Let G be a finite simple graph on the vertex set V(G) and let r be a positive integer. We consider the hypergraph \(\textrm{Con}_r(G)\) whose vertices are the vertices of G and the (hyper)edges are all \(A\subseteq V(G)\) such that \(|A|=r+1\) and the induced subgraph G[A] is connected. The (hyper)edge ideal \(I_r(G)\) of \(\textrm{Con}_r(G)\) is also the Stanley-Reisner ideal of a generalisation of the independence complex of G, called the r-independence complex \(\textrm{Ind}_r(G)\) . In this article we make extensive use of the Mayer-Vietoris sequence to find the graded Betti numbers of \(I_r(G_1*G_2)\) in terms of the graded Betti numbers of \(I_r(G_1)\) and \(I_r(G_2)\) , where \(G_1*G_2\) is the join of \(G_1\) and \(G_2\) . Moreover, we find formulas for the graded Betti numbers and the Castelnuovo-Mumford regularity of \(I_r(G)\) when G is a complete graph, complete multipartite graph, cycle graph, and the wheel graph.
