Glossary Of Graph Theory Basics Subgraph Path Cycle

While using this glossary of graph theory, please keep in mind that it is merely a starting point for beginners to get familiar with some basic concepts and terminologies, and is by no means a definitive presentation of what those concepts and terminolgies ought to be.

1 Basics

A graph G is consisted of two types of elements, namely vertex and edge, and can be modeled as a set system of 2-element sets (edges) over a ground set (vertices), or as a Boolean binary function over a set of vertices.

A vertex (basic element) is simply drawn as a node or a dot. The vertex set of G is usually denoted by V(G), or V when there is no danger of confusion. The order of a graph is the number of its vertices, i.e. |V(G)|.

An edge (a set of two elements) is drawn as a line connecting two vertices, called endvertices, or endpoints. An edge with endvertices x and y is denoted by xy without any mid-dot in between, that is, do not write x⋅y. The edge set of G is usually denoted by E(G), or E when there is no danger of confusion. The size of a graph is the number of its edges, i.e. |E(G)|.

A loop is an edge whose endvertices are the same vertex. An edge is multiple if there is another edge with the same endvertices; otherwise it is simple. The multiplicity of an edge is the number of multiple edges sharing the same endvertices; the multiplicity of a graph, the maximum multiplicity of its edges. A graph is a simple graph if it has no multiple edges or loops, a multigraph if it has multiple edges, but no loops, and a pseudograph if it contains both multiple edges and loops. When stated without any qualification, a graph is almost always assumed to be simple.

A label may be used on either an edge or a vertex to either uniquely identify it or otherwise indicate meaning. Graphs with labeled edges or vertices are known as labeled, those without as unlabeled. More specifically, graphs with labeled vertices only are vertex-labeled, those with labeled edges only are edge-labeled.

The example graph pictured to the right is a simple graph with vertex set V = {1, 2, 3, 4, 5, 6} and edge set E = 1,2}, {1,5}, {2,3}, {2,5}, {3,4}, {4,5}, {4,6}} (with the map w being the identity).

A hyperedge is an edge that is allowed to take on any number of vertices, even 2 or more. A graph that allows any hyperedge is called a hypergraph. A simple graph can be considered a special case of the hypergraph, namely the 2-uniform hypergraph. However, when stated without any qualification, an edge is always assumed to be consisted of at most 2 vertices, and a graph is never confused with a hypergraph.

The complement of a graph G is a graph with the same vertex set as G but with an edge set such that xy is an edge in if and only if xy is not an edge in G.

A graph is infinite if it has inifinitely many vertices or edges or both; otherwise the graph is finite. When stated without any qualification, a graph is usually assumed to be finite.

Two graphs G and H are said to be isomorphic, denoted by G ~ H, if there is an one-to-one correpondence, called isomorphism, between the vertices of the graph such that two vertices are adjacent in G if and only if their corresponding vertices are adjacent in H. Likewise, a graph G is said to be homomorphic to a graph H if there is a mapping, called homomorphism, from V(G) to V(H) such that if two vertices are adjacent in G then their corresponding vertices are adjacent in H.

2 Subgraph

A subgraph of a graph G is a graph whose vertex and edge sets are subsets of those of G. On the contrary, a supergraph of a graph G is a graph that contains G as a subgraph. We say a graph G contains another graph H if some subgraph of G is isomorphic to H.

A subgraph H is a spanning subgraph, or factor, of a graph G if it has the same vertex set as G. We say H spans G.

A subgraph H of a graph G is said to be induced if, for any pair of vertices x and y of H, xy is an edge of H if and only if xy is an edge of G. In other words, H is an induced subgraph of G if it has the most edges that appear in G over the same vertex set. If H is chosen based on a vertex subset S of V(G), then H can be written as G[S] and is said to be induced by S.

A graph that does not contain H as an induced subgraph is said to be H-free.

A universal graph in a class K of graphs is a simple graph in which every element in K can be embbeded as a subgraph.

2.1 Path

Traditionally, a path is graph consisted of a sequence of successively incident edges and their endvertices, where the terminating vertices are distinct. In modern literature, this definition usually refers to what is known as a trail, or open walk. When stated without any qualification, a path of n vertices, denoted by P_n, is usually assumed to be a simple path, or a simple trail in the modern sense, meaning every vertex is incident to at most two edges.

Two paths are internally disjoint (some people consider it independent) if they do not have any vertex in common, except the first and last ones.

The length of a path is the number of edges that the path uses. P_n has length n - 1. Some people count multiple edges multiple times. In the example graph, (1, 2, 5, 1, 2, 3) is a path with length 5, and (5, 2, 1) is a simple path of length 2.

A spanning path is also called a Hamiltonian path, or traceable path. A graph that contains a Hamiltonian path is a traceable graph; and one that, given any pair of (distinct) vertices, contains a Hamiltonian path using them as endvertices, a Hamiltonian connected graph.

2.2 Cycle

Traditionally, a cycle in a graph consisted of a sequence of successively incident edges and their endvertices, where the terminating vertices are identical. In modern literature, this definition usually refers to what is known as a circuit, or closed walk. When stated without any qualification, a cycle of n vertices, denoted by C_n, is usually assumed to be a simple cycle, or a simple circuit in the modern sense, meaning every vertex is incident to exactly two edges. In the above graph (1, 5, 2, 1) is a simple cycle.

The length of a cycle is the number of its edges. C_n has length n. A cycle, unlike a path, is not allowed to have length 0. Cycles of length 1 are loops. Cycles of length 2 are pairs of multiple edges. In the example graph, (1, 2, 3, 4, 5, 2, 1) is a cycle of length 6.

A cycle that has odd length is an odd cycle; otherwise it is an even cycle. A graph can be proved bipartite if there do not exist any odd cycle. See complete bipartite graph

The girth of a graph is the length of a shortest (simple) cycle in the graph; and the circumference, the length of a longest (simple) cycle. The girth and circumference of an acyclic graph are defined to be infinity ∞.

A graph is acyclic if it contains no cycles; unicyclic if it contains exactly one cycle; and pancyclic if it contains cycles of every possible length (from 3 to the order of the graph).

An Eulerian path in a graph is a path that uses each edge precisely once. If such a path exists, the graph is called traversable. An Eulerian cycle is a cycle which uses each edge precisely once.

The example graph does not contain an Eulerian path, but it does contain a Hamiltonian path.

C₃ is called a triangle.

A spanning cycle is also called a Hamiltonian cycle. A graph that contains a Hamiltonian cycle is a Hamiltonian graph.

2.3 Tree

A tree is a connected acyclic simple graph. A vertex of degree 1 is called a leaf, or pendant vertex. An edge incident to a leaf is an leaf edge, or pendant edge. (Some people define a leaf edge as a leaf and then define a leaf vertex on top of it. These two sets of definitions are often used interchangeably.) A non-leaf vertex is an internal vertex. Sometimes, one vertex of the tree is distinguished, and called the root. A rooted tree is a tree with a root. Rooted trees are often treated as directed acyclic graphs with the edges pointing away from the root.

Trees are commonly used as data structures in computer science (see tree data structureIn computer science, a tree is a widely-used computer data structure that emulates a tree structure with a set of linked nodes. Each node has zero or more child nodes, which are below it in the tree (in computer science, unlike in nature, trees grow down,).

A forest is a vertex-disjoint union of trees; or, equivalently, an acyclic graph.

A subtree of the graph G is a subgraph that is a tree.

A spanning tree is a spanning subgraph that is a tree. Every graph has a spanning forest. But only a connected graph has a spanning tree.

A special kind of tree called star is K_1,k. An induced star with 3 edges is a claw.

A k-ary tree is a rooted tree in which every internal vertex has k children. An 1-ary tree is just a path. A 2-ary tree is also called a binary tree.

2.4 Clique

The complete graphGraphs Graph theory A complete graph is a simple graph where an edge connects every pair of vertices. A complete graph on n vertices has n vertices and n ''n minus;1)/2 edges, and is indicated by the notation K''. It is a regular graph of degree n minus;1 K_n of order n is a simple graph with n vertices in which every vertex is adjacent to every other. The example graph is not complete. The complete graph on n vertices is often denoted by K_n. It has n(n-1)/2 edges (corresponding to all possible choices of pairs of vertices).

A cliqueIn computer science, the Clique Problem is an NP-complete problem in complexity theory. A clique in a graph is a set of pairwise adjacent vertices. In the graph at the right, vertices 1, 2 and 5 form a clique. The clique problem is the optimization proble (pronounced "click") in a graph is a set of pairwise adjacent vertices. Since any subgraph induced by a clique is a complete subgraph, the two terms and their notations are usually used interchangeably. A k-clique is a clique of order k. In the example graph above, vertices 1, 2 and 5 form a 3-clique, or a triangle.

The clique number ω(G) of a graph G is the order of a largest clique in G.

3 Adjacency and degree

In graph theory, degree, especially that of a vertex, is usually a measure of immediate adjacency.

An edge connects two vertices; these two vertices are said to be incident to that edge, or, equivalently, that edge incident to those two vertices. All degree-related concepts have to do with adajacency or incidence.

The degree, or valency, d_G(v) of a vertex v in a graph G is the number of edges incident to v, with loops being counted twice. A vertex of degree 0 is an isolated vertex. A vertex of degree 1 is a leaf. In the example graph vertices 1 and 3 have a degree of 2, vertices 2,4 and 5 have a degree of 3 and vertex 6 has a degree of 1. If E is finite, then the total sum of vertex degrees is equal to twice the number of edges.

A degree sequence is a list of degrees of a graph in non-increasing order (e.g. d₁ ≥ d₂ ≥ … ≥ d_n). A sequence of non-increasing integers is realizable if it is a degree sequence of some graph.

Two vertices u and v are considered adjacent if an edge exists between them. We denote by u ↓ v. In the above graph, vertices 1 and 2 are adjacent, but vertices 2 and 4 are not. The set of neighbors , called a (open) neighborhood N_G(v) for a vertex v in a graph G, consists of all vertices adjacent to v but not including v. When v is also included, it is called a closed neighborhood, denoted by N_G[v]. When stated without any qualification, a neighborhood is assumed to be open. The subscript G is usually dropped when there is no danger of confusion. In the example graph, vertex 1 has two neighbors: vertices 2 and 5. For a simple graph, the number of neighbors that a vertex has coincides with its degree.

A dominating set of a graph is a vertex subset whose closed neighborhood include all vertices of the graph. A vertex v dominates another vertex u if there is an edge from v to u. A vertex subset V dominates another vertex subset U if every vertex in U is adjacent to some vertex in V. The minimum size of a dominating set is the domination number γ(G).

In computers, a finite, directed or undirected graph (with n vertices, say) is often represented by its adjacency matrixIn mathematics and computer science, the adjacency matrix for a finite graph on n vertices is an n × n matrix in which entry a is the number of edges from v to v in. There exists a unique adjacency matrix for each graph, and it is not the adjacency matrix: an n-by-n matrixAbstract algebra Algebra Linear algebra In mathematics, a matrix (plural matrices is a rectangular table of numbers or, more generally, of elements of a fixed ring. In this article, if unspecified, the entries of a matrix are always real or complex number whose entry in row i and column j gives the number of edges from the i-th to the j-th vertex.

Spectral graph theoryIn mathematics, spectral graph theory is the study of properties of a graph in relationship to the eigenvalues and eigenvectors of its adjacency matrix. An undirected graph has a symmetric adjacency matrix and therefore has real eigenvalues and a complete studies relationships between the properties of the graph and its adjacency matrix.

The maximum degree Δ(G) of a graph G is the largest degree over all vertices; the minimum degree δ(G), the smallest.

A graph in which every vertex has the same degree is regularIn graph theory, a regular regular graph has all vertices of the same valency; which is to say, every vertex has k neighbors. A regular graph with vertices of valency k is called k-valent graph . A strongly regular graph is a regular graph where every adj. It is k-regular if every vertex has degree k. A 0-regular graph is an independent set. A 1-regular graph is a matching. A 2-regular graph is a vertex disjoint union of cycles. A 3-regular graph is said to be cubic, or trivalent.

A k-factor is a k-regular spanning subgraph. An 1-factor is a perfect matching. A partition of edges of a graph into k-factors is called a k-factorization. A k-factorable graph is a graph that admits a k-factorization.

A graph is biregular if it has unequal maximum and minimum degrees and every vertex has one of those two degrees.

A strongly regular graph is a regular graph such that any adjacent vertices have the same number of common neighbors as other adjacent pairs and that any nonadjacent vertice have the same number of common neighbors as other nonadajacent pairs.

3.1 Independence

In graph theory, the word independent usually carries the connotation of pairwise disjoint or mutually nonadjacent. In this sense, independence is a form of immediate nonadjacency. An isolated vertex is a vertex not incident to any edges. An independent set, or stable set or staset, is a set of isolated vertices. Since the graph induced by any independent set is an empty graph, the two terms are usually used interchangeably. In the example above, vertices 1, 3, and 6 form an independent set; and 3, 5, and 6 form another one.

The independence number α(G) of a graph G is a the size of a largest independent set of G.

A graph can be decomposed into independent sets in the sense that the entire vertex set of the graph can be partitioned into pairwise disjoint independent subsets. Such independent subsets are called partite sets, or simply parts.

A graph that can be decomposed into two partite sets but not fewer is bipartite; three sets but not fewer, tripartite; k sets but not fewer, k-partite; and an unknown number of sets, multipartite. An 1-partite graph is the same as an independent set, or an empty graph. A 2-partite graph is the same as a bipartite graph. A graph that can be decomposed into k partite sets is also said to be k-colorableIn graph theory, graph coloring is an assignment of colors almost always taken to be consecutive integers starting from 1 without loss of generality, to certain objects in a graph. Such objects can be vertices, edges, faces, or a mixture of the above..

A complete multipartite graph is a graph in which vertices are adjacent if and only if they belong to different partite sets. A complete bipartite graph is also referred to as a biclique.

A k-partite graph is semiregular if each of its partite set has a uniform degree; equipartite if each partite set has the same size; and balanced k-partite if each partite set differs in size by at most 1 with any other.

The matching number α′(G) of a graph G is a the size of a largest matching, or pairwise vertex disjoint edges, of G.

A spanning matching, also called a perfect matching is a matching that covers all vertices of a graph.

4 Connectivity

Connectivity extends the concept of adjacency and is essentially a form (and measure) of concatenated adjacency.

If it is possible to establish a path from any vertex to any other vertex of a graph, the graph is said to be connected; otherwise, the graph is disconnected. A graph is totally disconnected if there is no path connecting any pair of vertices. This is just another name to describe an empty graph or independent set.

A cut vertex, or articulation point, is a vertex whose removal disconnects a graph. A cut set, or vertex cut or separating set, is a set of vertices whose removal disconnects the graph.

If it is always possible to establish a path from any vertex to every other one even after removing any k - 1 vertices, then the graph is said to be k-connected. Note that a graph is k-connected if and only if it contains k internally disjoint paths between any two vertices. The example graph above is connected (and therefore 1-connected), but not 2-connected. The connectivity κ(G) of a graph G is the minumum number of vertices needed to disconnect G. By convention, K_n has connectivity n - 1; and a disconnected graph has connectivity 0.

A bridge, or cut edge or isthmus, is an edge whose removal disconnects a graph. (For example, a tree is made entirely of bridges.) A disconnecting set is a set of edges whose removal increases the number of components. An edge cut is the set of all edges having one endvertex in some proper vertex subset S and another endvertex in V(G)\S. Edges of K₃ form a disconnecting set but not an edge cut. Any two edges of K₃ form a minimal disconnecting set as well as an edge cut. An edge cut is necessarily a disconnecting set; and a minimal disconnecting set of an nonempty graph is necessarily an edge cut. A bond is a minimal (but not necessarily minimum), nonempty set of edges whose removal disconnects a graph.

A graph is k-edge-connectedif any subgraph formed by removing any k - 1 edges is still connected. The edge connectivity κ′(G) of a graph G is the minumum number of edges needed to disconnect G. One well-known result is that κ(G) ≤ κ′(G) ≤ δ(G).

A component is a maximally connected subgraph; a block, either a maximally 2-connected subgraph or a bridge with its endvertices; and a biconnected component, a maximal set of edges in which any two members lie on a common simple cycle.

5 Distance

The distance d_G(u, v) between two (not necessary distinct) vertices u and v in a graph G is the length of a shortest path between them. The subscript G is usually dropped when there is no danger of confusion. When u and v are identitcal, their distance is 0. When u and v are unreachable from each other, their distance is defined to be infinity ∞.

The eccentricity ε_G(v) of a vertex v in a graph G is the maximum distance from v to any other vertex. The diameter diam(G) of a graph G is the maximum eccentricity over all vertices in a graph; and the radius rad(G), the minimum. When there are two components in G, diam(G) and rad(G) defined to be infinity ∞. Trivially, diam(G) ≤ 2 rad(G). Vertices with maximum eccentricity are called peripheral vertex. Vertices of minimum eccentricity form the center. A tree has at most two center vertices.

The Wiener index of a vertex v in a graph G, denoted by W_G(v) is the sum of distances between v and all others. The Wiener index of a graph G, denoted by W(G), is the sum of distances over all pairs of vertices. An undirected graph's Wiener polynomial is defined to be Σ q^d(u,v) over all unordered pairs of vertices u and v. Wiener index and Wiener polynomial are of particular interests to mathematical chemists.

The k-th power G^k of a graph G is a supergraph formed by adding an edge between all pairs of vertices of G with distance at most k. A second power of a graph is also called a square.

A k-spanner is a spanning subgraph in which every two vertices are at most k times as far apart on S than on G. The number k is the dilation. k-spanner is used for studying geometric network optimization.

6 Genus

A crossing is a pair of intersecting edges. A graph is embeddable on a surface if its vertices and edges can be arranged on it without any crossing. The genus of a graph is the lowest genus of any surface on which the graph can embed.

A planar graph is one which can be drawn on the (Euclidean) plane without any crossing; and a plane graph, one which is drawn in such fashion. In other words, a planar graph is a graph of genus 0. The example graph is planar; the complete graph on n vertices, for n> 4, is not planar. Also, a tree is necessarily a planar graph.

When a graph is drawn without any crossing, any cycle that surrounds a region without any edge reaching from the cycle inside to such region forms a face. Two faces on a plane graph are adjacent if they share a common edge. A dual, or planar dual when the context needs to be clarified, G^* of a plane graph G is a graph whose vertices represent the faces, including any outerface, of G and are adjacent in G^* if and only if their corresponding faces are adjacent in G. The dual of a planar graph is always a planar pseudograph (e.g. consider the dual of a triangle). In the familiar case of a 3-connected simple planar graph G (isomorphic to a convex polyhedron P), the dual G^* is also a 3-connected simple planar graph (and isomorphic to the dual polyhedron P^*).

Furthermore, since we can establish a sense of "inside" and "outside" on a plane, we can identify an "outermost" region that contains the entire graph if the graph does not cover the entire plane. Such outermost region is called an outer face. An outerplanar graph is one which can be drawn in the planar fashion such that its vertices are all adjacent to the outer face; and an outerplane graph, one which is drawn in such fashion.

The minimum number of crossings that must appear when a graph is drawn on a plane is called the crossing number.

The minimum number of planar graphs needed to cover a graph is the thickness of the graph.

7 Weight

A weighted graph associates a real number label (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 word cost is used instead of weight. When stated without any qualification, a graph is always assumed to be unweighted.

8 Direction

An arc, or directed edge, is an ordered pair of endvertices. In such ordered pair, the first vertex is called a head, or initial vertex; and the second one, a tail, or terminal vertex. It can be thought of as an edge associated with a direction, namely designating a head and a tail to the endvertices. An undirected edge disregards any sense of direction and treats both endvertices interchangeably. A loop in a diagraph , however, keeps a sense of direction and treats both head and tail identically. A set of arcs are multiple, or parallel, if they share the same head and the same tail. A pair of arcs are anti-parallel if one's head/tail is the other's tail/head. A digraph, or directed graph, is analogous to an undirected graph except that it contains at least an arc. An oriented graph contains only arcs. When stated without any qualification, a graph is almost always assumed to be undirected. Also, a digraph is usually assumed to contain no undirected edges.

A digraph is called simple if it has no loops and at most one arc between any pair of vertices. When stated without any qualification, a digraph is usually assumed to be simple.

In a digraph Γ, we distinguish the out degree d_Γ⁺(v), the number of edges leaving a vertex v, and the in degree d_Γ^-(v), the number of edges entering a vertex v. The degree d_Γ(v) of a vertex v is equal to the sum of its out- and in- degrees. When the context is clear, the subscript Γ can be dropped. Maximum and minimum out degrees are denoted by Δ⁺(Γ) and δ^{+(Γ); and maximum and minimum in degrees, Δ^-(Γ) and δ^-(Γ).}

An out-neighborhood, or success set, N⁺_Γ(v) of a vertex v is the set of tails of arcs going from v. Likewise, an in-neighborhood, or predecessor set, N^-_Γ(v) of a vertex v is the set of heads of arc going into v.

A source is a vertex with 0 in-degree; and a sink, 0 out-degree.

A vertex v dominates another vertex u if there is an arc from v to u. A vertex subset S is out-dominating if every vertex not in S is dominated by some vertex in S; and in-dominating if every vertex in S is dominated by some vertex not in S.

A kernel is an independent out-dominating set. A digraph is kernel perfect if every induced sub-digraph has a kernel.

An Eulerian digraph is a digraph with equal in- and out-degrees at every vertex.

An orientation is an assignment of directions to the edges of an undirected or partially directed graph. When stated without any qualification, it is usually assumed that all undirected edges are replaced by a directed one in an orientation. Also, the underlying graph is usually assumed to be undirected and simple.

A tournament is a digraph in which each pair of vertices is connected by exactly one arc. In other words, it is an oriented complete graph.

A directed path, or just a path when the context is clear, is an oriented simple path such that all arcs go the same direction, meaning all internal vertices have in- and out-degrees 1. A vertex u is reachable from another vertex v if there is a directed path that starts from u and ends at v. Note that in general the condition that u is reachable from v does not imply that v is also reachable from u.

A digraph is strongly connected if every vertex is reachable from every other following the directions of the arcs. On the contrary, a diagraph is weakly connected if its underlying undirected graph is connected. A weakly connected graph can be thought of as a digraph in which every vertex is "reachable" from every other but not necessarily following the directions of the arcs. A strong orientation is an orientation that produces a strongly connected digraph.

A directed cycle, or just a cycle when the context is clear, is an oriented simple cycle such that all arcs go the same direction, meaning all vertices have in- and out-degrees 1. A digraph is acyclic if it does not contain any directed cycle. A finite, acyclic digraph with no isolated vertices necessarily contain at least one source and at least one sink.

An arborescence, or out-tree or branching, is an oriented tree in which all vertices are reachable from a single vertex. Likewise, an in-tree is an oriented tree in which a single vertex is reachable from every other one.

9 Various

A graph invariant is a property of a graph G, often a number, that does not depend upon the ordering of the vertices in G. Examples are graph genus and graph order .

10 See also

11 General references

Bollobás, Béla (1998). Modern graph theory. New York: Springer-Verlag. BooksEnthsiast.com [Packed with advanced topics followed by a historical overview at the end of each chapter.]
West, Douglas B. (2001). Introduction to graph theory (2ed). Upper Saddle River: Prentice Hall. BooksEnthsiast.com. [Tons of illustrations, references, and exercises. The most complete introductory guide to the subject.]
Eric W. Weisstein. "Graph." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Graph.html

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