# Daisy cubes and distance cube polynomial

## TL;DR

This paper introduces daisy cubes, a class of partial cubes including Fibonacci and Lucas cubes, and explores their distance cube polynomial, revealing key polynomial relationships and invariances in these structures.

## Contribution

It defines daisy cubes as a new class of partial cubes and establishes fundamental properties of their distance cube polynomials, linking them to existing cube polynomials.

## Key findings

- Daisy cubes include Fibonacci and Lucas cubes.
- Distance cube polynomial D G,0 n (x, y) equals the cube polynomial C G (x + y - 1).
- For any vertex u in a daisy cube, D G,u (x, -x) equals 1.

## Abstract

Let X $\subseteq$ {0, 1} n. Then the daisy cube Q n (X) is introduced as the sub-graph of Q n induced by the intersection of the intervals I(x, 0 n) over all x $\in$ X. Daisy cubes are partial cubes that include Fibonacci cubes, Lucas cubes, and bipartite wheels. If u is a vertex of a graph G, then the distance cube polynomial D G,u (x, y) is introduced as the bivariate polynomial that counts the number of induced subgraphs isomorphic to Q k at a given distance from the vertex u. It is proved that if G is a daisy cube, then D G,0 n (x, y) = C G (x + y -- 1), where C G (x) is the previously investigated cube polynomial of G. It is also proved that if G is a daisy cube, then D G,u (x, --x) = 1 holds for every vertex u in G.