Graph and Digraph Glossary


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Acyclic Digraph

A digraph is acyclic if it contains no cycles.


Arc

A directed edge of a digraph. Some authors use it as a synonym for an edge of a graph. Other synonyms for arc in a digraph are arrow, directed line, directed edge, and directed link.


Arc List

A representation of a digraph using the arcs of the digraph. Can be an unordered listing of the ordered pairs, or a pair of ordered lists with the starting vertex in one list and the ending vertex in the corresponding position of the second list.


Adjacency Matrix

A 0-1 square matrix whose rows and columns are indexed by the vertices. A 1 in the ij-th position of the matrix means that there is an edge (or arc) from vertex i to vertex j. A 0 indicates that there is no such edge (or arc). Can be used for both graphs and digraphs.


Adjacency Structure

A representation of a graph or digraph which lists, for each vertex, all the vertices that are adjacent to the given vertex.


Adjacent

Two vertices are adjacent if they are connected by an edge. We often call these two vertices neighbors. Two adjacent vertices:


Bipartite Graph

A graph is bipartite if the vertices can be partitioned into two sets, X and Y, so that the only edges of the graph are between the vertices in X and the vertices in Y. Trees are examples of bipartite graphs. If G is bipartite, it is usually denoted by G = (X, Y, E), where E is the edge set.


Binary Search Tree

A binary tree that has been labelled with numbers so that the right offspring and all of its descendants have labels smaller than the label of the vertex, and the left offspring and all its descendants have labels larger than that of the vertex. .


Bridge

An edge in a graph whose removal (leaving the vertices) results in a disconnected graph.


Chain

A chain in a graph is a sequence of vertices from one vertex to another using the edges. The length of a chain is the number of edges used, or the number of vertices used minus one. A simple chain cannot visit the same vertex twice. A closed chain is one where the first and last vertex are the same. Here is an example of a simple chain:

More formally, a chain is a sequence of vertices of the form <x0, x1, ..., xn> such that xi and xi+1 are adjacent for i=0,...,n-1. In a simple chain all the xi are distinct. In a closed chain, x0 = xn.


Chromatic Number

The chromatic number of a graph is the smallest k for which the graph is k-colorable. The chromatic number of the graph G is denoted by X(G). [X is the greek letter chi].


Circuit

In a graph, a circuit is a simple, closed chain.


Closure

The closure of a graph G with n vertices, denoted by c(G), is the graph obtained from G by repeatedly adding edges between non-adjacent vertices whose degrees sum to at least n, until this can no longer be done. Several results concerning the existence of hamiltonian circuits refer to the closure of a graph.


Complete

In a complete graph, all pairs of vertices are adjacent. They are denoted by Kn, where n is the number of vertices. (The K is in honor of Kuratowski, a pioneer in graph theory.) The corresponding concept for digraphs is called a complete symmetric digraph, in which every ordered pair of vertices are joined by an arc. Here is the complete graph on five vertices, K5:


Connected Component

In a graph, a (connected) component is a maximal, connected, induced subgraph. Maximal means that there is no larger connected, induced subgraph containing the vertices of the component.


Condensed Graph

Given a graph G, if two vertices of G are identified and any loops or multiple edges created by this identification removed, the resulting graph is called the condensed graph.


Connected

A connected graph is one in which every pair of vertices are joined by a chain. A graph which is not connected is called disconnected, and breaks up into connected components.


Cycle

In a digraph, a cycle is a simple closed path.


Decision Tree

A binary tree used to represent an algorithm for sorting by comparisons. The leaves of the tree represent the possible outcomes (orderings), while the other vertices represent test questions which have a yes or no answer.


Degree

The degree of a vertex is the size of its neighborhood. The degree of a graph is the maximum degree of all of its vertices.


Diameter

The diameter of a graph is the length of the longest chain you are forced to use to get from one vertex to another in that graph. You can find the diameter of a graph by finding the distance between every pair of vertices and taking the maximum of those distances.


Digraph

A digraph is a graph in which the edges are directed and called arcs. More formally, a digraph is a set of vertices together with a set of ordered pairs of the vertices, called arcs. Here is a digraph on 5 vertices:


Distance

The distance between two vertices is the length of the shortest chain between them.


Edge

An edge connects two vertices in a graph. We call those two vertices the endpoints of the edge. Other synonyms for edge are arc, link and line. Here are the edges of a graph (in red):


Forest

A graph which contains no circuits. The connected components of a forest are trees.


Graph

A graph is basically a collection of dots, with some pairs of dots being connected by lines. The dots are called vertices, and the lines are called edges.

More formally, a graph is two sets. The first set is the set of vertices. The second set is the set of edges. The vertex set is just a collection of the labels for the vertices, a way to tell one vertex from another. The edge set is made up of unordered pairs of vertex labels from the vertex set.

Here is a diagram of a graph, and the sets that the graph is made from:

V={A,B,C,D}
--The vertex set.
E={(A,B) , (A,C) , (B,C) , (B,D)}
--The edge set.
A graph diagram.The sets that make up a graph.

Hamiltonian

A chain or circuit in a graph is said to be hamiltonian if each vertex of the graph appears in it precisely once. Paths and cycles of digraphs are called hamiltonian if the same condition holds. A graph containing a hamiltonian circuit, or a digraph containing a hamiltonian cycle is referred to as a hamiltonian graph or digraph.


Height

The height of a rooted tree is the length of the longest simple chain starting at the root of the tree.


Homeomorphic

Two graphs are homeomorphic if they can both be obtained from a common graph by a sequence of replacing edges by simple chains. In appearance, homeomorphic graphs look like ones that have extra vertices added to or removed from edges.


Incidence Matrix

A 0-1 matrix whose rows are indexed by the vertices of a graph and whose columns are indexed by the edges. A 1 in the ij-th position of the matrix means that the vertex i is on the edge j. A 0 indicates that it is not.


Isomorphic

Two graphs are isomorphic if they are they same graphs, drawn differently. Two graphs are isomorphic if you can label both graphs with the same labels so that every vertex has exactly the same neighbors in both graphs. Here are two isomporphic graphs:


k-Colorable

A graph is said to be k-colorable if each of its vertices can be assigned one of k colors in such a way that no two adjacent vertices are assigned the same color. The assignment is called a coloring.


Label

Labels are just the names we give vertices and edges so we can tell them apart. Usually, we use the integers 1, 2, ..., n as the labels of a graph or digraph with n vertices. The assignment of label to vertex is arbitrary.


Leaf

A vertex of degree 1. Also known as a pendant vertex.


Level

In a rooted tree, the vertices at the same distance from the root are said to be at the same level. The root is considered to be at level 0 and the height of the tree is the maximum level.


Loop

An edge or arc from a vertex to itself is called a loop. Loops are not allowed in simple graphs or digraphs.


m-ary Tree

A rooted tree in which every vertex has either 0 or m offspring. When m = 2, these are called binary trees.


Matching

A matching in a graph is a set of edges such that every vertex of the graph is on at most one edge in the set.


Neighborhood

The neighborhood of a vertex is all the vertices that it is adjacent to (all of the vertex's neighbors). Here we have a vertex (in blue) and the vertices in its neighborhood (in red):


Node

Another word for vertex.


Offspring

In a rooted tree, the vertices adjacent to a given vertex at the next higher level are called the offspring of the given vertex. They are sometimes called sons. The descendents of a vertex are the vertices in the set of vertices which are offspring, or offspring of offspring, etc. of the given vertex..


Order

The order of a graph is the number of vertices it has.


Orientation

An assignment of a direction to each edge of a graph. A graph which has been given an orientation is called an oriented graph, and is a digraph.


Path

A path in a digraph is a sequence of vertices from one vertex to another using the arcs. The length of a path is the number of arcs used, or the number of vertices used minus one. A simple path cannot visit the same vertex twice. A closed path has the same first and last vertex. Here is an example of a path:

More formally, a path is a sequence of vertices in a digraph of the form <x0, x1, ..., xn> such that xi and xi+1 are adjacent for i=0,...,n-1. In a simple path all the xi are distinct. In a closed path, x0 = xn.

Path is used by some authors to mean a simple chain in a graph.


Pendant Vertex

A vertex of degree 1. Also known as a leaf.


Perfect Matching

In a graph with 2n vertices, a matching with n edges is said to be perfect. Every vertex of the graph is saturated by a perfect matching. Another term for a perfect matching is a 1-factor.


Planar

A planar graph is a graph that you can draw on a flat surface, or plane, without any of the edges crossing. Graphs that cannot be drawn on the plane without crossed edges are called non-planar graphs. Any graph that has either of the following graphs as subgraphs are non-planar:


Reduced Graph

If an edge, a, is removed from a given graph G, the resulting graph, denoted G'a is referred to as a reduced graph.


Regular

In a regular graph, each vertex has the same degree. If this common degree is k, then we say that the graph is k-regular.


Rooted Tree

A tree in which one vertex has been distinguished. The distinguished vertex is called the root of the tree.


Saturated vertex

A vertex in a graph which is on an edge of a matching is said to be saturated. Given a matching M, if X is a set of vertices saturated by M, then M is said to be an X-saturating matching.


Size

The size of a graph is the number of edges it has.


Spanning Subgraph

A subgraph of the graph G which contains all of the vertices of G.


Spanning Tree

A spanning subgraph of a graph which is also a tree.


Strongly Connected

In a digraph there are many degrees of connectedness. A strongly connected digraph is one in which any vertex can be reached from any other vertex by a path.


Subgraph

A subgraph of a graph is some smaller portion of that graph. Here is an example of a subgraph:

A graphA subgraph
An induced (generated) subgraph is a subset of the vertices of the graph together with all the edges of the graph between the vertices of this subset. The induced subgraph of the above example is:

Topological Order

A topological ordering of a digraph is a labelling of the vertices with consecutive integers so that every arc is directed from a smaller label to a larger label.


Tournament

A tournament is a digraph in which there is exactly one arc between any two vertices. A tournament is said to be transitive if whenever (a,b) and (b,c) are arcs of the tournament, then (a,c) is also an arc.


Tree

A connected graph containing no circuits.


Vertex

A vertex is a 'dot' in a graph. The plural of vertex is 'vertices', as in, 'this graph has five vertices'. Other synonyms for vertex are node and point. Here are the vertices of a graph (in red):


Acyclic Digraph | Arc | Arc List | Adjacency Matrix | Adjacency Structure | Adjacent | Bipartite Graph | Binary Search Tree | Bridge | Chain | Chromatic Number | Circuit | Closure | Complete | Component | Condensed Graph | Connected | Cycle | Decision Tree | Degree | Diameter | Digraph | Distance | Edge | Forest | Graph | Hamiltonian | Height | Homeomorphic | Incidence Matrix | Isomorphic | k-Colorable | Label | Leaf | Level | Loop | m-ary Tree | Matching | Neighborhood | Node | Offspring | Order | Orientation | Path | Pendant Vertex | Perfect Matching | | Planar | Reduced Graph | Regular | Rooted Tree | Saturated Vertex | Size | Spanning Subgraph | Spanning Tree | Strongly Connected | Subgraph | Topological Order | Tournament | Tree | Vertex