Every incidence structure ${\mathcal J}$ (understood as a triple of sets $(G, M, I)$, ${I}\subseteq G \times M$) admits for every positive integer $p$ an incidence structure ${\mathcal J}^p=(G^p, M^p, \mathrel {{\mathrm I}^p})$ where $G^p$ ($M^p$) consists of all independent $p$-element subsets in $G$ ($M$) and $\mathrel {{\mathrm I}^p}$ is determined by some bijections. In the paper such incidence structures ${\mathcal J}$ are investigated the ${\mathcal J}^p$’s of which have their incidence graphs of the simple join form. Some concrete illustrations are included with small sets $G$ and $M$.
A perfect independent set I of a graph G is defined to be an independent set with the property that any vertex not in I has at least two neighbors in I. For a nonnegative integer k, a subset I of the vertex set V (G) of a graph G is said to be k-independent, if I is independent and every independent subset I' of G with |I' | ≥ |I| − (k − 1) is a subset of I. A set I of vertices of G is a super k-independent set of G if I is k-independent in the graph G[I, V (G) − I], where G[I, V (G) − I] is the bipartite graph obtained from G by deleting all edges which are not incident with vertices of I. It is easy to see that a set I is 0-independent if and only if it is a maximum independent set and 1-independent if and only if it is a unique maximum independent set of G. In this paper we mainly investigate connections between perfect independent sets and k-independent as well as super k-independent sets for k = 0 and k = 1.