graph theory In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...

, a path in a

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...

is a finite or infinite

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...

edges Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed by ...

which joins a sequence of vertices which, by most definitions, are all distinct (and since the vertices are distinct, so are the edges). A directed path (sometimes called dipathGraph Structure Theory: Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Graph Minors, Held June 22 to July 5, 1991
p.205
/ref>) in a

directed graph In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. Definition In formal terms, a directed graph is an ordered pai ...

is a finite or infinite sequence of edges which joins a sequence of distinct vertices, but with the added restriction that the edges be all directed in the same direction. Paths are fundamental concepts of graph theory, described in the introductory sections of most graph theory texts. See e.g. Bondy and Murty (1976), Gibbons (1985), or Diestel (2005). Korte et al. (1990) cover more advanced

algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...

ic topics concerning paths in graphs.

Definitions

Walk, trail, and path

* A walk is a finite or infinite

which joins a sequence of vertices. : Let be a graph. A finite walk is a sequence of edges for which there is a sequence of vertices such that ''ϕ''(''e''_''i'') = for . is the ''vertex sequence'' of the walk. The walk is ''closed'' if ''v''₁ = ''v''_''n'', and it is ''open'' otherwise. An infinite walk is a sequence of edges of the same type described here, but with no first or last vertex, and a semi-infinite walk (or

ray Ray may refer to: Fish * Ray (fish), any cartilaginous fish of the superorder Batoidea * Ray (fish fin anatomy), a bony or horny spine on a fin Science and mathematics * Ray (geometry), half of a line proceeding from an initial point * Ray (g ...

) has a first vertex but no last vertex. * A trail is a walk in which all edges are distinct. * A path is a trail in which all vertices (and therefore also all edges) are distinct. If is a finite walk with vertex sequence then ''w'' is said to be a ''walk from'' ''v''₁ ''to'' ''v''_''n''. Similarly for a trail or a path. If there is a finite walk between two ''distinct'' vertices then there is also a finite trail and a finite path between them. Some authors do not require that all vertices of a path be distinct and instead use the term simple path to refer to such a path where all vertices are distinct. A

weighted graph This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by lines or edges Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * ...

associates a value (''weight'') with every edge in the graph. The ''weight of a walk'' (or trail or path) in a weighted graph is the sum of the weights of the traversed edges. Sometimes the words ''cost'' or ''length'' are used instead of weight.

Directed walk, directed trail, and directed path

* A directed walk is a finite or infinite

directed in the same direction which joins a sequence of vertices. : Let be a directed graph. A finite directed walk is a sequence of edges for which there is a sequence of vertices such that for . is the ''vertex sequence'' of the directed walk. The directed walk is ''closed'' if ''v''₁ = ''v''_''n'', and it is ''open'' otherwise. An infinite directed walk is a sequence of edges of the same type described here, but with no first or last vertex, and a semi-infinite directed walk (or

) has a first vertex but no last vertex. * A directed trail is a directed walk in which all edges are distinct. * A directed path is a directed trail in which all vertices are distinct. If is a finite directed walk with vertex sequence then ''w'' is said to be a ''walk from'' ''v''₁ ''to'' ''v''_''n''. Similarly for a directed trail or a path. If there is a finite directed walk between two ''distinct'' vertices then there is also a finite directed trail and a finite directed path between them. Some authors do not require that all vertices of a directed path be distinct and instead use the term simple directed path to refer to such a directed path. A weighted directed graph associates a value (''weight'') with every edge in the directed graph. The ''weight of a directed walk'' (or trail or path) in a weighted directed graph is the sum of the weights of the traversed edges. Sometimes the words ''cost'' or ''length'' are used instead of weight.

Examples

* A graph is connected if there are paths containing each pair of vertices. * A directed graph is

strongly connected In the mathematical theory of directed graphs, a graph is said to be strongly connected if every vertex is reachable from every other vertex. The strongly connected components of an arbitrary directed graph form a partition into subgraphs that ...

if there are oppositely oriented directed paths containing each pair of vertices. * A path such that no graph edges connect two nonconsecutive path vertices is called an induced path. * A path that includes every vertex of the graph without repeats is known as a

Hamiltonian path In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vertex ...

. * Two paths are ''vertex-independent'' (alternatively, ''internally vertex-disjoint'') if they do not have any internal vertex in common. Similarly, two paths are ''edge-independent'' (or ''edge-disjoint'') if they do not have any internal edge in common. Two internally vertex-disjoint paths are edge-disjoint, but the converse is not necessarily true. * The

distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...

between two vertices in a graph is the length of a shortest path between them, if one exists, and otherwise the distance is infinity. * The diameter of a connected graph is the largest distance (defined above) between pairs of vertices of the graph.

Finding paths

Several algorithms exist to find shortest and longest paths in graphs, with the important distinction that the former problem is computationally much easier than the latter.

Dijkstra's algorithm Dijkstra's algorithm ( ) is an algorithm for finding the shortest paths between nodes in a graph, which may represent, for example, road networks. It was conceived by computer scientist Edsger W. Dijkstra in 1956 and published three years ...

produces a list of shortest paths from a source vertex to every other vertex in directed and undirected graphs with non-negative edge weights (or no edge weights), whilst the

Bellman–Ford algorithm The Bellman–Ford algorithm is an algorithm that computes shortest paths from a single source vertex to all of the other vertices in a weighted digraph. It is slower than Dijkstra's algorithm for the same problem, but more versatile, as it i ...

can be applied to directed graphs with negative edge weights. The

Floyd–Warshall algorithm In computer science, the Floyd–Warshall algorithm (also known as Floyd's algorithm, the Roy–Warshall algorithm, the Roy–Floyd algorithm, or the WFI algorithm) is an algorithm for finding shortest paths in a directed weighted graph with p ...

can be used to find the shortest paths between all pairs of vertices in weighted directed graphs.

References