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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.
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
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]
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 ...
An undirected acyclic graph is the same thing as a forest. An acyclic directed graph, which is a digraph without directed cycles, is often called a directed acyclic graph, especially in computer science. [2] 2. An acyclic coloring of an undirected graph is a proper coloring in which every two color classes induce a forest. [3] adjacency matrix
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 ...
Therefore, an acyclic orientation is a unique sink orientation if and only if there is no other acyclic orientation with a smaller sum. Additionally, a k -regular subgraph of the given graph forms a face of the polytope if and only if its vertices form a lower set for at least one acyclic unique sink orientation.