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An acyclic orientation of a complete graph is called a transitive tournament, and is equivalent to a total ordering of the graph's vertices. In such an orientation there is in particular exactly one source and exactly one sink. More generally, an acyclic orientation of an arbitrary graph that has a unique source and a unique sink is called a ...
An acyclic orientation is an orientation that results in a directed acyclic graph. Every graph has an acyclic orientation; all acyclic orientations may be obtained by placing the vertices into a sequence, and then directing each edge from the earlier of its endpoints in the sequence to the later endpoint.
In graph theory, a bipolar orientation or st-orientation of an undirected graph is an assignment of a direction to each edge (an orientation) that causes the graph to become a directed acyclic graph with a single source s and a single sink t, and an st-numbering of the graph is a topological ordering of the resulting directed acyclic graph. [1] [2]
Different total orders may lead to the same acyclic orientation, so an n-vertex graph can have fewer than n! acyclic orientations. The number of acyclic orientations is equal to |χ(−1)|, where χ is the chromatic polynomial of the given graph. [19] The yellow directed acyclic graph is the condensation of the blue directed
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 undirected graph is ...
Such a graph can be obtained by applying an orientation to an undirected graph. Tournaments are oriented graphs obtained by choosing a direction for each edge in undirected complete graphs. A tournament is a semicomplete digraph. [4] A directed graph is acyclic if it has no directed cycles. The usual name for such a digraph is directed acyclic ...
When the orientation has the additional property that the orientation forms a directed acyclic graph, which happens when unique sink orientations are used to model LP-type problems, it is possible to find the sink using a randomized algorithm in expected time exponential in the square root of d (Gärtner 2002).
A bipartite graph may be oriented from one side of the bipartition to the other. The longest path in this orientation has length one, with only two vertices. Conversely, if a graph is oriented without any three-vertex paths, then every vertex must either be a source (with no incoming edges) or a sink (with no outgoing edges) and the partition of the vertices into sources and sinks shows that ...
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