Title: Graphs Basic Concepts
1Graphs Basic Concepts
- William T. Trotter and Mitchel T. Keller
- Math 3012 Applied Combinatorics
- Spring 2009
2What is a Graph?
3What 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 2-element 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
4Notation and Terminology
- Usually, we write xy is an edge in G, or xy ÃŽ
E rather than x,y ÃŽ 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.
5Data Files for Graphs
- graph_data.txt
- 6
- 2
- 6
- 6
- 1
- 3
- 6 2
6Subgraphs 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
7Induced 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
8Isomorphic 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.
9Paths in Graphs
- A path in a graph (from x to y) is a sequence
x0, x1, x2, , xt such that - x x0
- y xt and
- xi xi1 is an edge for every i 0,1,2,, t
-1.
10A Path from 18 to 12
11Connected Graphs
A graph G is connected if there is a path from
x to y for every distinct pair of vertices in
G.
12A Connected Graph
13Components 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.
14A Disconnected Graph with 3 components
15Cycles 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 - x x0
- y xt and
- xi xi1 is an edge for every i 0,1,, t
-1 - xt x0 is an edge.
16Lenth of Paths and Cycles
- When (x0, x1, x2, , xt) is a path, there are
t1 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
t1 vertices in the sequence, but now we say that
the length of the cycle is t1. 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.
17A cycle of length 8
18Complete and Independent Graphs
19Cliques 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
?(G).
20Triangles in Graphs
A clique of size 3 is called a triangle.
1,2,8 is a triangle, but ?(G) 4
21?(G) 6
22Caution Are we certain that ?(G) 6
- To show that ?(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!
23Determining ?(G)
Alice claims that ?(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 ?(G) as an integer on the first
line followed by ?(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.
24Independent 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 a(G).
25a(G) 12
26Are you certain that a(G) 12
Same caution as before. We have shown only that
a(G) 12. It requires much more work to show
that G does not contain an independent set of
size 13.
27Graph 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 ?(G), is
the least positive integer t for which G has
a t coloring.
28A 6 - Coloring
This shows that ?(G) 6
29?(G) 4
30A Trivial Inequality
?(G) ?(G)
The chromatic number of a graph is at least as
large as the maximum clique size.
31?(G) ?(G) 4
32How Good is this Inequality?
?(G) ?(G)
Maybe, just maybe, the chromatic number of a
graph is always equal to the maximum clique size?!
33Not Always Equal
3 ?(G) gt ?(G) 2
34Very large ? and small ?
Theorem. For every t 3, there exists a
graph Gt so that ?(G) t and
?(G) 2
35Posets and Cover Graphs
Cover graphs of posets are triangle-free, i.e.,
?(G) 2
36Height and Chromatic Number
If G is the cover graph of a poset P with
height (P) h, then
?(G) h Since any
partition of P into h antichains is also a
coloring of G using h colors.
37Posets and large ?, small ?
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
?(Gt) t and ?(Gt) 2
38The First Case
P3
3 height(P3) ?(G3) while ?(G3) 2