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An Eulerian trail, [note 1] or Euler walk, in an undirected graph is a walk that uses each edge exactly once. If such a walk exists, the graph is called traversable or semi-eulerian. [3] An Eulerian cycle, [note 1] also called an Eulerian circuit or Euler tour, in an undirected graph is a cycle that uses each edge exactly once
The Euler tour technique (ETT), named after Leonhard Euler, is a method in graph theory for representing trees. The tree is viewed as a directed graph that contains two directed edges for each edge in the tree. The tree can then be represented as a Eulerian circuit of the directed graph, known as the Euler tour representation (ETR) of the tree
For example, connectedness of zones might be enforced, or concurrency of curves or multiple points might be banned, as might tangential intersection of curves. In the adjacent diagram, examples of small Venn diagrams are transformed into Euler diagrams by sequences of transformations; some of the intermediate diagrams have concurrency of curves.
Euler's formula can also be proved as follows: if the graph isn't a tree, then remove an edge which completes a cycle. This lowers both e and f by one, leaving v – e + f constant. Repeat until the remaining graph is a tree; trees have v = e + 1 and f = 1, yielding v – e + f = 2, i. e., the Euler characteristic is 2.
Treat each cross as a graph with 5 vertices and 4 edges. In the starting position with n crosses, we have a planar graph with v = 5n vertices, e = 4n edges, f = 1 face, and k = n connected components. The Euler characteristic for connected planar graphs is 2. In a disconnected planar graph, we get + = + After m moves, we have:
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When the graph has an Eulerian circuit (a closed walk that covers every edge once), that circuit is an optimal solution. Otherwise, the optimization problem is to find the smallest number of graph edges to duplicate (or the subset of edges with the minimum possible total weight) so that the resulting multigraph does have an Eulerian circuit. [1]
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