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  2. Tree (graph theory) - Wikipedia

    en.wikipedia.org/wiki/Tree_(graph_theory)

    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.

  3. Spanning tree - Wikipedia

    en.wikipedia.org/wiki/Spanning_tree

    Almost all graph theory books and articles define a spanning forest as a forest that spans all of the vertices, meaning only that each vertex of the graph is a vertex in the forest. A connected graph may have a disconnected spanning forest, such as the forest with no edges, in which each vertex forms a single-vertex tree. [8] [9] A few graph ...

  4. Pseudoforest - Wikipedia

    en.wikipedia.org/wiki/Pseudoforest

    The names are justified by analogy to the more commonly studied trees and forests. (A tree is a connected graph with no cycles; a forest is a disjoint union of trees.) Gabow and Tarjan [2] attribute the study of pseudoforests to Dantzig's 1963 book on linear programming, in which pseudoforests arise in the solution of certain network flow ...

  5. Kruskal's algorithm - Wikipedia

    en.wikipedia.org/wiki/Kruskal's_algorithm

    Test whether adding the edge to the current forest would create a cycle. If not, add the edge to the forest, combining two trees into a single tree. At the termination of the algorithm, the forest forms a minimum spanning forest of the graph. If the graph is connected, the forest has a single component and forms a minimum spanning tree.

  6. Polytree - Wikipedia

    en.wikipedia.org/wiki/Polytree

    A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest. In other words, if we replace its directed edges with undirected edges, we obtain an undirected graph that is acyclic. A polytree is an example of an oriented graph. The term polytree was coined in 1987 by Rebane and ...

  7. Minimum spanning tree - Wikipedia

    en.wikipedia.org/wiki/Minimum_spanning_tree

    A planar graph and its minimum spanning tree. Each edge is labeled with its weight, which here is roughly proportional to its length. A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight. [1]

  8. Arboricity - Wikipedia

    en.wikipedia.org/wiki/Arboricity

    The arboricity of a graph is a measure of how dense the graph is: graphs with many edges have high arboricity, and graphs with high arboricity must have a dense subgraph. In more detail, as any n-vertex forest has at most n-1 edges, the arboricity of a graph with n vertices and m edges is at least ⌈ / ⌉. Additionally, the subgraphs of any ...

  9. Cayley's formula - Wikipedia

    en.wikipedia.org/wiki/Cayley's_formula

    The complete list of all trees on 2,3,4 labeled vertices: = tree with 2 vertices, = trees with 3 vertices and = trees with 4 vertices. In mathematics, Cayley's formula is a result in graph theory named after Arthur Cayley.