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Given a graph, the eccentricity of a vertex v is defined as the greatest distance from v to any other vertex. A center of a graph is a vertex with minimal eccentricity. A graph can have an arbitrary number of centers. However, Jordan (1869) has proved that for trees, there are only two possibilities: The tree has precisely one center (centered ...
The eccentricity ϵ(v) of a vertex v is the greatest distance between v and any other vertex; in symbols, = (,). It can be thought of as how far a node is from the node most distant from it in the graph. The radius r of a graph is the minimum eccentricity of any vertex or, in symbols,
A recursive tree is a labeled rooted tree where the vertex labels respect the tree order (i.e., if u < v for two vertices u and v, then the label of u is smaller than the label of v). In a rooted tree, the parent of a vertex v is the vertex connected to v on the path to the root; every vertex has a unique parent, except the root has no parent. [24]
2. A peripheral vertex is a vertex whose eccentricity is maximum. In a tree, this must be a leaf. Petersen 1. Julius Petersen (1839–1910), Danish graph theorist. 2. The Petersen graph, a 10-vertex 15-edge graph frequently used as a counterexample. 3. Petersen's theorem that every bridgeless cubic graph has a perfect matching. planar
Equivalently, it is the set of vertices with eccentricity equal to the graph's radius. [3] Thus vertices in the center ( central points ) minimize the maximal distance from other points in the graph. This is also known as the vertex 1-center problem and can be extended to the vertex k-center problem .
On the left a centered tree, on the right a bicentered one. The numbers show each node's eccentricity. To give another class of examples, every free tree T has a separator S consisting of a single vertex, the removal of which partitions T into two or more connected components, each of size at most n ⁄ 2.
In mathematics, the eccentricity of a conic section is a non-negative real number that uniquely characterizes its shape. One can think of the eccentricity as a measure of how much a conic section deviates from being circular. In particular: The eccentricity of a circle is 0. The eccentricity of an ellipse which is not a circle is between 0 and 1.
That is, each graph vertex is contained in at least one tree node. If X i and X j both contain a vertex v, then all nodes X k of T in the (unique) path between X i and X j contain v as well. Equivalently, the tree nodes containing vertex v form a connected subtree of T. For every edge (v, w) in the graph, there is a subset X i that contains ...