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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
Euler diagram illustrating that the set of "animals with four legs" is a subset of "animals", but the set of "minerals" is a disjoint set (it has no members in common) with "animals" Euler diagram showing the relationships between different Solar System objects
An Eulerian circuit (also called an Eulerian cycle or an Euler tour) is a closed walk that uses every edge exactly once. An Eulerian graph is a graph that has an Eulerian circuit. For an undirected graph, this means that the graph is connected and every vertex has even degree.
Robbins' theorem states that a graph has a strong orientation if and only if it is 2-edge-connected; disconnected graphs may have totally cyclic orientations, but only if they have no bridges. [5] An acyclic orientation is an orientation that results in a directed acyclic graph. Every graph has an acyclic orientation; all acyclic orientations ...
The Lagrangian and Eulerian specifications of the kinematics and dynamics of the flow field are related by the material derivative (also called the Lagrangian derivative, convective derivative, substantial derivative, or particle derivative). [1] Suppose we have a flow field u, and we are also given a generic field with Eulerian specification F ...
The mean curvature is an extrinsic invariant. In intrinsic geometry, a cylinder is developable, meaning that every piece of it is intrinsically indistinguishable from a piece of a plane since its Gauss curvature vanishes identically. Its mean curvature is not zero, though; hence extrinsically it is different from a plane.
Since the graph corresponding to historical Königsberg has four nodes of odd degree, it cannot have an Eulerian path. An alternative form of the problem asks for a path that traverses all bridges and also has the same starting and ending point. Such a walk is called an Eulerian circuit or an Euler tour. Such a circuit exists if, and only if ...
It has no crossings, so every polyhedral graph is also a planar graph. Additionally, by Balinski's theorem , it is a 3-vertex-connected graph . According to Steinitz's theorem , these two graph-theoretic properties are enough to completely characterize the polyhedral graphs: they are exactly the 3-vertex-connected planar graphs.