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In graph theory, a branch of mathematics, a linear forest is a kind of forest where each component is a path graph, [1]: 200 or a disjoint union of nontrivial paths. [2]: 246 Equivalently, it is an acyclic and claw-free graph. [3]: 130, 131 An acyclic graph where every vertex has degree 0, 1, or 2 is a linear forest.
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 ] A directed tree, [ 3 ] oriented tree, [ 4 ] [ 5 ] polytree , [ 6 ] or singly connected network [ 7 ] is a directed acyclic graph (DAG) whose underlying ...
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 ...
quasi-line graph A quasi-line graph or locally co-bipartite graph is a graph in which the open neighborhood of every vertex can be partitioned into two cliques. These graphs are always claw-free and they include as a special case the line graphs. They are used in the structure theory of claw-free graphs. quasi-random graph sequence
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 theory authors define a spanning forest to be a maximal acyclic subgraph of the given graph, or equivalently a subgraph consisting of a spanning tree in each connected component of the ...
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 ...
The general graph Steiner tree problem can be approximated by computing the minimum spanning tree of the subgraph of the metric closure of the graph induced by the terminal vertices, as first published in 1981 by Kou et al. [18] The metric closure of a graph G is the complete graph in which each edge is weighted by the shortest path distance ...
The model is defined as a cellular automaton on a grid with L d cells. L is the sidelength of the grid and d is its dimension. A cell can be empty, occupied by a tree, or burning. The model of Drossel and Schwabl (1992) is defined by four rules which are executed simultaneously: A burning cell turns into an empty cell