Peripheral \texorpdfstring{$Θ$}{Theta}-classes and forbidden partial cube-minors of daisy cubes

Abstract

Daisy cubes are partial cubes whose vertices can be represented by a down-set of a Boolean lattice. This paper gives a label-free characterization: a finite partial cube is a daisy cube if and only if every Djoković--Winkler $Θ$-class is peripheral. The proof orients each $Θ$-class toward a peripheral halfspace and shows that the resulting $Θ$-coordinate labels are closed downward. The characterization turns recognition into a condition on the halfspace structure and gives an exact obstruction formulation: the minimal forbidden pc-minors for daisy cubes are precisely the pc-minor-minimal partial cubes containing a non-peripheral $Θ$-class. We also give an infinite product family of such obstructions. For all $r\ge 2$ and $s\ge 1$, the graph obtained from $P_3^{\square r}\sq Q_s$ by deleting the two opposite corners is a minimal forbidden partial cube-minor for the class of daisy cubes.