Summary: Let \(K_m\) be the complete graph on \(m\) vertices and let \(H\) be a subgraph of it. Let \(K_m-H\) be the graph obtained from \(K_m\) by removing the edge set \(E(H)\) of \(H\). We use the symbol \(Z_4\) to denote \(K_4-P_2\). A sequence \(S\) is potentially \(K_m-H\)-graphical if it has a realization containing a \(K_m-H\) as a subgraph. Let \(\sigma (K_m-H,n)\) denote the smallest degree sum such that every \(n\)-term graphical sequence \(S\) with \(\sigma (S)\geq \sigma (K_m-H,n)\) is potentially \(K_m-H\)-graphical. In this paper, we determine the values of \(\sigma (K_{r+1}-Z,n)\) for \(n\geq 5r+19\), \(r+1\geq k\geq 5\), \(j\geq 5\), where \(Z\) is a graph on \(k\) vertices and \(j\) edges which contains a graph \(Z_4\) but does not contain a cycle on 4 vertices. We also determine the values of \(\sigma (K_{r+1}-Z_4,n)\), \(\sigma (K_{r+1}-(K_4-e),n)\) and \(\sigma (K_{r+1}-K_4,n)\) for \(n\geq 5r+16\), \(r\geq 4\).