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Graphs Basic Concepts

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Title: Graphs Basic Concepts


1
Graphs Basic Concepts
  • William T. Trotter and Mitchel T. Keller
  • Math 3012 Applied Combinatorics
  • Spring 2009

2
What is a Graph?
3
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 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

4
Notation 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.

5
Data Files for Graphs
  • graph_data.txt
  • 6
  • 2
  • 6
  • 6
  • 1
  • 3
  • 6 2

6
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
7
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
8
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.

9
Paths 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.

10
A Path from 18 to 12
11
Connected Graphs
A graph G is connected if there is a path from
x to y for every distinct pair of vertices in
G.
12
A Connected Graph
13
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.
14
A Disconnected Graph with 3 components
15
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
  • x x0
  • y xt and
  • xi xi1 is an edge for every i 0,1,, t
    -1
  • xt x0 is an edge.

16
Lenth 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.

17
A cycle of length 8
18
Complete and Independent Graphs
19
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
    ?(G).

20
Triangles in Graphs
A clique of size 3 is called a triangle.
1,2,8 is a triangle, but ?(G) 4
21
?(G) 6
22
Caution 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!

23
Determining ?(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.
24
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 a(G).

25
a(G) 12
26
Are 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.
27
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 ?(G), is
    the least positive integer t for which G has
    a t coloring.

28
A 6 - Coloring
This shows that ?(G) 6
29
?(G) 4
30
A Trivial Inequality
?(G) ?(G)
The chromatic number of a graph is at least as
large as the maximum clique size.
31
?(G) ?(G) 4
32
How Good is this Inequality?
?(G) ?(G)
Maybe, just maybe, the chromatic number of a
graph is always equal to the maximum clique size?!
33
Not Always Equal
3 ?(G) gt ?(G) 2
34
Very large ? and small ?
Theorem. For every t 3, there exists a
graph Gt so that ?(G) t and
?(G) 2
35
Posets and Cover Graphs
Cover graphs of posets are triangle-free, i.e.,
?(G) 2
36
Height 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.
37
Posets 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
38
The First Case
P3
3 height(P3) ?(G3) while ?(G3) 2
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