  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 $(mw_i)_{i\geq 1}$ 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
embeddable into the Cartesian product of $i$ paths. The sequence $(lw_i)_{i\geq
1}$ 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.
