Daisy cubes: a characterization and a generalization

05/17/2019

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by   Andrej Taranenko, et al.

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Daisy cubes are a recently introduced class of isometric subgraphs of hypercubes Q_n. They are induced with intervals between chosen vertices of Q_n and the vertex 0^n∈ V(Q_n). In this paper we characterize daisy cubes in terms of an expansion procedure thus answering an open problem proposed by Klavžar and Mollard, 2018, in the introductory paper of daisy cubes KlaMol-18. To obtain such a characterization several interesting properties of daisy cubes are presented. For a given graph G isomorphic to a daisy cube, but without the corresponding embedding into a hypercube, we present an algorithm which finds a proper embedding of G into a hypercube in O(mn) time. Finally, daisy graphs of a rooted graph are introduced and shown to be a generalization of daisy cubes.