Fig. 3

The same example of Fig. 2 but modeled using end-point weight graphs. Each node is represented using an arrow-like shape with two-matching sides. Only opposite sides having the same weight can be matched. The numbers inside the shapes represent the end-point weights; the extra darker square contains the node identifier. An arrow oriented from left-to-right models a node with positive sign; vice versa, an arrow oriented from right-to-left models a node with negative sign. Gray edges represent edges that cannot be traversed without changing the orientation of one of the two connected nodes. Black edges represent edges that can be traversed. Lastly, we highlight in red the edges that belong to paths in a graph cover. The example in a corresponds to that of Fig. 2b where no node has changed orientation and, therefore, we have three paths in the cover: \((+4 \rightarrow +2)\), \((+3)\), and \((+1)\). Other two different covers are shown in b and c. In b the cover contains the single path \((+1 \rightarrow -3 \rightarrow +4 \rightarrow +2)\) and corresponds to the example of Fig. 2c where the node 3 was changed orientation from \(+\) to—(shown in yellow color). In c the cover contains the two paths \((+2 \rightarrow -3 \rightarrow +4)\) and the singleton path \((+1)\)