  A median graph is a connected graph, such that for any three vertices $u,v,w$
there is exactly one vertex $x$ that lies simultaneously on a shortest
$(u,v)$-path, a shortest $(v,w)$-path and a shortest $(w,u)$-path. Examples of
median graphs are trees and hypercubes.
  We introduce and study a generalisation of tree decompositions, to be called
median decompositions, where instead of decomposing a graph $G$ in a treelike
fashion, we use general median graphs as the underlying graph of the
decomposition. We show that the corresponding width parameter $\text{mw}(G)$,
the medianwidth of $G$, is equal to the clique number of the graph, while a
suitable variation of it is equal to the chromatic number of $G$.
  We study in detail the $i$-medianwidth $\text{mw}_i(G)$ of a graph, for which
we restrict the underlying median graph of a decomposition to be isometrically
embeddable to the Cartesian product of $i$ trees. For $i\geq 1$, the parameters
$\text{mw}_i$ constitute a hierarchy starting from treewidth and converging to
the clique number. We characterize the $i$-medianwidth of a graph to be,
roughly said, the largest "intersection" of the best choice of $i$ many tree
decompositions of the graph.
  Lastly, we extend the concept of tree and median decompositions and propose a
general framework of how to decompose a graph $G$ in any fixed graphlike
fashion.
