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In mathematics, and, in particular, in graph theory, a rooted graph is a graph in which one vertex has been distinguished as the root. [1] [2] Both directed and undirected versions of rooted graphs have been studied, and there are also variant definitions that allow multiple roots. Examples of rooted graphs with some variants.
If H is a two-vertex complete graph K 2, then for any graph G, the rooted product of G and H has domination number exactly half of its number of vertices. Every connected graph in which the domination number is half the number of vertices arises in this way, with the exception of the four-vertex cycle graph.
A rooted tree T that is a subgraph of some graph G is a normal tree if the ends of every T-path in G are comparable in this tree-order (Diestel 2005, p. 15). Rooted trees, often with an additional structure such as an ordering of the neighbors at each vertex, are a key data structure in computer science; see tree data structure.
The term arborescence comes from French. [6] Some authors object to it on grounds that it is cumbersome to spell. [7] There is a large number of synonyms for arborescence in graph theory, including directed rooted tree, [3] [7] out-arborescence, [8] out-tree, [9] and even branching being used to denote the same concept. [9]
graph intersection: G 1 ∩ G 2 = (V 1 ∩ V 2, E 1 ∩ E 2); [1] graph join: . Graph with all the edges that connect the vertices of the first graph with the vertices of the second graph. It is a commutative operation (for unlabelled graphs); [2] graph products based on the cartesian product of the vertex sets:
An example for an undirected Graph with a vertex r and its corresponding level structure For the concept in algebraic geometry, see level structure (algebraic geometry) In the mathematical subfield of graph theory a level structure of a rooted graph is a partition of the vertices into subsets that have the same distance from a given root vertex.
In graph theory, Edmonds' algorithm or Chu–Liu/Edmonds' algorithm is an algorithm for finding a spanning arborescence of minimum weight (sometimes called an optimum branching) [1]. It is the directed analog of the minimum spanning tree problem.
In the graph shown below, the tree with edges 1–3, 2–3, and 3–4 is a Trémaux tree when it is rooted at vertex 1 or vertex 2: every edge of the graph belongs to the tree except for the edge 1–2, which (for these choices of root) connects an ancestor-descendant pair.