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
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]
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.
An Eulerian circuit is a directed closed trail that visits 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).
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.
Euler's argument shows that a necessary condition for the walk of the desired form is that the graph be connected and have exactly zero or two nodes of odd degree. This condition turns out also to be sufficient—a result stated by Euler and later proved by Carl Hierholzer. Such a walk is now called an Eulerian trail or Euler walk in his honor ...
The secret to this easy recipe lies in the tangy lemon-and-garlic drizzle that picks up the savory flavors left in the pan. Pan-searing chicken tenders locks in moisture while crisping up the outside.
Similarly, an Eulerian circuit or Eulerian cycle is a Eulerian trail which starts and ends on the same vertex. we see that in the disconnected case the sets of graphs satisfying either of the two definitions aren't disjoint either: consider the graph with two vertices and a single loop - it clearly satisfies both definitions.