Daisy cubes and distance cube polynomial

Abstract

Let X⊆{0,1}n. The daisy cube Qn(X) is introduced as the subgraph of Qn induced by the union of the intervals I(x,0n) over all x∈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 DG,u(x,y) is introduced as the bivariate polynomial that counts the number of induced subgraphs isomorphic to Qk at a given distance from the vertex u. It is proved that if G is a daisy cube, then DG,0n(x,y)=CG(x+y−1), where CG(x) is the previously investigated cube polynomial of G. It is also proved that if G is a daisy cube, then DG,u(x,−x)=1 holds for every vertex u in G.