Graphs/Definitions

Dietel Chapter 1: Graph Definitions and Concepts

Outline:

• Definitions
• graph,
• degree,
• path,
• cycle,
• connectivity,
• tree, forest,
• k-partite,
• contraction,
• Euler tours

Graph definitions

A graph G consist of a set of nodes (vertices) V and edges E, denoted $G(V,E)$

Vertex set on graph G is denoted $V(G)$

Edge set on graph G is denoted $E(G)$

Number of vertices in a graph is called the order and is denoted $|G|$

Number of edges in a graph is denoted $||G||$

Vertex $v\in V$ and edge $e\in E$ are incident if the edge connects to the vertex.

Set of all edges at a particular vertex v is denoted $E(v)$

Two vertices x, y are adjacent on a graph if there is an edge with endpoints x and y

If all vertices are pairwise adjacent, the graph is complete

For two graphs $G=(V,E)$ and $G'=(V',E')$, the graphs are isomorphic if there exists a biijection from G to G'.

If we have two graphs G and G', we say that G' is a subgraph of G if all V' subset of V and all E' subset of E

A subgraph G' is a spanning subgraph of G if all V' span all of G (if V' = V)

Degree definitions

Set of neighbors of a vertex v is denoted $N(v)$

Degree of a vertex v is denoted $d(v)$ and is equal to the number of edges $|E(v)|$ at v

Vertex of degree 0 is isolated

A graph G where all of the vertices have the same degree, k, is called k-regular (or, just regular).

The vertex on the graph with the smallest degree $\delta (G)=\min \left(d(v)|v\in V\right)$ is the minimum degree of G

The vertex on the graph with the largest degree $\Delta (G)=\max \left(d(v)|v\in V\right)$ is the maximum degree of G

The average degree of G is given by the expression $d(G)={\dfrac {1}{|V|}}\sum _{v\in V}d(v)$

Ratio of edges to vertices on a graph is $\epsilon (G)={\dfrac {|E|}{|V|}}$

If we define edges as having two endpoints, then adding up the degrees of all vertices will lead to twice the number of edges. Mathematically: $|E|={\dfrac {1}{2}}\sum _{v\in V}d(v)={\dfrac {1}{2}}d(G)|V|$

This leads to the identity $\epsilon (G)={\dfrac {1}{2}}d(G)$ and the theorem that the number of vertices of odd degree in a graph must always be even. Contrawise proof: if the number of vertices of odd degree is odd, the number of edges is not be an integer.

Path and Cycle Definitions

A path P on a graph G is a non-empty graph that contains vertices and edges that are in G: $V=\{x_{1},x_{2},\dots ,x_{k}\}$ and $E=\{x_{0}x_{1},x_{1}x_{2},\dots ,x_{k-1}x_{k}\}$

A path is usually referred to by the sequence of vertices it visits, or as a path "from/between x1 to xk"

Independent paths are paths containing no common (internal) vertices. Independent paths may share endpoints though.

We can denote parts of a path using special notation: if a path $P=x_{0}\dots x_{k}$, then the following notation is used to denote only a part of that path:

${\begin{aligned}Px_{i}=x_{0}\dots x_{i}\qquad 0\leq i\leq k\\x_{i}P=x_{i}\dots x_{k}\\x_{i}Px_{j}=x_{i}\dots x_{j}\end{aligned}}$

We can also connect paths using unions, or by using more shorthand:

$Px\bigcup xQy\bigcup yR=PxQyR$

A cycle C consists of a path whose final edge connects the last node to the first node. Given a path $P=x_{0}\dots x_{k-1}$ the cycle is then $C=P+x_{k-1}x_{0}$

A k-cycle is denoted $C^{k}$ and is a cycle of length k.

The girth of a cycle is the number of edges or vertices in a cycle in a graph G. The circumference of a graph is the maximum length of a cycle in a graph G.

The distance of two vertices x and y is the length of the shortest path from x to y $d_{G}(x,y)$.

A vertex is central if greatest distance from any other vertex is as small as possible. This minimum distance is the radius of the graph G. Formally:

$rad(G)=\min _{x\in V(G)}\max _{y\in V(G)}d_{G}(x,y)$

Note that the radius of a graph is different from the minimum/average degree.

Connectivity

A graph is connected if any two arbitrary vertices are connected.

If the graph is directed, a connected graph means that for any two arbitrary vertices u and v, there is an edge connecting u to v or v to u. A strongly connected graph means that for any two arbitrary nodes u and v, there is an edge connecting u to v and another edge connecting v to u.

Suppose we have two sets of vertices A and B, and a third set of vertices X. Further suppose that any path that connects a vertex from A to a vertex from B contains a vertex from X. Then we say that X separates the vertex sets A and B.

A subgraph of G that is maximally connected (contains every vertex in G) is a component of G. If a component is connected, it is always non empty.

Vertex connectivity: A graph G is k-connected if it has more than k vertices and if no two vertices of G are separated by fewer than k vertices. The maximum value of k such that G is k-connected is the connectivity of G and is denoted $\kappa (G)$.

Edge connectivity: A graph G is l-edge-connected if every vertex is connected with fewer than l edges (this is a bit unclear). The edge connectivity is denoted $\lambda (G)$.

Theorem due to Mader 1972: Every graph of average degree at least 4k has a k-connected subgraph. (Can prove inductively.)

Trees and Forests

Acyclic graphs are called forests. Connected forests are called trees.

A connected graph with n vertices is a tree if and only if it has n-1 edges.

We can (but don't have to) pick a particular node to be special - the root of the tree. In that case it is a rooted tree. When we pick a root, this imposes an ordering (assuming vertices can be compared). Given two nodes x and y, we say that $x\leq y{\mbox{ if }}x\in rTy$.

A rooted tree T is called normal if any two vertices that are adjacent in the graph are comparable. Every graph has a normal spanning tree.

Bipartite Graphs

A k-partite graph is a graph where the set of vertices V can be partitioned into k classes, such that every edge that starts in one partition will end in a different partition.

If we can select any two vertices from two different classes and they are connected, the k-partite graph is complete.

Bipartite graphs cannot contain cycles of odd lengths. This is always true, so that we can identify bipartite graphs using this property: a graph is bipartite iff it contains no odd cycle.

Edge Contraction

Given a graph G with vertex set V and edge set E. Let e be an edge connecting vertex x to vertex y. Then $G/e$ denotes the graph obtained by contracting the edge into a new vertex, which is now adjacent to all former neighbors of x and y.

Euler Tours

Main article: Graphs/Euler Tours

An Euler tour is a closed walk on a graph that traverses each edge of the graph exactly once. Some graphs have an Euler tour (and are called Eulerian), other graphs are not.

A connected graph is Eulerian if and only if every vertex has even degree. This is because any vertex appearing k times in an Euler tour must have degree 2k.

Other Definitions

A hypergraph is a pair of disjoint sets (V,E) where the elements of the edge sets are non-empty subsets of V. (That is, a given edge e in the set E can connect multiple vertices.)

A directed graph is a pair of disjoint sets (V,E) along with two maps: the init map from V to E (denoting the initialization or origin of the directed edge) and the term map from V to E (denoting the termination of the directed edge).

A multigraph is a pair of disjoint sets (V,E) together with a map from E to V or to V^2. In other words, it is a graph in which we can have edges that begin and end at the same vertex.