Degree Sequences & Digraphs
Degree Sequences & Digraphs
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Graph Theory: Degree Sequences, Graphic Sequences, and Digraphs, Slides of Computer Science
Various concepts in graph theory, including degree sequences, graphic sequences, and digraphs. It explains that the degree sequence of a graph is the list of its vertex degrees in non-increasing order, and that a graphic sequence is a list of non-negative numbers that is the degree sequence of some simple graph. The document also covers the concept of 22-switches, which can be used to transform one simple graph into another, and the orientation of a digraph, which is an orientation of a graph obtained by choosing an orientation for each edge. The document also mentions that every tournament, an orientation of a complete graph, has a king vertex.
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2012/2013
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Degree Sequences & Digraphs
Degree Sequences & Digraphs
Degree Sequence
Degree Sequence
The The
degree sequence degree sequence
of a graph is the list of a graph is the list
of vertex degrees, usually written in non of vertex degrees, usually written in non-
increasing order, as d increasing order, as d
1 1
d
d
n n
Graphic Sequence
Graphic Sequence
A
A
graphic sequence graphic sequence
is a list of non is a list of non-
-negative
negative
numbers that is the degree sequence of numbers that is the degree sequence of
some simple graph. some simple graph.
- A simple graph with degree sequence
A simple graph with degree sequence
d d realizes
realizes
d. d.
Graphic: necessary & sufficient
Graphic: necessary & sufficient
For n>1, the non For n>1, the non-
-negative integer list
negative integer list
d d
of of
size size
n n
is graphic if and only if is graphic if and only if
d
d
is graphic, is graphic,
where where
d d
is the list of size is the list of size
n n-
obtained obtained
from from
d d
by deleting its largest element by deleting its largest element
, and , and
subtracting 1 from its subtracting 1 from its
next largest next largest
elements. elements.
[
[Havel
Havel 1955,
1955, Hakimi
Hakimi 1962]
1962]
Orientation of a Digraph
Orientation of a Digraph
An An
orientation orientation
of a graph G is a digraph D of a graph G is a digraph D
obtained from G by choosing an orientation obtained from G by choosing an orientation
(x (x
Æ
Æ
y or y y or y
Æ
Æ
x) for each edge x) for each edge
xy xy
E(G).
E(G).
A
A
tournament tournament
is an orientation of a
is an orientation of a
complete graph. complete graph.
King
King
A
A
king king
is a vertex from which every vertex is is a vertex from which every vertex is
reachable by a path of length at most 2. reachable by a path of length at most 2.
Every tournament has a king. Every tournament has a king.