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 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.