Path (graph theory)

Last updated October 19, 2024

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 directed path (sometimes called dipath^[1]) in a directed graph 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.

Definitions

Walk, trail, and path

A walk is a finite or infinite sequence of edges which joins a sequence of vertices.^[2]

Let G = (V, E, ϕ) be a graph. A finite walk is a sequence of edges (e₁, e₂, …, e_{n − 1}) for which there is a sequence of vertices (v₁, v₂, …, v_n) such that ϕ(e_i) = {v_i, v_{i + 1}} for i = 1, 2, …, n − 1. (v₁, v₂, …, v_n) is the vertex sequence of the walk. The walk is closed if v₁ = 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) has a first vertex but no last vertex.

A trail is a walk in which all edges are distinct.^[2]
A path is a trail in which all vertices (and therefore also all edges) are distinct.^[2]

If w = (e₁, e₂, …, e_{n − 1}) is a finite walk with vertex sequence (v₁, v₂, …, v_n) then w is said to be a walk fromv₁tov_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 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 sequence of edges directed in the same direction which joins a sequence of vertices.^[2]

Let G = (V, E, ϕ) be a directed graph. A finite directed walk is a sequence of edges (e₁, e₂, …, e_{n − 1}) for which there is a sequence of vertices (v₁, v₂, …, v_n) such that ϕ(e_i) = (v_i, v_{i + 1}) for i = 1, 2, …, n − 1. (v₁, v₂, …, v_n) is the vertex sequence of the directed walk. The directed walk is closed if v₁ = 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 ray) has a first vertex but no last vertex.

A directed trail is a directed walk in which all edges are distinct.^[2]
A directed path is a directed trail in which all vertices are distinct.^[2]

If w = (e₁, e₂, …, e_{n − 1}) is a finite directed walk with vertex sequence (v₁, v₂, …, v_n) then w is said to be a walk fromv₁tov_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.

A "simple directed path" is a path where all vertices are distinct.

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 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.
Two paths are vertex-independent (alternatively, internally disjoint or internally vertex-disjoint) if they do not have any internal vertex or edge in common. Similarly, two paths are edge-independent (or edge-disjoint) if they do not have any edge in common. Two internally disjoint paths are edge-disjoint, but the converse is not necessarily true.
The distance 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 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 can be applied to directed graphs with negative edge weights. The Floyd–Warshall algorithm can be used to find the shortest paths between all pairs of vertices in weighted directed graphs.

The path partition problem

The k-path partition problem is the problem of partitioning a given graph to a smallest collection of vertex-disjoint paths of length at most k.^[3]

Notes

↑ McCuaig 1992, p. 205.
1 2 3 4 5 6 Bender & Williamson 2010, p. 162.
↑ Chen, Yong; Goebel, Randy; Lin, Guohui; Su, Bing; Xu, Yao; Zhang, An (2019-07-01). "An improved approximation algorithm for the minimum 3-path partition problem". Journal of Combinatorial Optimization. 38 (1): 150–164. doi:10.1007/s10878-018-00372-z. ISSN 1382-6905.

Related Research Articles

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 which are connected by edges. A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics.

A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight. That is, it is a spanning tree whose sum of edge weights is as small as possible. More generally, any edge-weighted undirected graph has a minimum spanning forest, which is a union of the minimum spanning trees for its connected components.

In graph theory, a cycle in a graph is a non-empty trail in which only the first and last vertices are equal. A directed cycle in a directed graph is a non-empty directed trail in which only the first and last vertices are equal.

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 is capable of handling graphs in which some of the edge weights are negative numbers. The algorithm was first proposed by Alfonso Shimbel, but is instead named after Richard Bellman and Lester Ford Jr., who published it in 1958 and 1956, respectively. Edward F. Moore also published a variation of the algorithm in 1959, and for this reason it is also sometimes called the Bellman–Ford–Moore algorithm.

In computer science, the Floyd–Warshall algorithm is an algorithm for finding shortest paths in a directed weighted graph with positive or negative edge weights. A single execution of the algorithm will find the lengths of shortest paths between all pairs of vertices. Although it does not return details of the paths themselves, it is possible to reconstruct the paths with simple modifications to the algorithm. Versions of the algorithm can also be used for finding the transitive closure of a relation $, or widest paths between all pairs of vertices in a weighted graph.$

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.

<span class="mw-page-title-main">Graph (discrete mathematics)</span> Vertices connected in pairs by edges

In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a set of objects where some pairs of the objects are in some sense "related". The objects are represented by abstractions called vertices and each of the related pairs of vertices is called an edge. Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges.

In graph theory, an Eulerian trail is a trail in a finite graph that visits every edge exactly once. Similarly, an Eulerian circuit or Eulerian cycle is an Eulerian trail that starts and ends on the same vertex. They were first discussed by Leonhard Euler while solving the famous Seven Bridges of Königsberg problem in 1736. The problem can be stated mathematically like this:

In the mathematical field of graph theory, a spanning treeT of an undirected graph G is a subgraph that is a tree which includes all of the vertices of G. In general, a graph may have several spanning trees, but a graph that is not connected will not contain a spanning tree. If all of the edges of G are also edges of a spanning tree T of G, then G is a tree and is identical to T.

In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path connecting them. This is also known as the geodesic distance or shortest-path distance. Notice that there may be more than one shortest path between two vertices. If there is no path connecting the two vertices, i.e., if they belong to different connected components, then conventionally the distance is defined as infinite.

In the mathematical discipline of graph theory, Menger's theorem says that in a finite graph, the size of a minimum cut set is equal to the maximum number of disjoint paths that can be found between any pair of vertices. Proved by Karl Menger in 1927, it characterizes the connectivity of a graph. It is generalized by the max-flow min-cut theorem, which is a weighted, edge version, and which in turn is a special case of the strong duality theorem for linear programs.

Johnson's algorithm is a way to find the shortest paths between all pairs of vertices in an edge-weighted directed graph. It allows some of the edge weights to be negative numbers, but no negative-weight cycles may exist. It works by using the Bellman–Ford algorithm to compute a transformation of the input graph that removes all negative weights, allowing Dijkstra's algorithm to be used on the transformed graph. It is named after Donald B. Johnson, who first published the technique in 1977.

Edge disjoint shortest pair algorithm is an algorithm in computer network routing. The algorithm is used for generating the shortest pair of edge disjoint paths between a given pair of vertices. For an undirected graph G(V, E), it is stated as follows:

Run the shortest path algorithm for the given pair of vertices
Replace each edge of the shortest path by a single arc directed towards the source vertex
Make the length of each of the above arcs negative
Run the shortest path algorithm (Note: the algorithm should accept negative costs)
Erase the overlapping edges of the two paths found, and reverse the direction of the remaining arcs on the first shortest path such that each arc on it is directed towards the destination vertex now. The desired pair of paths results.

In computer science, graph traversal refers to the process of visiting each vertex in a graph. Such traversals are classified by the order in which the vertices are visited. Tree traversal is a special case of graph traversal.

In the mathematical field of graph theory, graph operations are operations which produce new graphs from initial ones. They include both unary and binary operations.

In mathematics, and more specifically in graph theory, a directed graph is a graph that is made up of a set of vertices connected by directed edges, often called arcs.

<span class="mw-page-title-main">Squaregraph</span> Planar graph with quadrilateral faces

In graph theory, a branch of mathematics, a squaregraph is a type of undirected graph that can be drawn in the plane in such a way that every bounded face is a quadrilateral and every vertex with three or fewer neighbors is incident to an unbounded face.

In the mathematical discipline of graph theory, a graph $C$ is a covering graph of another graph $G$ if there is a covering map from the vertex set of $C$ to the vertex set of $G$ . A covering map $f$ is a surjection and a local isomorphism: the neighbourhood of a vertex $v$ in $C$ is mapped bijectively onto the neighbourhood of ⁠ $⁠$ in $G$ .

In theoretical computer science and network routing, Suurballe's algorithm is an algorithm for finding two disjoint paths in a nonnegatively-weighted directed graph, so that both paths connect the same pair of vertices and have minimum total length. The algorithm was conceived by John W. Suurballe and published in 1974. The main idea of Suurballe's algorithm is to use Dijkstra's algorithm to find one path, to modify the weights of the graph edges, and then to run Dijkstra's algorithm a second time. The output of the algorithm is formed by combining these two paths, discarding edges that are traversed in opposite directions by the paths, and using the remaining edges to form the two paths to return as the output. The modification to the weights is similar to the weight modification in Johnson's algorithm, and preserves the non-negativity of the weights while allowing the second instance of Dijkstra's algorithm to find the correct second path.

In graph theory, a matching in a hypergraph is a set of hyperedges, in which every two hyperedges are disjoint. It is an extension of the notion of matching in a graph.

References

Bender, Edward A.; Williamson, S. Gill (2010). Lists, Decisions and Graphs. With an Introduction to Probability.
Bondy, J. A.; Murty, U. S. R. (1976). Graph Theory with Applications. North Holland. p. 12-21. ISBN 0-444-19451-7.
Diestel, Reinhard (2005). Graph Theory. Springer-Verlag. pp. 6–9. ISBN 3-540-26182-6.
Gibbons, A. (1985). Algorithmic Graph Theory. Cambridge University Press. pp. 5–6. ISBN 0-521-28881-9.
Korte, Bernhard; Lovász, László; Prömel, Hans Jürgen; Schrijver, Alexander (1990). Paths, Flows, and VLSI-Layout . Springer-Verlag. ISBN 0-387-52685-4.
McCuaig, William (1992). "Intercyclic Digraphs" . In Robertson, Neil; Seymour, Paul (eds.). Graph Structure Theory. AMS–IMS–SIAM Joint Summer Research Conference on Graph Minors, Seattle, June 22 – July 5, 1991. American Mathematical Society. p. 205.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[FOOTNOTEMcCuaig1992205-1] McCuaig 1992, p. 205.

[FOOTNOTEBenderWilliamson2010162-2] 1 2 3 4 5 6 Bender & Williamson 2010, p. 162.

[3] Chen, Yong; Goebel, Randy; Lin, Guohui; Su, Bing; Xu, Yao; Zhang, An (2019-07-01). "An improved approximation algorithm for the minimum 3-path partition problem". Journal of Combinatorial Optimization. 38 (1): 150–164. doi:10.1007/s10878-018-00372-z. ISSN 1382-6905.

[1]

[2]

[3]