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PM contains at least one crossing edge, which guarantees that an empty match does not qualify.
This paper presents a novel noise robust edge detector based upon the automatic anisotropic Gaussian kernels (ANGKs), which also addresses the current problem that the seminal Canny edge detector may miss some obvious crossing edge details.
The second condition says that two local partial matches share at least one common crossing edge that corresponds to the same query edge.
Furthermore, they share a common crossing edge (overrightarrow{008,003}), where 008 and 003 match query vertices (v_3) and (v_5) in the two local partial matches, respectively.
Suppose at some step (A, B) there is no crossing edge and (B) is not empty, this would imply (G[A cup B] = G[ H_k] is not connected, which contradicts the assumption.
Then at each step, we move one node from B to A, by choosing the crossing edge(one endpoint in A, the other in B) whose degree of the node in A is minimum compared with other crossing edges.
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Every time the while loop is executed, however, the number of crossing edges increases by at least one, and there are E edges, so the loop terminates in at most E iterations.
The number of crossing edges in the original network was then compared with the distribution of number of crossing edges seen in 1,000 randomized networks.
For these data, the optimal partitioning had 102 crossing edges, which, although it appears to be a large number, is significantly smaller than the minimum number of crossing edges seen in the randomized networks (P < 0.001).
We applied a force-directed layout to minimize the number of crossing edges and to enhance readability of node labels.
Nevertheless, this arrangement problem is in general much smaller than the complexity of minimizing all crossing edges at once.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com