Brešar, Boštjan; Imrich, Wilfried; Klavžar, Sandi Tree-like isometric subgraphs of hypercubes. (English) Zbl 1055.05129 Discuss. Math., Graph Theory 23, No. 2, 227-240 (2003). Summary: 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 may contain larger classes 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 the cube graph of a tree-like partial cube is dismantlable. This in particular implies that every tree-like partial cube \(G\) contains a cube that is invariant under every automorphism of \(G\). We also show that weak retractions preserve tree-like partial cubes, which in turn implies that every contraction of a tree-like partial cube fixes a cube. The paper ends with several Frucht-type results and a list of open problems. Cited in 2 ReviewsCited in 7 Documents MSC: 05C75 Structural characterization of families of graphs 05C12 Distance in graphs 05C05 Trees Keywords:isometric embeddings; graph automorphisms; automorphism groups; dismantlable graphs; tree-like partial cubes; median graphs; trees; open problems PDFBibTeX XMLCite \textit{B. Brešar} et al., Discuss. Math., Graph Theory 23, No. 2, 227--240 (2003; Zbl 1055.05129) Full Text: DOI Link