Search results
Results from the WOW.Com Content Network
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
In 1736, Euler showed that G has an Eulerian circuit if and only if G is connected and the indegree is equal to outdegree at every vertex. In this case G is called Eulerian. We denote the indegree of a vertex v by deg(v). The BEST theorem states that the number ec(G) of Eulerian circuits in a connected Eulerian graph G is given by the formula
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.
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.
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]
People who bought the snacks with the “non-GMO ingredients” graphic in the U.S. between Feb. 2, 2017, through Dec. 6, 2024, can “submit a valid timely” claim form by July 28, 2025.
Of course Aaron Rodgers and Davante Adams walked through that door on Sunday. Of course the rendition of grumpy old men morphed into the dynamic duo, when the postseason is no longer on the line ...
A trail is a walk in which all edges are distinct. [2] A path is a trail in which all vertices (and therefore also all edges) are distinct. [2] If w = (e 1, e 2, …, e n − 1) is a finite walk with vertex sequence (v 1, v 2, …, v n) then w is said to be a walk from v 1 to v n. Similarly for a trail or a path.