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But if G is a directed acyclic graph (DAG), then no negative cycles can be created, and a longest path in G can be found in linear time by applying a linear time algorithm for shortest paths in −G, which is also a directed acyclic graph. [4] For a DAG, the longest path from a source vertex to all other vertices can be obtained by running the ...
Grundy number of a directed graph. [3]: GT56 Hamiltonian completion [3]: GT34 Hamiltonian path problem, directed and undirected. [2] [3]: GT37, GT38, GT39 Induced subgraph isomorphism problem; Graph intersection number [3]: GT59 Longest path problem [3]: ND29 Maximum bipartite subgraph or (especially with weighted edges) maximum cut.
A three-dimensional hypercube graph showing a Hamiltonian path in red, and a longest induced path in bold black. In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices which, by most definitions, are all distinct (and since the vertices are distinct, so are the edges).
A graph that contains a Hamiltonian cycle is called a Hamiltonian graph. Similar notions may be defined for directed graphs, where each edge (arc) of a path or cycle can only be traced in a single direction (i.e., the vertices are connected with arrows and the edges traced "tail-to-head").
The first three stages of Johnson's algorithm are depicted in the illustration below. The graph on the left of the illustration has two negative edges, but no negative cycles. The center graph shows the new vertex q, a shortest path tree as computed by the Bellman–Ford algorithm with q as starting vertex, and the values h(v) computed at each other node as the length of the shortest path from ...
The transitive reduction of a finite directed graph G is a graph with the fewest possible edges that has the same reachability relation as the original graph. That is, if there is a path from a vertex x to a vertex y in graph G, there must also be a path from x to y in the transitive reduction of G, and vice versa. Specifically, if there is ...
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 ...
The algorithm terminates if either the target node t is reached, or the length of the path so far exceeds n, the number of nodes in the graph. The complement of st-connectivity , known as st-non-connectivity , is also in the class NL, since NL = coNL by the Immerman–Szelepcsényi theorem .