  We study the NP-hard Minimum Shared Edges (MSE) problem on graphs: decide
whether it is possible to route $p$ paths from a start vertex to a target
vertex in a given graph while using at most $k$ edges more than once. We show
that MSE can be decided on bounded (i.e. finite) grids in linear time when both
dimensions are either small or large compared to the number $p$ of paths. On
the contrary, we show that MSE remains NP-hard on subgraphs of bounded grids.
Finally, we study MSE from a parametrised complexity point of view. It is known
that MSE is fixed-parameter tractable with respect to the number $p$ of paths.
We show that, under standard complexity-theoretical assumptions, the problem
parametrised by the combined parameter $k$, $p$, maximum degree, diameter, and
treewidth does not admit a polynomial-size problem kernel, even when restricted
to planar graphs.
