In previous work, we introduced median decompositions, a generalisation of tree decompositions where a graph can be modelled after any median graph, along with a hierarchy of i-medianwidth parameters (mw_i)_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 lw_i, for which we restrict the modelling median graph of a decomposition to be isometrically

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... able into the Cartesian product of i paths. The sequence (lw_i)_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.