Basic graph theory (CSCI 2824, Spring 2015)

In this lecture, we will study graphs and some very basic properties of graphs. We will conclude by studying the concept of Eulerian tours.

Directed Graphs

We have already encountered graphs before when we studied relations. We viewed graphs as ways of picturing relations over sets.

Definition: Directed Graphs

A directed graph consists of a finite set of nodes (or vertices) and a set of arcs (or edges) E subseteq V times V .

Each arc is a tuple where are vertices of the graph.
The set of arcs is a relation over .

We draw a graph by drawing circles to represent each of its vertices and arrows to represent arcs. The arc (u,v) is represented by an arrow from to . The direction of the arrow points from to . The arrows have a direction and therefore the graph is a directed graph.

Example 1

Take V={a,b,c,d} and E = {(a,b), (b,c), (a,c), (a,d), (d,b), (d,a), (d,c) } . Here is how we draw this graph:

Example 2

Take V={hello, world, discrete, math} and E = {( hello, hello), (hello, world), (hello, discrete), (hello, math), (math, math), (math, hello), (math, world) } .

Given an arc

The arc (hello,hello) and the arc (math, math) are called self-loops, since they point from a vertex to itself.

Why Study Graphs?

Graphs are useful in a variety of situations. Graph models are really common representations of networks (computer networks, social networks, protein networks,…). It is very useful to model a variety of entities as graphs and study their structure:

Eg., set of nodes = people on facebook and edges = friendship. Prof. Clauset's group at CU Boulder specializes on such graphs, among many others.
Eg., set of nodes = proteins and edges = protein-protein interactions. Our bioinformatics research group (Prof. Goldberg) looks at such interactions in graphs.
Eg., set of nodes = web pages and edges = links between pages.

Undirected Graphs

So far, we have studied directed graphs, which are just representations of relations over finite sets (assume that there are no self-loops). An undirected graph is a special kind of directed graph that occurs when the arc relation is symmetric.

Definition: Undirected Graphs

An undirected graph is a set of vertices along with a set of edges E subseteq V times V such that the relation is symmetric:

Whenever the edge exists in an undirected graph then so does the edge .
For simplicity we would store the edge by instead of storing both and .

As a result, we draw an undirected graph by not drawing placing any arrows on the edges. Edges are simply straight-lines. Alternatively, we could have represented each edge by a double arrow, one in each direction or two sets of arrows. These are all equivalent.

Example 3

Consider the undirected graph : V={a,b,c,d} and E = {(a,b), (b,a), (b,c), (c,b), (a,c), (c,a),(a,d),(d,a), (d,b), (b,d), (d,c),(c,d) } . Verify that the relation represented by is indeed symmetric. Here is how we draw this graph:

Example 4

Take V={hello, world, discrete, math} and E = { (hello, world), (world,hello), (hello, discrete), (discrete, hello), (hello, math), (math,hello), (math, world), (world,math) } .

Once again, this graph has self loops. But we will silently assume, henceforth, that there are no self-loops.

Representing Graphs

There are two ways of representing a graph inside a computer: adjacency list or a adjacency matrix. You should already be aware of adjacency list and matrix representations of graphs from your data-structures class (please email me if you are not). We will not go into these concepts here.

Degrees

In/Out Degree

Let be a directed graph with node set and arc et .

Define the set of all predecessors of :

$delta_{mathsf{in}}(v) = { uin V | (u,v) in E } ,.$

In-Degree

For any vertex v in V , the in-degree of is the number of predecessors of .

Similarly, we can define the set of all successors of :

$delta_{mathsf{out}}(v) = { yin V | (v,y) in E } ,.$

Out-Degree

For any vertex v in V , the out-degree of is the number of successors of .

Note For a undirected graph, the set of incoming edges is the same as the set of out-going edges for any vertex.

Degree

The degree of a vertex in an undirected graph is the number of (we can say either incoming or outgoing) edges that are incident on .

Note that the concepts of in-degree and out-degree coincide with that of degree for an undirected graph.

Degree Sequences

Assumption

In the following, wee assume that any graph we look at does not have self loops. Most of the results discussed below applies to graphs without self-loops.

Going through the vertices of the graph, we simply list the degree of each vertex to obtain a sequence of numbers. Let us call it the degree sequence of a graph. The degree sequence is simply a list of numbers, often sorted.

Example 5

Consider the undirected graph : V={a,b,c,d} and E = {(a,b), (b,a), (b,c), (c,b), (a,c), (c,a),(a,d),(d,a), (d,b), (b,d), (d,c),(c,d) } .

Node	Degree
a	3
b	3
c	3
d	3

The degree sequence is 3,3,3,3 .

Example 6

Here is a graph with degree sequence 1,2,2,3 .

Example 7

Can you construct a graph with a degree sequence 0,1,1 ?

It needs to have three vertices a,b,c and a single edge between and and no edges to .

textcircled{a} quad textcircled{b} - textcircled{c}

Properties of Degree Sequences

Given a undirected graph without self-loops, what can we say about its degree sequence?

Example 8

Can there be an undirected graph (no self-loops allowed) with degree sequence 1,2,3,4,5 ?

Answer

There can be no such graph.

Let us try to construct such a graph. How many nodes does it need to have? from the degree sequence, we know that it has nodes. Now note that there must be a node with degree according to the given degree sequence. However, every node can have between and edges.

Sum of Degree Sequence

For an undirected graph without self-loops, the sum of all the numbers in its degree sequence is exactly twice the number of edges.

In other words, let $V = {v_1,ldots,v_n}$ be the vertex set of an undirected graphs with no self-loops and be the edge set. Let us write the degree of a node v_i as $mathsf{degree}(v_i)$ . We conclude that

$sum_{v in V} mathsf{degree}(v) = 2 |E|$

Proof

By summing up the degree of each vertex, we are counting all edges that are incident on that vertex. In this summation, therefore each edge (u,v) in the graph contributes to a value of (one for the degree of and one for the degree of ). Therefore, we conclude that the summation is twice the number of edges.

As a consequence, the summation of a degree sequence must be even.

Example 9

Is it possible to have a graph (no self-loops allowed, remember) with the following degree sequence 1,1,1,2,2,4,4 ?

Answer

Answer is no since the sum of the degree sequence is 3 + 4 + 8 = 15 which is an odd number.

Degree Sequence and Pigeon Hole Principle

Let be an undirected graph so that

has no self-loops
every vertex of has some edge that incident on it (there are no degree vertices). At least two vertices in the graph must have the same degree.

Proof

The proof is simple application of the pigeon-hole principle. In , every vertex can have a degree between 1,ldots,(n-1) , where n = |V| is the total number of vertices. This means that there are n-1 possible degrees (holes) and possible vertices (pigeons). Therefore two vertices must have the same degree.

In/Out degress for directed Graphs

For a directed graph with vertices $V={v_1,ldots,v_n}$ and edges E subseteq V times V , we observe that

$sum_{v in V} |delta_mathsf{in}(v)|= sum_{v in V} |delta_mathsf{out}(v)| = |E|$ .

In other words, the sum of in-degrees of each vertex coincided with the sum of out-degrees, both of which equal the number of arcs in the graph.

This is because, every arc is incoming to exactly one node and outgoing to exactly one node. Therefore summing up all the in-degrees, counts very arc (u,v) precisely once, when we add up mathsf{incoming}(v) . Similar reasoning applies to out-degrees too.

Walk

Given a graph (can be directed or undirected), with vertices and edges , a walk of the graph is a sequence of alternating vertices and edges such that

We can start our walk at any vertex and end at any vertex.

A single step of the walk takes an outgoing edge from current vertex to visit a neighbouring vertex.

A walk has to respect the edge direction. In other words, if we go from vertex to vertex in a single step then the edge (u,v) must be present. This is important for directed graphs and is trivial for undirected graphs.

Example 10

Take the graph:

Here is a walk:

hello rightarrow hello rightarrow math rightarrow hello rightarrow discrete

Here is another example of a walk:

hello rightarrow hello rightarrow math rightarrow math

Here is an example of a sequence that is not a walk:

hello rightarrow discrete rightarrow world rightarrow world

There is no edge from discrete to world . So our walk cannot go in one step from to world .

Example 11

Take the graph:

Here are some walks

(just start and stop at ).
.
.

Paths

A path in a graph is a walk that does not repeat any vertices. The length of a path is the number of edges traversed by the path and one less than the number of vertices traversed.

Consider, again, the graph below:

Examples of paths include:

(it is a path of length 3)
(it is a path of length 1)
(trivially it is a path of length 0)

Non-examples of paths include:

. This is a walk but not a path since it repeats the vertex .

Eulerian Tour

Read about the Koenigsberg bridge problem here: Seven Bridges of Koenigsberg.

Here is the map of Koenigsberg in Germany where the famous mathematician Leonard Euler lived:

The green ovals show the bridges. Question is can we take a tour of each of the bridges:

starting anywhere we like as long as we return to our starting point
traversing each of the bridges exactly once.

We can ask the same question on the following graph which represents the topology:

Is the connection between the original “topology” of the bridges and graphs clear?

If so, then the problem can be stated as:

Let us “walk” the graph starting from any vertex and traversing any edge that takes us to a neighbouring vertex and so on, such that

Each edge can be traversed at least once in either direction.
We have to end the walk at the same vertex where we began.

Eulerian Tour

Definition:Eulerian Tour

An Eulerian tour is a special walk of the graph with the following conditions:

The walk starts and stops at the same vertex
Every edge in the graph is traversed exactly once during the tour.

Example 12

Does this graph have an Eulerian Tour:

Yes, here is a tour: 2 rightarrow 1 rightarrow 2 rightarrow 3 rightarrow 4 rightarrow 2 .

We started from and ended at . Also note that we have traversed each of the six edges in the graph exactly once.

Example 13

What about this graph?

It does not have a tour.

This is the graph, we derived from the Konigsberg bridge problem. Turns out that we cannot have an Eulerian tour here. To see this, let us focus on the vertex labelled .

has three edges incident on it.
The walk must traverse each of the edges. Let us assume that the walk does not start at . If it does, we can simply do the following reasoning about the node , instead.
We must enter by one of the edges for the first time and leave it by another edge. This leaves one edge untraversed.
But we can only use the untraversed edge to enter and thence we will be stuck (since all the other edges are traversed).

Theorem

An undirected, connected graph (without self-loops) has an Eulerian tour if and only if every vertex in the graph has an even degree.

Since the graph of Konigsberg bridge problem has vertices a,b with odd degree, it cannot have an Eulerian tour. Interestingly, there is an efficient algorithm called Fleury's Algorithm that can be used to construct an Eulerian tour if the given graph has all vertices with even degree.