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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
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. If such a cycle exists, the graph is called Eulerian or unicursal. [4] The term "Eulerian graph" is also sometimes used in a weaker sense to denote a graph where every vertex has even degree.
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.
Euler diagrams consist of simple closed shapes in a two-dimensional plane that each depict a set or category. How or whether these shapes overlap demonstrates the relationships between the sets. Each curve divides the plane into two regions or "zones": the interior, which symbolically represents the elements of the set, and the exterior, which ...
In graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one path, or equivalently a connected acyclic undirected graph. [1] A forest is an undirected graph in which any two vertices are connected by at most one path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees. [2]
2. A k-tree is a graph formed by gluing (k + 1)-cliques together on shared k-cliques. A tree in the ordinary sense is a 1-tree according to this definition. tree decomposition A tree decomposition of a graph G is a tree whose nodes are labeled with sets of vertices of G; these sets are called bags.
Euler stated the fundamental results for this problem in terms of the number of odd vertices in the graph, which the handshaking lemma restricts to be an even number. If this number is zero, an Euler tour exists, and if it is two, an Euler path exists. Otherwise, the problem cannot be solved.
A tree structure, tree diagram, or tree model is a way of representing the hierarchical nature of a structure in a graphical form. It is named a "tree structure" because the classic representation resembles a tree , although the chart is generally upside down compared to a biological tree, with the "stem" at the top and the "leaves" at the bottom.