This document provides definitions and theorems related to domination and strong domination of graphs. It begins with introductions to graph theory concepts like vertex degree. It then defines different types of domination like dominating sets, connected dominating sets, and k-dominating sets. Further definitions include total domination, strong domination, and dominating cycles. Theorems are provided that relate strong domination number to independence number and domination number. The document concludes by discussing applications of domination in fields like communication networks and distributing computer resources.

1. G.Nithya M.sc.,M.Phil.,(Ph.D).,
Associate Professor ,Department of Mathematics,
Sri Adi Chunchanagiri Women’s College,
Kumali Main road,Cumbum-625533,Tamil
Nadu,India.
nithyarajkumar15@gmail.com

2. INTRODUCTION
 In this paper we consider simple graph G finite, undirected, connected ,
domination of graphs and Strongly Domination of graphs. The vertex set and edge
set of the graph G is denoted by V (G) and E(G) respectively. We denote the
degree of a vertex v in a graph G by deg(v). The maximum and minimum degree
of the graph G is denoted by △(G) and δ(G) respectively.
 A graph G = (V, E) consists of a vertex set V and edge set E. Let n = |V(G)| denote
the order of G. In a graph G, the degree of a vertex v is the number of vertices
adjacent to v, denoted by dG(v). The minimum and maximum degree of a graph
are denoted by δ(G) and ∆(G) respectively. A vertex v is an isolated vertex if and
only if dG(v)= 0. A graph is connected if for every pair of vertices u and v there is
a u — v path in the graph. If G is connected, then the distance between two
vertices u and v is the minimum length of a u — v path in G, denoted by dG(u, v).
Let NG(v) denote the set of neighbours of a vertex vϵV(G), and let NG[v] = NG(v)
U {v} be the closed neighbourhood of v in G. Let dG[v] = |NG[v] | = dG(v) + 1.

3. DEFINITIONS
 A dominating set D is a set of vertices such that each vertex of
G is either in D or has at least one neighbour in D. The
minimum cardinality of such a set is called the domination
number of G, γ(G).
 Connected dominating set is a dominating set which induces a
connected subgraph. Since each dominating set has at least one
vertex in each component of G, only connected graphs have a
connected dominating set. Therefore, here we may assume all
graphs are connected. The minimum cardinality of a connected
dominating set is called connected domination number γc(G).
In Figure 9 filled vertices form a connected dominating set of
minimum size and therefore, γc(G)= 3.

4.  A dominating cycle is a cycle in which every vertex is in the
neighbourhood of a vertex on the cycle. The minimum
cardinality of a dominating cycle is denoted by γcy(G). In
Figure 3, filled vertices form a dominating cycle of
minimum size and hence, γcy(G) = 3.

5.  A k-dominating set is a set of vertices D such that
each vertex in V(G) - D is dominated by at least k
vertices in D for a fixed positive integer k. The
minimum cardinality of a k-dominating set is called k-
domination number γk(G). In k-domination Fink and
Jacobson presented the upper bound in terms of
order, maximum degree and k.

6.  Total dominating set is a set of vertices such that each
vertex vεV is in open neighbourhood of a vertex in the
set. Note that in total domination vertex v does not
dominate itself and so it is required that there be no
isolated vertex. The minimum cardinality of a total
dominating set is called the total domination number
γt(G). The decision problem to determine the total
domination number of a graph is known to be NP-
complete

7.  If uv is an edge of G then, u strongly dominates v if
dG(u) ≥ dG(v). A set S ⊆V(G)is a strong dominating set
(sd − set) if every vertex v ∈ V(G) − S is strongly
dominated by some u in S. The minimum cardinality
of a strong dominating set is called the strong
domination number of G and it is denoted by γst(G).
Analogously, one can define a weak dominating set
(wd −set).

8.  In Figure , S = {v1, v2, v3, v4, v5} is strong dominating set
of the graph C5 ◦P2 and γst(C5 ◦P2) = 5. The strong
dominating set is shown with solid vertices.

9. Theorem :
For any graph G with no isolated vertex, γst(G)≤α(G)+γ(G).
Proof.
Let D be a γ(G)-set. Let I be an α(G)-set such that |D∩I| is at its maximum among
all α(G)-sets. Notice that for any x∈D∩I, epn(x,D∪I)∪ipn(x,D∪I)⊆epn(x,I).
(1) We next define a set S⊆V(G) of minimum cardinality among
the sets satisfying the following properties.
(a) D∪I⊆S.
(b) For every vertex x∈D∩I,
(b1) if epn(x,D∪I)≠⌀, then S∩epn(x,D∪I)≠⌀;
(b2) if epn(x,D∪I)=⌀, ipn(x,D∪I)≠⌀ and epn(x,I)ipn(x,D∪I)≠⌀, then either
epn(x,I)D=⌀ or S∩epn(x,I)D≠⌀;
(b3) if epn(x,D∪I)=⌀ and epn(x,I)=ipn(x,D∪I)≠⌀, then S∩N(epn(x,I)){x}≠⌀;
(b4) if epn(x,D∪I)=ipn(x,D∪I)=⌀, then N(x)(D∪I)=⌀ or S∩N(x)(D∪I)≠⌀.
Since D and I are dominating sets, from (a) and (b) we conclude that S is a TDS. From now on,
let v∈V(G)S. Observe that there exists a vertex u∈N(v)∩I⊆N(v)∩S, as I⊆S is an α(G)-set.
We claim that ipn(u,D∪I)={w}., suppose that there exists w′∈ipn(u,D∪I){w}. Notice that w′∈D.
By (1) and the fact that all vertices in epn(u,I) form a clique, we prove that ww′∈E(G), and
so w∉ipn(u,D∪I), which is a contradiction. Therefore, ipn(u,D∪I)={w} and, as a
result,epn(u,D∪I)∪{w}⊆epn(u,I). ………..2
∪I)≠⌀. By (2), (b1),

10. Subcase 2.1. . epn(u,D∪I)≠⌀. By (2), (b1), and the fact that all vertices
in epn(u,I) form a clique, we conclude that w is adjacent to some vertex
in S{u}⊆S′, as desired.
Subcase 2.2. epn(u,D∪I)=⌀ and epn(u,I){w}≠⌀. By (2), (b2), and the fact that
all vertices in epn(u,I) form a clique, we show that w is dominated by some
vertex in S{u}⊆S′, as desired.
Subcase 2.3. epn(u,D∪I)=⌀ and epn(u,I)={w}. In this case, by (b3) we deduce
that w is dominated by some vertex in S{u}⊆S′, as desired.
According to the two cases above, we can conclude that S′ is a TDS
of G, and so S is a STDS of G. Now, by the the minimality of |S|, we show
that |S|≤|D∪I|+|D∩I|=|D|+|I|. Therefore, γst(G)≤|S|≤|I|+|D|=α(G)+γ(G), which
completes the proof.

11. Proposition: If H is a spanning subgraph (with no isolated
vertex) of a graph G, thenγst(G)≤γst(H).
Lemma :1 Let M be a maximum matching of a graph G. The
following statements hold.
i) N(u)⊆VM for every u∈V(G)VM.
(ii) If u∈V(G)VM is adjacent to v∈VM,
then N(v′)⊆VM∪{u}, where v′ is the partner of v.
Lemma :2
For any graph G with L(G)≠∅, there exists a
maximum matching M such that for each vertex x∈S(G) there
exists y∈L(G) such that xy∈M.

12.  Theorem : Let G be a graph of order n.
If γtoc(G)≤n−2, then γst(G)≤⌊γ toc(G)+n2⌋.
 Proof. We assume that γtoc(G)≤n−2. Let D be
a γtoc(G)-set and S a γ(⟨V(G)D⟩)-set. Since D is a
TDS of G, D∪S is a TDS as well. Furthermore, every
vertex u∈V(G)(D∪S) is dominated by some
vertex v∈S, and D⊆(D∪S∪{u}){v} is a TDS of G.
Hence, D∪S is a STDS of G, which implies
that γst(G)≤|D∪S|=|D|+|S|. Now, since ⟨V(G)D⟩ is a
connected nontrivial graph, we have
that |S|=γ(⟨V(G)D⟩)≤|V(G)D|2=n−γtoc(G)2.
Therefore, γst(G)≤⌊γtoc(G)+n2⌋, which completes the
proof.

13. Theorem . If G is a connected graph, then the following
statements are equivalent.
i)γst(G)=γt(G).
ii)γst(G)=2.
iii)G has two universal vertices.
Proof.
The result above is an important tool to characterize
all graphs with γst(G)=3
Note:
The bound above is tight. For instance, it is achieved
for the wheel graph G≅N1+C4 and for G≅N2+P3. In both
cases γst(G)=3 and γtoc(G)=2.

14. Applications of Domination Graphs:
 The concept of domatic partition arises in various areas. In particular, in
the problem of communication networks. Domatic number of a graph
represents the maximum number of disjoint transmitting groups.
 Another application of domatic number is related to the task of
distributing resources in a computer network in the most economic
way.. Distance Domination The concept of domination can be extended
into distance version which is more applicable in practical problems.
 For example consider the communication network problem. Here, a
transmitting group is a subset of those cities that are able to transmit
messages to every city in the network, via communication links, by at
most l hops. Suppose, for another example, resources are to be
distributed in a computer network in such a way that expensive
services are quickly accessible in the neighbourhood of each vertex.
 If every vertex can serve a single resource only, then the maximum
number of resources that can be supported equals the domatic number
in the graph representing the network.

15. CONCLUSIONS
 In this paper to study the total domination number of a graph. We study the
properties of this parameter in order to obtain its exact value or general
bounds. Among our main contributions we highlight the following.
 We show that γst(G)≤α(G)+γ(G). Since γ(G)≤α(G), this result improves the
bound γst(G)≤2α(G)
 We characterize the graphs with γst(G)=3.
 We show that if G is a {K1,3,K1,3+e}-free graph G with no isolated vertex,
then γst(G)≤min{γt(G),γr(G)}+γ(G)≤γs(G)+γ(G).
 We study the relationship that exists between the secure total domination
number and the matching number of a graph.

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17. Thank you