On semicube graphs

Eppstein [D. Eppstein, The lattice dimension of a graph, European J. Combin. 26 (2005) 585–592] introduced semicube graphs as the key tool for efficient computation of the lattice dimension of a graph. In this paper it is shown that, roughly speaking, every graph can be realized as the semicube graph of some partial cube. Semicube graphs of trees are studied in detail. In particular the chromatic number, the independence number and the domination number of semicube graphs of trees are determined in terms of related invariants of trees.