Transcript Slide 1

Graphs – Basic Concepts
William T. Trotter and Mitchel T. Keller
Math 3012 Applied Combinatorics
Spring 2009
What is a Graph?
What is a Graph?
 A graph G is a pair (V, E) where V is a set (almost
always finite in this course) and E is a collection of 2element subsets of V.
 Elements of V are called vertices and V is the vertex
set.
 Elements of E are called edges and E is the edge set
Notation and Terminology
 Usually, we write xy is an edge in G, or xy  E rather
than {x,y}  E.
 Of course, xy is an edge if and only if yx is an edge.
 When xy is an edge, we say x and y are adjacent.
 When x and y are distinct and xy is not an edge, we
say that x and y are non-adjacent.
Data Files for Graphs
graph_data.txt
6
3 2
1 6
5 6
2 1
1 3
6 2
Subgraphs – Two kinds
H is a subgraph of G when every vertex of H is a
vertex in G, and every edge in H is an edge in G.
NO
YES
Induced Subgraphs
H is an induced subgraph of G when every vertex of H
is a vertex in G, and every edge in G with both
endpoints in H is an edge in H.
NO
YES
Isomorphic Graphs
G and H are isomorphic when there is a bijection
f : V(G) V(H) between their vertex sets so that
x and y are adjacent in G if and only if f(x) and
f(y) are adjacent in H.
Paths in Graphs
A path in a graph (from x to y) is a sequence
x0, x1, x2, …, xt such that
1.
x = x0 ;
2.
y = xt; and
3.
xi xi+1 is an edge for every i = 0,1,2,…, t -1.
A Path from 18 to 12
Connected Graphs
A graph G is connected if there is a path from x to
y for every distinct pair of vertices in G.
A Connected Graph
Components of Disconnected
Graphs
When a graph is disconnected (not connected), an
induced subgraph H is called a component of G when:
(1) H is connected; and
(2) there is no connected subgraph of G containing the
vertex set of H and having more vertices than H.
A Disconnected Graph with
3 components
Cycles in Graphs
A cycle in a graph G (from x to y) is a
sequence x0, x1, x2, …, xt of distinct vertices
from G with t ≥ 2 such that
1. x = x0;
2. y = xt; and
3. xi xi+1 is an edge for every i = 0,1,…, t -1;
4. xt x0 is an edge.
Lenth of Paths and Cycles
 When (x0, x1, x2, …, xt) is a path, there are t+1
vertices in the sequence, but we say that the length of
the path is t. This counts the number of edges.
 For a positive integer n, it is customary to denote a path
on n vertices as Pn. The length of Pn is then n-1.
 When (x0, x1, x2, …, xt) is a cycle, there are t+1
vertices in the sequence, but now we say that the length
of the cycle is t+1. This again counts the number of
edges since the last vertex is also adjacent to the first.
 For a positive integer n, it is customary to denote a
cycle on n vertices as Cn. The length of Cn is then n.
A cycle of length 8
Complete and Independent Graphs
Cliques in Graphs
 A set S of vertices in a graph G is called a clique when every
distinct pair of vertices in S is adjacent.
 A clique in G is just a set of vertices that induces a complete
subgraph of G.
 The maximum clique size of G is denoted by ω(G).
Triangles in Graphs
A clique of size 3 is called a triangle.
{1,2,8} is a triangle, but ω(G) = 4
ω(G) = 6
Caution: Are we certain that
ω(G) = 6
 To show that ω(G) ≥ 6, it is enough to show that G
contains a clique of size 6.
 But how do we show that G does not contain a clique of
size 7 without testing every subset S consisting of 7
vertices of G? If G contains 34824125 vertices, this
could take a long time!
Determining ω(G)
Alice claims that ω(G) = 257. How does she verify
this claim for a graph G with 10342 vertices?
Can you write C code that will accept a graph data
file as input and output a text file, which has ω(G) as
an integer on the first line followed by ω(G) integers,
one per line, immediately below? If you can do this
and have your algorithm run in time which is
polynomial in the input size, then you are guaranteed
an A++ in this course!! Please share your code with
Professor Trotter before going public.
Independent Sets in Graphs
 A set S of vertices in a graph G is called an
independent set (also a stable set) when no distinct pair
of vertices in S is adjacent in G.
 The maximum size of an independent set of vertices in
G is called the independence number of G and is
denoted α(G).
α(G) = 12
Are you certain that α(G) = 12
Same caution as before. We have shown only that
α(G) ≥ 12. It requires much more work to show
that G does not contain an independent set of size
13.
Graph Coloring
 A t - coloring of a graph is a function f which assigns to
each vertex x an integer f(x) from {1,2,…,t} so that
f(x) ≠ f(y) whenever xy is an edge of G.
 The chromatic number of G, denoted Χ(G), is the least
positive integer t for which G has a t – coloring.
A 6 - Coloring
This shows that Χ(G) ≤ 6
Χ(G) ≤ 4
A Trivial Inequality
Χ(G) ≥ ω(G)
The chromatic number of a graph is at least as large as
the maximum clique size.
Χ(G) = ω(G) = 4
How Good is this Inequality?
Χ(G) ≥ ω(G)
Maybe, just maybe, the chromatic number of a graph is
always equal to the maximum clique size?!
Not Always Equal
3 = Χ(G)
> ω(G) = 2
Very large Χ and small ω
Theorem.
so that
For every t ≥ 3, there exists a graph Gt
Χ(G) = t and ω(G) = 2
Posets and Cover Graphs
Cover graphs of posets are triangle-free, i.e., ω(G) ≤ 2
Height and Chromatic Number
If G is the cover graph of a poset P with
height (P) = h,
then:
Χ(G) ≤ h
Since any partition of P into h antichains is also a
coloring of G using h colors.
Posets and large Χ, small ω
Theorem. For every t ≥ 3, there exists a poset Pt with
height (Pt) = t so that if Gt is the cover graph of Pt, then
Χ(Gt) = t and ω(Gt) = 2
The First Case
P3
3 = height(P3) = Χ(G3)
while ω(G3) = 2