Fig. 5

Illustration of the recursion for the partition function. Given a bistructure X with exposed vertices \(v_1\) and \(v_2\), we decomposed X into a loop \(L(v_1, u_1, u_2, u_3)\) and a substructure \(X'(v_1, u_1, u_2, u_3, v_2)\). Here we assume \(v_1\), \(u_1\), \(u_2\), \(u_3\), and \(v_2\) are contained in loops of \(X'\), hence \(v_1\) and \(v_2\) remain exposed and \(u_1\), \(u_2\), and \(u_3\) are newly exposed vertices