Let us create a small imaginary trail graph in Python with NetworkX, and play around with plotting and paths. Suppose we have a graph with nodes from 0 to 15:

import networkx as nx import matplotlib.pyplot as plt G=nx.Graph() edges=[(0,1,1),(1,2,1),(2,3,1),(3,4,1),(4,5,1),(5,6,1),(6,7,1), (8,3,2),(9,4,1),(9,5,1),(10,7,2), (8,9,2),(9,10,2), (8,11,2),(12,10,2), (11,12,2),(12,13,2),(13,7,2), (11,14,2),(14,15,4),(15,13,1)] G.add_weighted_edges_from(edges)

The tuple (8,11,2) means an edge goes from node 8 to node 11, with weight 2.

Now we specify where the nodes are located.

#pos=nx.spring_layout(G) does not give satisfactory results. # Draw node positions the hard way. pos=[(0,4),(1,4),(2,4),(3,4),(4,4),(5,4),(6,4),(7,4), (1,3),(3,3),(5,3), (3,2),(6,2),(8,2), (4,1),(7,1)]

We are almost ready to draw the network. Since we want to draw weights for the edges, we have an extra step.

options = { "font_size": 16, "node_size": 1000, "node_color": "white", "edgecolors": "black", "linewidths": 5, "width": 5, } nx.draw_networkx(G, pos,**options) labels = {e: G.edges[e]['weight'] for e in G.edges} nx.draw_networkx_edge_labels(G, pos, edge_labels=labels, font_color="red") plt.show()

And here is our result.

Think how we can convert this into a Euler graph, so all nodes have an even number of edges. We can try a Networkx library function, but should not expect it will do what we want, because it will create a multigraph, with more than one edge between nodes.

H = nx.eulerize(G)

Our command to plot the resulting graph has to be a bit more complicated, to allow for showing more than one edge between pairs of nodes. [hat-tip]

nx.draw_networkx_nodes(H, pos, node_color="white", node_size=1000) ax = plt.gca() for e in H.edges: ax.annotate("", xy=pos[e[0]], xycoords='data', xytext=pos[e[1]], textcoords='data', arrowprops=dict(arrowstyle="->", color="0.5", shrinkA=5, shrinkB=5, patchA=None, patchB=None, connectionstyle="arc3, rad=rrr". replace('rrr',str(0.3*e[2]) ), ), ) plt.axis('off') plt.show()

Well, we expected to eliminate nodes 0-2, as a dead-end on our Euler path. After that, there are 4 pairs of nodes with multiple edges. If we remove every double-edge, we hope to still have an Euler graph. Let’s write code to remove all double-edges. Python does not like when we modify a graph during a for-loop, so a better practice is to create a new graph, copying all edges that are not double-edges.

J = nx.Graph() for node in H.nodes(): # We look for adjacent nodes neighbors = H.neighbors(node) #Find all neighbors of node n for adj_node in neighbors: if len(H[node][adj_node]) == 1: # get weight of the edge w = H.edges[node,adj_node,0]['weight'] J.add_edge(node, adj_node, weight = w)

A subtlety here for newbie NetworkX users like myself is to use

>> *H.edges[node,adj_node,0][‘weight’]*

instead of

>> *H.edges[node,adj_node][‘weight’]*

because H is a multigraph, and could possibly have more than one edge between each pair of nodes. Since J is a simple graph, it should not matter if we add an edge twice, since the edge would just get overwritten. Plotting the result:

The output is not ideal, because our naive algorithm disconnected the graph.

Alternatively, manually choose better edges to delete , and defer the max-path algorithm to another post.

G.remove_edges_from([(0,1),(1,2),(2,3),(4,5),(8,9),(9,10), (11,12),(7,13)]) G.remove_nodes_from([0,1,2]) print("Is G an Eulerian graph? : ",nx.is_eulerian(G)) print(list(nx.eulerian_path(G,source=3)))

```
Is G an Eulerian graph? : True
[(3, 4), (4, 9), (9, 5), (5, 6), (6, 7), (7, 10), (10, 12), (12, 13), (13, 15), (15, 14), (14, 11), (11, 8), (8, 3)]
```