Star structure connectivities of pancake graphs and burnt pancake graphs

Let H be a connected subgraph of a graph G. The H-structure connectivity of G is the cardinality of a minimum set of subgraphs in G, whose deletion disconnects G and every element in the set is isomorphic to H. Similarly, the H-substructure connectivity of G is the cardinality of a minimum set of subgraphs in G, whose deletion disconnects G and every element in the set is isomorphic to a connected subgraph of H. Structure connectivity and substructure connectivity generalise the classic connectivity. Let and be the n-dimensional pancake graph and n-dimensional burnt pancake graph, respectively. In this paper we show , and .