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Tree-like isometric subgraphs of hypercubes
Boštjan Brešar, Wilfried Imrich, Sandi Klavžar, 2003, original scientific article

Abstract: Tree-like isometric subgraphs of hypercubes, or tree-like partial cubes as we 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 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.
Keywords: mathematics, graph theory, Isometric embeddings, partial cubes, expansion procedures, trees, median graphs, graph automorphisms, automorphism groups, dismantlable graphs
Published: 31.03.2017; Views: 561; Downloads: 208
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