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Graph (graph theory)

In graph theory, a graph is a (usually finite) nonempty set of vertices that are joined by a number (possibly zero) of edges. Graphs are frequently represented graphically, with the vertices as points and the edges as smooth curves joining pairs of vertices.

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Formally, a graph $G$ is a pair, $G = (V, E)$, of a set $V$ of vertices together with a class of subsets $E$ made up of pairs of elements from $V$. Note that this definition describes simple, loopless graphs: there is at most one edge joining two vertices, no edge may join a vertex to itself, and the edges are not directed. For graphs with multiple edges, see multigraph. If the edges are directed, then $E$ may be defined using ordered pairs from the product set $V \times V$.

Definitions

  • If $v \in V$, $e \in E$ and $v \in e$ then we say $e$ and $v$ are incident. If $e, f \in E$ and $v \in e, f$ we say the edges $e$ and $f$ are coincident at $v$.
  • The number of edges in $E$ containing $v$ is the degree of $v$ and is often denoted $d(v)$.
  • A vertex $v$ is isolated if $d(v) = 0$, i.e. if there are no edges incident to $v$.
  • If $G_1 = (V_1, E_1)$ and $G_2 = (V_2, E_2)$ are graphs such that $V_2 \subseteq V_1$ and $E_2 \subseteq E_1$ then we say $G_2$ is a subgraph of $G_1$. If $E_2 = \{\{v_1, v_2\} \mid \{v_1, v_2\} \in E \textrm{ and } v_1, v_2 \in V_2$ (informally, if $E_2$ contains all those edges of $E_1$ whose vertices are in $V_2$) then we say that $G_2$ is an induced subgraph of $G_1$.

Types of Graphs and Subgraphs

Complete Graph or Clique

A complete graph is a graph in which there is an edge joining every pair of vertices is connected. The complete graph on $n$ vertices is denoted $K_n$, and has $\binom{n}{2} = \frac{n(n-1)}{2}$ edges. If $H$ is a complete subgraph of $G$ then the vertices of $H$ are said to form a clique in $G$.

Complementary Graphs

If $G_1 = (V, E_1)$ and $G_2 = (V, E_2)$ are two graphs on the same vertex set such that $G = (V, E_1 \cup E_2)$ is a complete graph and $E_1 \cap E_2 = \emptyset$ then $G_2$ is said to be the complement of $G_1$ and vice-versa.

Null Graph or Independent Set

A null graph (or independent set) is the complement of a complete graph. Equivalently, a null graph is a graph in which every vertex is isolated. When drawn in the usual fashion, a null graph is simply a collection of scattered points (the vertices) with no edges connecting them. The terminology "independent set" is used most frequently to refer to a subgraph. In other words, one says $V_1 \subseteq V_1$ is an independent set in $G= (V, E)$ if and only if $V_1$ is a clique in the complement of $G$.

Connected Graph

An undirected graph $G$ is connected if for all $u, v \in V$, there exists a path from $u$ to $v$ using only edges in $G$. That is, there are no isolated vertices with no paths coming from them, nor can the vertex set be partitioned into two parts with no edge between them. A related concept is a connected component, which is a maximally connected subgraph of a graph.

Strongly Connected Graph

A directed graph $G$ is strongly connected if for all $u, v \in V$, there exists a directed path from $u$ to $v$ using only edges in $G$.

Planar Graphs

A graph is said to be planar if it can be drawn in a plane with no intersecting edges. For example, $K_1,K_2,K_3,$ and $K_4$ are planar, but $K_5$ and $K_{3,3}$ are not planar.

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In a planar graph, we can define faces of the graph, or the smallest regions bounded by edges. (An alternate definition is the regions bounded by edges which do not have any edges going through them.) Note that the area outside the planar graph is also a face, called the unbounded face. The degree of the face is the number of edges that bound the face. (Note that the same term is used for vertices, which can become confusing)

All planar graphs have dual graphs, which involve turning the planes of one graph into vertices, and the vertics into planes, with edges connecting if two planes are adjacent. The dual of the dual of a graph is returns the original graph.

An interesting result is Euler's Polyhedral Formula, which states that in a planar graph with $V$ vertices, $E$ edges, and $F$ faces, then \[V-E+F=2\] The proof of this is simple using induction, but the derivation of the formula is much trickier.

Other interesting results for planar graphs are that:

  • $E\le 3V-6$
  • if the the sum of the degrees of the faces of the graph is $F$, then $F\le \frac{2}{3}E$.

Bipartite graph

A graph is called bipartite if its vertex set can be split into two disjoint subsets $L$ and $R$ such that every edge connects a vertex in $L$ to a vertex in $R$ (by this definition, the empty graph on $n$ vertices is bipartite). A graph $G$ is bipartite if and only if it has no odd cycles, if and only if $G$ is 2-colorable. Bipartite graphs have many applications including matching problems.

The complete bipartite graph (denoted $K_{m,n}$ for integers $m$ and $n$) is a bipartite graph where $|L| = m$, $|R| = n$, and there is an edge connecting every $u \in L$ to every $v \in R$ (so that $K_{m,n}$ has $mn$ edges).

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Walks

A walk is the general process of moving along the edges of a graph. A path does not go through any vertex more than once, while a trail does not go through any edge more than once

Paths and Cycles

A path in a graph $G = (V, E)$ is a sequence $v_0, e_1, v_1, \ldots, e_n, v_n$ such that $v_i \in V$, $e_i \in E$ and $e_i = \{v_{i - 1}, v_i\}$ for all $i$. A cycle is a path in which the initial and final vertices are the same.

Euler Trail

A Euler trail is a graph where it is possible to form a trail which uses all the edges. A Euler trail has at most two vertices with odd degrees. The sum of all the degrees of the vertices equals twice the number of edges in the graph.

Trees and Forests

A forest is a graph which does not have any cycles. A tree is a connected forest.

Weighted Graphs

The edges of a graph can have weights assigned to them that represent some value or "cost" (such as distance). For example, Dijkstra's algorithm, which computes the shortest path from a source vertex $s \in V$ to all vertices in $v \in V$, runs on a graph whose edge weights are non-negative.

Hypergraph

A hypergraph is an extension of the concept of a graph where the edges can encompass more than two vertices, and essentially become sets themselves. Hypergraph theory is often difficult to visualize, and thus is often studied based on the sets that make it up.


See Also

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