generalisation of the classical Cops and Robber game, where the robber plays against not just one, but i cop players. Depending on whether the robber is visible or not, we show a direct connection to i-medianwidth or i-latticewidth, respectively.