The Sierpiński product of graphs

In this paper we introduce a product-like operation that generalizes the construction of generalized Sierpiński graphs. Let \(G,H\) be graphs and let \(f: V(G) \to V(H)\) be a function. Then the Sierpiński product of \(G\) and \(H\) with respect to \(f\) is defined as a pair \((K,\varphi)\), where \(K\) is a graph on the vertex set \(V(G) \times V(H)\) with two types of edges: -- \(\{(g,h),(g,h')\}\) is an edge in \(K\) for every \(g\in V(G)\) and every \(\{h,h'\}\in E(H)\), -- \(\{(g,f(g'),(g',f(g))\}\) is an edge in \(K\) for every edge \(\{g,g'\} \in E(G)\); and \(\varphi: V(G) \to V(K)\) is a function that maps every vertex \(g \in V(G)\) to the vertex \((g,f(g)) \in V(K)\). Graph \(K\) will be denoted by \(G\otimes_f H\). Function \(\varphi\) is needed to define the product of more than two factors. By applying this operation \(n\) times to the same graph we obtain the \(n\)-th generalized Sierpiński graph. Some basic properties of the Sierpiński product are presented. In particular, we show that \(G \otimes_f H\) is connected if and only if both \(G\) and \(H\) are connected and we present some necessary and sufficient conditions that \(G,H\) must fulfill in order for \(G \otimes_f H\) to be planar. As for symmetry properties, we show which automorphisms of \(G\) and \(H\) extend to automorphisms of \(G \otimes_f H\). In many cases we can also describe the whole automorphism group of \(G\otimes_f H\).