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A graph with a loop on vertex 1. In graph theory, a loop (also called a self-loop or a buckle) is an edge that connects a vertex to itself. A simple graph contains no loops. Depending on the context, a graph or a multigraph may be defined so as to either allow or disallow the presence of loops (often in concert with allowing or disallowing ...
In graph theory, reachability refers to the ability to get from one vertex to another within a graph. A vertex s {\displaystyle s} can reach a vertex t {\displaystyle t} (and t {\displaystyle t} is reachable from s {\displaystyle s} ) if there exists a sequence of adjacent vertices (i.e. a walk ) which starts with s {\displaystyle s} and ends ...
In an empty graph, each vertex forms a component with one vertex and zero edges. [3] More generally, a component of this type is formed for every isolated vertex in any graph. [4] In a connected graph, there is exactly one component: the whole graph. [4] In a forest, every component is a tree. [5] In a cluster graph, every component is a ...
The two queries partition the vertex set into 4 subsets: vertices reached by both, either one, or none of the searches. One can show that a strongly connected component has to be contained in one of the subsets. The vertex subset reached by both searches forms a strongly connected component, and the algorithm then recurses on the other 3 subsets.
A directed pseudoforest is a directed graph in which each vertex has at most one outgoing edge; that is, it has outdegree at most one. A directed 1-forest – most commonly called a functional graph (see below), sometimes maximal directed pseudoforest – is a directed graph in which each vertex has outdegree exactly one. [8]
In the case of a directed graph, each edge has an orientation, from one vertex to another vertex. A path in a directed graph is a sequence of edges having the property that the ending vertex of each edge in the sequence is the same as the starting vertex of the next edge in the sequence; a path forms a cycle if the starting vertex of its first ...
Then one endpoint of edge e is in set V and the other is not. Since tree Y 1 is a spanning tree of graph P, there is a path in tree Y 1 joining the two endpoints. As one travels along the path, one must encounter an edge f joining a vertex in set V to one that is not in set V.
It is possible to find the maximum clique, or the clique number, of an arbitrary n-vertex graph in time O (3 n/3) = O (1.4422 n) by using one of the algorithms described above to list all maximal cliques in the graph and returning the largest one. However, for this variant of the clique problem better worst-case time bounds are possible.