On the Medianwidth of Graphs

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, grids 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 median graphs as the underlying graph of the decomposition. We show that the corresponding width parameter mw(G), themedianwidth 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 mwi(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 ≥ 1, the parameters mwi 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 in any fixed graphlike fashion.