Recognizing generalized Sierpiński graphs

Abstract

Let H be an arbitrary graph with vertex set V (H) = [nH] = {l,?, nH}. The generalized Sierpi?ski graph SnH , n ? N, is defined on the vertex set [nH]n, two different vertices u = un ?u1 and v = vn ? v1 being adjacent if there exists an h? [n] such that (a) ut = vt, for t > h, (b) uh ? vh and uhvh ? E(H), and (c) ut = vh and vt = uh for t < h. If H is the complete graph Kk, then we speak of the Sierpi?ski graph Sn k . We present an algorithm that recognizes Sierpi?ski graphs Sn k in O(|V (Sn k )|1+1=n) = O(|E(Sn k )|) time. For generalized Sierpi?ski graphs SnH we present a polynomial time algorithm for the case when H belong to a certain well defined class of graphs. We also describe how to derive the base graph H from an arbitrarily given SnH .