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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 tree can then be represented as a Eulerian circuit of the directed graph, known as the Euler tour representation (ETR) of the tree. The ETT allows for efficient, parallel computation of solutions to common problems in algorithmic graph theory. It was introduced by Tarjan and Vishkin in 1984. [1]
A directed circuit is a non-empty directed trail (e 1, e 2, ..., e n) with a vertex sequence (v 1, v 2, ..., v n, v 1). A directed cycle or simple directed circuit is a directed circuit in which only the first and last vertices are equal. [1] n is called the length of the directed circuit resp. length of the directed cycle.
SPOILERS BELOW—do not scroll any further if you don't want the answer revealed. The New York Times Today's Wordle Answer for #1310 on Sunday, January 19, 2025
circuit A circuit may refer to a closed trail or an element of the cycle space (an Eulerian spanning subgraph). The circuit rank of a graph is the dimension of its cycle space. circumference The circumference of a graph is the length of its longest simple cycle. The graph is Hamiltonian if and only if its circumference equals its order. class 1.
Get ready for all of today's NYT 'Connections’ hints and answers for #582 on Monday, January 13, 2025. Today's NYT Connections puzzle for Monday, January 13, 2025 The New York Times
Get ready for all of today's NYT 'Connections’ hints and answers for #584 on Wednesday, January 15, 2025. Today's NYT Connections puzzle for Wednesday, January 15, 2025 The New York Times
For planar graphs, the properties of being Eulerian and bipartite are dual: a planar graph is Eulerian if and only if its dual graph is bipartite. As Welsh showed, this duality extends to binary matroids: a binary matroid is Eulerian if and only if its dual matroid is a bipartite matroid, a matroid in which every circuit has even cardinality.