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3.4   Result
and Analysis

Now
we explore some domination related results on digraphs analogous to those of
undirected graphs. First we look at some common b?unds  for ?(D).
One of the earliest b?unds for the domination number for
any undirected graph
was proposed by Ore.

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Theorem
3.41(Ore
) For any graph G without isolates, ?(G) ?

 , where
n is the number of vertices.

This result does not hold for directed
graphs; ? counter example is the digraph K1,n,
n ? 2, with its arcs directed from the end vertices towards the central
vertex. The general bound which holds for digraphs is not very good for ?
majority of digraphs. We assume our digraphs to be those whose underlying
graphs are connected.

 

Observation
3.42   For any digraph’
with n vertices, ?(D) ? n -1.

This
bound is sharp because the domination number of the digraph ?1,n for

n ? 2 with its arcs directed
from the end vertices towards the central vertex is
n. Since very few graphs agree
with this bound we find other bounds which are
tighter for ? significant number of digraphs.

 

Theorem3.43
For any digraph D on n vertices,

 ? ?(D) ? n –

 (D),
wh???  

 (D) denotes
the maximum
outdegree.

Proof.  For the upper bound we
form ? dominating set of D by including the vertex ? of maximum outdegree and
all the other vertices in the digraph which are not dominated by v. This set is
clearly ? dominating set and has cardinality n –

 (D).

Note that any vertex in D can dominate at most 1 +

 (D) vertices.
In ? minimum dominating set S of D there
are ?(D) vertices, so they can dominate at most ?(D)(1 +

 (D)) vertices. Since S is dominating this
number has to be at least n. Thus
we get the lower bound.

??
 get another bound we look for certain
characteristics in ? digraph.

Observation 3.44 For any digraph D on n vertices, which has ? hamiltonian
circuit, ?(D) ?

 

Proof.
Let
D contain ? hamiltonian circuit
?. ?? dominate the vertices of D
it suffices to dominate the cycle ?. We know that the domination number
of ?
circuit is bounded above by

and so the same holds for the
digraph D.

Theorem
3.45
For ? strongly connected digraph D on
n vertices, ?(D) ?

 

In
addition to ?(D) we introduce some domination related parameters in digraphs,
in particular, the irredundance
number, the u???r irredundance
number and the u???? domination
number, analogous to those for undirected graphs. Recall that ? set S ?V(D) of ? digraph D is
? dominating set if for all

v

S, vis ? successor of some vertex in S. ? dominating set S is
? minimal dominating set if for
every v

S,Ov-OS-v

 If u

 O? – OS – ? , then u will be called ? private outneighbor (??n) of ? with
respect to S. See 38 for another characterization of minimal dominating sets
in digraphs.

Let ?(D), the upper domination
??????, denote the maximum cardinality of ? minimal dominating set. As in the
undirected case, we define an irredundant
set S ?V(D) to be ? set such that every ?

 S has ? private outneighbor. The irredundance number ir(D) and the ????? irredundance number IR(D) are,
respectively, the minimum and maximum cardinalities of ? maximal irredundant
set.

The notion of ?
solution also yields parameters which are new to the field of digraphs. Let
i(D) and

(D) denote respectively the minimum and maximumcardinalities
of an independent dominating set. It must be pointed out that not ?ll digraphs
have independent dominating sets. As these are special cases of solutions,
these exist in digraphs which admit at least one solution. It must be mentioned
here that due to the concepts defined above the following string of
inequalities hold for any digraph D with ? solution,

r(D) ? ?(D) ? i(D) ? ?(D) ? ?(D) ? IR(D).             

Researchers
interested in domination theory for undirected graphs are quite  familiar with the corresponding inequality
chain. This chain raises some interesting  questions about the structural properties of
digraphs D (having ? solution),
for which

 

1.?(D) = i(D),

2. ir(D) = ?(D),

?.?(D) = ?(D) = IR(D), or

4. i(D) ? ?(D).

 

The
following theorem is an interesting result for transitive digraphs.

Theorem
3.46   In ? transitive digraph
D, we have ?(D) = i(D) =
?(D)= ?(D) = IR(D).

Proof.  Note that if D is ? transitive digraph so is its
reversal D-1. It is
then known that ? solution exists in D.
Moreover, from Berge’s theorem, we see that in D, every minimal absorbant set is independent and the kernel is
unique. This implies that ?(D) = I(D)
= ?(D)= ?(D).

?? show ? (D) = IR(D), suppose that S is an irredundant set with ?S? = IR(D). We will call such ? set an IR-set. Amongst all IR-sets let S
contain the minimum number of arcs in it. If S has no arcs, then certainly S is
independent and ? (D) ? IR(D) implying
? (D) = IR(D). So suppose that
< S > contains an arc (?, ?). Since S is irredundant y must have ?
private outneighbor ?1

S. But D is ? transitive digraph, so (? ,?1) must be an arc,
contradicting that y1 is ? private neighbor of x. Hence S is
independent and the result follows.

3.5.
Applications

3.51. Game
Theory (Von Neumann,
Morgenstern )

Suppose
that ? players, denoted by (1),(2),…,(n) can discuss together to
select ? point x from ? set ? (the “situations”). If
player (i) prefers situation ? to situation b, we shall write ? ?I
b. The individual preferences might not be compatible, and consequently it is
necessary to introduce the concept of effective preference. The
situation ? is said to be effectively preferred to b, or ? >
b, if there is ? set of players who prefer ? to b and who are ?ll
together capable of enforcing their preference for ?. However, effective
preference is not transitive; i.?., ?> b and b > ? does not
necessarily imply that ? > ?.

Consider the digraph D = (V, ?) where O(x)
denotes the set of situations effectively preferred to x. Let S be ? kernel of D.
Von Neumann and Morgen-stern suggested that the selection be confined to
the elements of S. Since S is independent, n? situation in S is effectively
preferred to any other situation in S. Since S is absorbant, for every
situation x

 there is ? situation in S that is effectively
preferred to x, so that x can be immediately discarded.

3.52. Problem
in Logic (Berge,
Rao 06)

Let
us consider ? set of properties P = {?1,p2} and ?
set of theorems of the type: “property ?i implies property ?j’
. These theorems can be represented by ? directed graph D = (V, ?) with vertex
set P, where (?i, ?j) is an ??? if and only if it
follows from one or more of the existing theorems that ?i implies ?j
Suppose we want to show that n? arc of the complementary graph

is good to represent ? true
implication of that kind: more precisely, with each ??? (?, q) with ? ? q and

we assign ? student who has to find
an example where ? is fulfilled but not q (i.?., ? counter-example to the
statement that p implies q).

In
10 they determined the minimum number of students needed to show that ?ll the
possible (pairwise) implications are already represented in the di-graph D. It
was found that this number corresponded to the cardinality of the unique kernel
of the transitive digraph under study.

 

3.5
3   Facility Location

Let D = (V, ?) be ? digraph where the
vertices represent “locations” and there is an ??? from location ?
to location ? if location ? can be “reached?’ from
location ?. Assume that each “location” has ? weight
associated with it which represents some parameter pertinent to the study.

Choose ? subset of “locations” such that
those outside the set have an ??? incident from ? member of the set, which
means that ?ll the “locations” can be “serviced’ by the members
of the set S. Let w(S) denote the sum of the weights of the members of S. The
problem of finding such ? set S such that w(S) is minimized. The relevant graph
theoretic concept is that of directed domination.

3.6. 
 Conclusions   and
Open Problems

Domination and other related topics in undirected
graphs are extensively stud-ied,
both theoretically and algorithmically. However, the corresponding topics on
digraphs have not received much attention, even though digraphs come up ??r?
naturally in modelling real world problems. With this view in mind, we have
made an attempt to survey some of the existing results on domination related
concepts on digraphs. We have also introduced some parameters on di-graphs analogous to domination
parameters on undirected graphs. As ? matter of fact, it seems that almost all
domination related problems on undirected graphs, if they make sense in
digraphs, may be investigated. Algorithmic aspects of these problems on
digraphs will be another good area, of research.

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