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Abstract:
Suppose
 D= (V, A) be a digraph. A subset S of V
is called a dominating set of D if for every vertex v  in V – S, there exists a vertex u in S such
that (u, v)?A.We
use the notation

(D) to
represent the domination number of ? digraph, i.?., the minimum cardinality of
? set S ?V(D) which is dominating.In
this paper we present results concerning domination in digraphs along with
application in game theory. In terms of applications, the questions of Facility Location, Assignment Problems etc.
are very much related to the idea, of domination or independent domination on
digraph.

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Keywords: Digraphs

1.       Introduction

Domination and other related concepts in undirected
graphs are well studied. The pioneering work in digraphs in this area can be
ascribed to Berge, ??????, Konig, Grundy and Richardson, amongst others. This
paper detailing the results on digraphs has been naturally influenced by the
book ????? and JJypergraphs by Berge .In this paper some results on
domination in digraphs; the concept and results concerning solutions in ?
digraph and the application of some of these ideas to game theory. The
significant works in these areas by Blidia, Duchet, Galeana-Sanchez, Kwasnik,
Meyneil,Neumann-Lara,Roth, Smith, ???? and others are recorded in this
endeavor. We begin this journey with definitions of the major concepts.

 

2.     Definitions

Perhaps
no other area of domination has as great ? need to standardize defini-tions and notation as that of
directed domination. Different terms are chosen for the same concept and the
same term is occasionally chosen for different con-cepts. We have tried to clarify the situation by giving
common alternate terms and pointing out differences in definitions. For this paper,
unless otherwise mentioned, ? graph D = (V, ?) consists of ? finite vertex set
V and an ??? set ? ??, where P is the set of all ordered pairs of distinct vertices of V.
That is, D has no multiple
loops and no multiple arcs (but pairs of opposite arcs are allowed). For this paper
we assume that the underlying graph of the digraph D is connected. In the terminology of Berge we are considering
connected 1-graphs without loops. Let D
= (V, ?) be such ? digraph. If ? = P then the digraph is complete.
Following Berge, ? subset S ?
V is absorbant if for every
vertex ? ? S there is ? vertex y

 S
such that y is ? successor of x. We define ? set S ? V of ? digraph D to be ? dominating.

3.
Domination in Digraphs

Although the concept of
domination in graphs has received extensive attention as evidenced by this
volume, the same concept has been somewhat sparsely studied for digraphs. Even
bounds undirected graphs have not been considered and compared with their counterparts
for digraphs. In terms of applications, the questions of Facility Location, Assignment Problems etc.
are very much related to the idea, of domination or independent domination on
digraph. There have been over the year’s ? few papers on the domination number
of digraphs. These and other related concepts are presented below. We use the
notation

(D) to
represent the domination number of ? digraph, i.?., the minimum cardinality of
? set S ?V(D) which is dominating

4. New results

In
this section 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.

 

Theorem
4.1(Ore
85) 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
4.2  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 endvertices 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.

 

Theorem4.3
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).

 

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