Abstract

We fill in the details of the algorithm sketched in [Sh] and determine its complexity. As a part of this main algorithm, we also describe an algorithm which recognizes graphs which are isometric subgraphs of halved cubes. We discuss possible further applications of the same ideas and give a nice example of a non ` 1 -graph allowing a highly isometric embedding into a halved cube. 1 Introduction For a set \Omega\Gamma let 2\Omega denote the set of all the subsets of \Omega\Gamma We turn 2\Omega into an n-dimensional cube graph Q k , where k = j\Omega j, by making two subsets A and B adjacent whenever the symmetric difference A4B has size 1. The graph Q k is bipartite, the bipartite half of Q k being known as the halved cube graph (we denote it HQ k ). Therefore, HQ k can be defined as the graph on the even size subsets in 2\Omega , in which two such subsets A and B are adjacent whenever jA4Bj = 2. We usually identify a graph \Gamma with its set of vertices, and we use the same notati...

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