Let G be a finite simple graph on the vertex set V(G) and let r be a positive integer. We consider the hypergraph Conr(G) whose vertices are the vertices of G and the (hyper)edges are all A⊆V(G) such that |A|=r+1 and the induced subgraph G[A] is connected. The (hyper)edge ideal Ir(G) of Conr(G) is also the Stanley-Reisner ideal of a generalisation of the independence complex of G, called the r-independence complex Indr(G). In this article we make extensive use of the Mayer-Vietoris sequence to find the graded Betti numbers of Ir(G1∗G2) in terms of the graded Betti numbers of Ir(G1) and Ir(G2), where G1∗G2 is the join of G1 and G2. Moreover, we find formulas for the graded Betti numbers and the Castelnuovo-Mumford regularity of Ir(G) when G is a complete graph, complete multipartite graph, cycle graph, and the wheel graph.