Is often a maximum matching of G . Direct the edges in G as Ed = a, r M M ( a, r). All edges amongst G and G get started at the variables in G and finish in the equations in G . For the under-constrained variables in G , no feasible path enters G. For that reason, no new under-constrained node might be observed in G . Moreover, no new augment path exists in G , simply because all nodes in G are covered by the maximum matching M M . No new matching edge might be found. As a result, the under-constrained nodes in G are still within the under-constrained aspect of G . Therefore, the under-constrained aspect of G equals the under-constrained aspect of G , and Lemma three is proven. four.1.2. Construction from the Dummy Model To get a hierarchical EoM m = ( A, S, R), the dummy model is Zingerone NF-��B constructed according to the decomposing outcome of every single element. The equations within the dummy model are a subset in the equations within the flattened model m. The structural analysis on the dummy model can reveal the structural singularity from the original model m. Definition 9. The dummy model of an NLAE model m = ( A, S, R) is defined as u u u u ^ m = A iS Ai , , R iS Ri , exactly where Ai and Ri are variables and equations in the under-constrained component of each and every element, respectively. The pseudocode of constructing the dummy model for an NLAE model is presented as follows. Our remarks on Algorithm two are as follows: 1. 2. In line 3, the function constructs a bipartite graph for the component mi . In line 4, the function decompose(mi) is an implementation of Algorithm 1. It decomposes a element mi and returns the variable set along with the equation set inside the under-constrained element. In the event the element set is empty, the dummy model is equivalent to the flattened model.3.Mathematics 2021, 9,13 of^ Algorithm 2. Building in the dummy model m. ^ Input a model m = ( A, S, R); output the dummy model m. 1: 2: three: 4: 5: six: 7: ^ ^ Let A = A, R = R; for every mi = ( Ai , Si , Ri) S, do Gi = bipartiteGraph(mi) u u Ai , Ri = decompose( Gi); ^ = A Au ; ^ let A i ^ = R Ru ; ^ let R i ^ ^ ^ ^ let m be the dummy model of m, m = ( A, , R);Theorem 1. For an NLAE model, the structural singularities from the dummy model and the flattened model are equivalent. Proof. Assume an NLAE model m = ( A, S, R). The corresponding flattened model is denoted as m = A, , R , where A and R Lactacystin Epigenetic Reader Domain represent the union sets in the variables and ^ ^ ^ equations in m and its components. The dummy model of m is denoted as m = A, , R . ^ If m is a key model, the element set S is empty. Clearly, A = A = A and ^ R = R = R. The dummy model along with the flattened model include the same variables and equations. Their structural singularities are equivalent. If m is often a first-level model, each element mi = ( Ai , , Ri) S is usually a major model. Algorithm 1 decomposes each component mi into the under-constrained component u u w w Giu = Ai Ri , Eiu and also the well-constrained portion Giw = Ai Ri , Eiw . u Aw and R = Ru Rw , the According to Definition 9 along with the assumptions Ai = Ai i i i i ^ dummy model m satisfies the following: ^ A = A( ^ = R( Ru w w i S Ai) = A ( i S Ai) – ( i S Ai) = A – ( i S Ai) , w u w iS Ri) = R ( iS Ri) – ( iS Ri) = R – ( iS Ri).(five)For each and every element mi , a well-constrained model mw = (Aw , , Rw) is usually built i i i with all the variables and equations inside the well-constrained aspect. According to Lemma two, ^ the over-constrained element of m is often a subgraph from the over-constrained element of your model ^ ^ mw = (Aw A, , Rw R). Beneath the assumption that th.