Before revision:  It generalizes a recent technique developed for the case that G is a cycle and the embedding is a simple polygon Akitaya et al., 2016 , and combines local constraints on the orientation of subgraphs directly, thereby eliminating the need for solving large systems of linear equations.
After revision:  It generalizes a recent technique developed for the case that G is a cycle and the embedding is a simple polygon Akitaya et al., 2016 combines local constraints on the orientation of subgraphs directly, thereby eliminating the need for solving large systems of linear equations.