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Test if a strongly connected mixed graph is Eulerian or unicursal

Contributed by:
Peter Cullen Burbery

Eulerian graphs have applications in arc routing and operations research.

A mixed graph that is strongly connected is Eulerian, or unicursal, if and only if the graph is even (meaning all nodes/vertices are connected to an even number of other nodes/vertices and have an even degree) and the graph satisfied the balanced set condition, which states that for any subset of vertices/nodes, the difference between the number of arcs entering and the number of arcs leaving is less than or equal to the number of edges incident with the vertices/nodes.

Test if a mixed graph is Eulerian:

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Although this graph is even, the graph violates the balanced set condition and is therefore not Eulerian:

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A graph that is beyond a limit of around 14 nodes can take an unreasonable amount of time because the number *n* of subsets grows exponentially according to O(2^{n}):

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- 1.0.0 – 23 August 2022

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