Path graph
Encyclopedia
In the mathematical
field of graph theory
, a path graph or linear graph is a particularly simple example of a tree
, namely a tree with two or more vertices
that is not branched at all, that is, contains only vertices of degree
2 and 1. In particular, it has two terminal vertices (vertices that have degree 1), while all others (if any) have degree 2.
A path in a graph
is a sequence
of vertices
such that from each of its vertices there is an edge to the next vertex in the sequence. A path may be infinite, but a finite path always has a first vertex, called its start vertex, and a last vertex, called its end vertex. Both of them are called end or terminal vertices of the path. The other vertices in the path are internal vertices. A cycle is a path such that the start vertex and end vertex are the same. Note that the choice of the start vertex in a cycle is arbitrary.
Paths and cycles 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 algorithmic topics concerning paths in graphs.
s, with the edges being directed from each vertex to the following one. Often the terms directed path and directed cycle are used in the directed case.
A path with no repeated vertices is called a simple path, and a cycle with no repeated vertices or edges aside from the necessary repetition of the start and end vertex is a simple cycle. In modern graph theory
, most often "simple" is implied; i.e., "cycle" means "simple cycle" and "path" means "simple path", but this convention is not always observed, especially in applied graph theory. Some authors (e.g. Bondy and Murty 1976) use the term "walk" for a path in which vertices or edges may be repeated, and reserve the term "path" for what is here called a simple path.
A path such that no graph edges connect two nonconsecutive path vertices is called an induced path
.
A simple cycle that includes every vertex, without repetition, of the graph is known as a Hamiltonian cycle.
A cycle with just one edge removed in the corresponding spanning tree
of the original graph is known as a Fundamental cycle.
Two paths are independent (alternatively, internally vertex-disjoint) if they do not have any internal vertex in common.
The length of a path is the number of edges that the path uses, counting multiple edges multiple times. The length can be zero for the case of a single vertex.
A weighted graph associates a value (weight) with every edge in the graph. The weight of a 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
field of graph theory
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
, a path graph or linear graph is a particularly simple example of a tree
Tree (graph theory)
In mathematics, more specifically graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one simple path. In other words, any connected graph without cycles is a tree...
, namely a tree with two or more vertices
Vertex (graph theory)
In graph theory, a vertex or node is the fundamental unit out of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges , while a directed graph consists of a set of vertices and a set of arcs...
that is not branched at all, that is, contains only vertices of degree
Degree (graph theory)
In graph theory, the degree of a vertex of a graph is the number of edges incident to the vertex, with loops counted twice. The degree of a vertex v is denoted \deg. The maximum degree of a graph G, denoted by Δ, and the minimum degree of a graph, denoted by δ, are the maximum and minimum degree...
2 and 1. In particular, it has two terminal vertices (vertices that have degree 1), while all others (if any) have degree 2.
A path in a graph
Graph (mathematics)
In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...
is a sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
of vertices
Vertex (graph theory)
In graph theory, a vertex or node is the fundamental unit out of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges , while a directed graph consists of a set of vertices and a set of arcs...
such that from each of its vertices there is an edge to the next vertex in the sequence. A path may be infinite, but a finite path always has a first vertex, called its start vertex, and a last vertex, called its end vertex. Both of them are called end or terminal vertices of the path. The other vertices in the path are internal vertices. A cycle is a path such that the start vertex and end vertex are the same. Note that the choice of the start vertex in a cycle is arbitrary.
Different types of path graphs
The same concepts apply both to undirected graphs and directed graphDirected graph
A directed graph or digraph is a pair G= of:* a set V, whose elements are called vertices or nodes,...
s, with the edges being directed from each vertex to the following one. Often the terms directed path and directed cycle are used in the directed case.
A path with no repeated vertices is called a simple path, and a cycle with no repeated vertices or edges aside from the necessary repetition of the start and end vertex is a simple cycle. In modern graph theory
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
, most often "simple" is implied; i.e., "cycle" means "simple cycle" and "path" means "simple path", but this convention is not always observed, especially in applied graph theory. Some authors (e.g. Bondy and Murty 1976) use the term "walk" for a path in which vertices or edges may be repeated, and reserve the term "path" for what is here called a simple path.
A path such that no graph edges connect two nonconsecutive path vertices is called an induced path
Induced path
In the mathematical area of graph theory, an induced path in an undirected graph G is a path that is an induced subgraph of G. That is, it is a sequence of vertices in G such that each two adjacent vertices in the sequence are connected by an edge in G, and each two nonadjacent vertices in the...
.
A simple cycle that includes every vertex, without repetition, of the graph is known as a Hamiltonian cycle.
A cycle with just one edge removed in the corresponding spanning tree
Spanning tree
Spanning tree can refer to:* Spanning tree , a tree which contains every vertex of a more general graph* Spanning tree protocol, a protocol for finding spanning trees in bridged networks...
of the original graph is known as a Fundamental cycle.
Two paths are independent (alternatively, internally vertex-disjoint) if they do not have any internal vertex in common.
The length of a path is the number of edges that the path uses, counting multiple edges multiple times. The length can be zero for the case of a single vertex.
A weighted graph associates a value (weight) with every edge in the graph. The weight of a 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.
See also
- Glossary of graph theoryGlossary of graph theoryGraph theory is a growing area in mathematical research, and has a large specialized vocabulary. Some authors use the same word with different meanings. Some authors use different words to mean the same thing. This page attempts to keep up with current usage....
- Shortest path problemShortest path problemIn graph theory, the shortest path problem is the problem of finding a path between two vertices in a graph such that the sum of the weights of its constituent edges is minimized...
- Traveling salesman problem
- Cycle spaceCycle spaceIn graph theory, an area of mathematics, a cycle space is a vector space defined from an undirected graph; elements of the cycle space represent formal combinations of cycles in the graph....
- Path (graph theory)Path (graph theory)In graph theory, a path in a graph is a sequence of vertices such that from each of its vertices there is an edge to the next vertex in the sequence. A path may be infinite, but a finite path always has a first vertex, called its start vertex, and a last vertex, called its end vertex. Both of them...
- Caterpillar treeCaterpillar treeIn graph theory, a caterpillar or caterpillar tree is a tree in which all the vertices of the caterpillar are within distance 1 of a central path....
- Cycle graphCycle graphIn graph theory, a cycle graph or circular graph is a graph that consists of a single cycle, or in other words, some number of vertices connected in a closed chain. The cycle graph with n vertices is called Cn...
- Complete graphComplete graphIn the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge.-Properties:...
- Null graphNull graphIn the mathematical field of graph theory, the null graph may refer either to the order zero graph, or alternatively, to any edgeless graph .-Order zero graph:...
- Path decompositionPath decompositionIn graph theory, a path decomposition of a graph G is, informally, a representation of G as a "thickened" path graph, and the pathwidth of G is a number that measures how much the path was thickened to form G...