every (row,column) position contains exactly two vertices, which we refer to as the *true vertex* and the *false vertex*, and we say that these two vertices are *mates*. The first $n$ rows form the *variable block* of the graph and we think of row $i$ as representing the variable $v_i$. The next $5m$ rows are made up of $m$ *clause blocks* ${\mathcal C}^{k,1}, {\mathcal C}^{k,2}, \ldots, {\mathcal C}^{k,m}$, each of five rows. Every true vertex of the variable and clause blocks is joined by an edge to every false vertex in the same row except its mate. Hence the vertices of each row induce a complete bipartite graph minus a matching. In the final row, each column contains a single vertex, and each of these vertices is joined by an edge to every other vertex in the same column. We call this row the *dominating block*. We complete the definition of the clause graph by describing how we add further edges so that it contains the constraint graph $J$ as an induced subgraph. Let the literals of $C_k$ be $x_{\ell_1},x_{\ell_2},x_{\ell_3}$. We choose vertices from the first three columns of the variable block of ${\mathcal C}^k$ that we will denote $X_1^k, 