Tree-like isometric subgraphs of hypercubes

Tree-like isometric subgraphs of hypercubes, or tree-like partial cubes as we shall call them, are a generalization of median graphs. Just as median graphs they capture numerous properties of trees, but encompass a larger class of graphs that may be easier to recognize than the class of median graphs. We investigate the structure of tree-like partial cubes, characterize them, and provide examples of similarities with trees and median graphs. For instance, we show that every treelike partial cube G contains a cube that is invariant under every automorphism of G. This property is shared with median graphs and generalizes the existence of a center in trees. Furthermore, we establish a connection with dismantlable graphs, and show that every finite group is isomorphic to the automorphism group of a tree-like partial cube and, more restrictively, to that of a median graph.