CHAPTER 1 INTRODUCTION. In this chapter some basic definitions of graph theory and domination are given. A bird s

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1 CHAPTER INTRODUCTION. Basic Definitions In this chapter some basic definitions of graph theory and domination are given. A bird s eye view of brief survey of domination results and varieties of domination are discussed. Motivation and few applications are cited. Recently, Sampathkumar and Pushpalatha introduced strong domination and mixed domination. J.H.Hattingh, R.C.Laskar, Henningh and others have worked on this domain and many papers are found in literature and few of the important results are quoted. For standard notations and definition refer [ and 2]. Definition.. A graph G V, X ) consists of a nonempty set V whose elements are called vertices or points) and a possibly empty set X of unordered pairs u, v) of distinct vertices whose elements are called edges or lines). Then of G respectively. V p and X q are called order and size Definition..2 Degree of a vertex v, denoted as dv) is the number of edges incident on v. Similarly, the degree of an edge x uv, denoted as d e x) is the number of edges adjacent to the edge x. Equivalently, x) d u) d v) 2. The maximum degree, the minimum degree, d e the maximum edge degree and the minimum edge degree of G are respectively denoted by G), G), e G), G). If G) G) k, then G is said to be k-regular. A vertex v e is called an isolated vertex if d v) 0 and a pendant vertex if d v). An edge incident on a pendant vertex is called a pendant edge.

2 Definition..3 For any v V, the set N v) { u V uv X} is the open neighbourhood of the vertex v ; while the set N[ v] N v) { v} is the closed neighbourhood of v. Similarly, for any edge x uv, [ ]. For any set S V, N S) N v) is called the open neighbourhood of the set S. Definition..4 A subgraph of G is a graph having all of its vertices and edges in G. A vs spanning subgraph is a subgraph containing all the vertices of G. For any set S V, the induced subgraph S is the maximal subgraph of G with vertex set S. Thus, two vertices of S are adjacent if, and only if, they are adjacent in G. Definition..5 If an edge is incident on a vertex, we say that the vertex and the edge cover each other. A set S V is said to be a vertex cover, if the vertices of S cover all the edges of G. The vertex covering number ) is the minimum cardinality of a vertex cover of G. 0 0 G Definition..6 A set S V is said to be independent if no two vertices are adjacent. The independence number ) is the maximum cardinality of an independent set. 0 0 G Definition..7 A set L X is said to be an edge cover if the edges of L cover all the vertices of G. The edge covering number G) is the minimum cardinality of an edge cover of G. Definition..8 A set L X is said to be edge independent if no two edges are adjacent. The edge independence number G) is the maximum cardinality of an edge independent set of G. An edge independent set is also called a matching. The above four parameters are related by classical Gallai s Theorem [3]. 2

3 Theorem..9 For any isolate free graph G V, X ) with p vertices, 0 0 p p The independent sets play an important role in the concept of graph coloring. Definition..0 A coloring of a graph is an assignment of colors to its vertices. A proper coloring is a coloring so that no two adjacent vertices have same color. The set of all vertices with one color is independent and is called a color class. The chromatic number is the minimum number of colors required to properly color the vertices of a graph G..2 Different types of graphs Definition.2. A complete graph K has every pair of its p p vertices adjacent. The complement G of a graph G has V G) V G) and uv X G) if, and only if, uv X G). In particular, K p has p vertices and no edges. Also, ) p and K p ). For any graph K p with p vertices we have G G K. p Definition.2.2 A bipartite graph G is a graph whose vertex set can be partitioned into two subsets V and V 2 such that every edge of G joins a vertex of V with a vertex of V 2. If every vertex of V is joined with every vertex of V 2, then G is said to be complete bipartite graph and we write where V m and V2 n. In particular, a complete bipartite G K m, n graph is called a star. For any bipartite graph G, G) 2. K, n 3

4 Definition.2.3 A wheel W n invented by the eminent graph theorist W.T. Tutte, is defined as C n K. Definition.2.4 A tree T is a connected acyclic graph. Every tree is a bipartite graph. Any tree has at least two pendant vertices. Any two vertices of a tree are joined by a unique path. A tree with p vertices has p edges. Further, T) 2 for any tree T. Definition.2.5 The corona of two graphs G and G 2 as defined by Frucht and Harary [4] is the graph G = G G 2 formed from one copy of G and V G ) copies of G 2, where the i th vertex of G is adjacent to every vertex in the i th copy of G 2. A Corona C 3 C 3 A spider A wounded spider. A Caterpillar Fig..2.6 Definition.2.7 A subdivision of an edge uv is obtained by replacing the edge uv with a new vertex w and the edges uw and vw. Definition.2.8 A spider is a tree on 2n + vertices obtained by subdividing each edge of a star K, n. Definition.2.9 A wounded spider is the graph formed by subdividing at most n edges of a star. Thus, K, K, n and the corona K, n ) K ) are the examples of wounded spider. K, n Definition.2.0 A caterpillar C is a tree, the deletion of whose end vertices results in a path 4

5 is some times called spine of C. Fig..2.6 provides the examples of a corona C3 C 3, a spider, a wounded spider and a caterpillar..3 Concept of Domination The mathematical study of dominating sets in graphs began around 960. Claude Berge [5] wrote a book on Graph Theory in which he defined the concept of domination number of a graph calling it as coefficient of external stability. In 962, Oystein Ore [6] published his book on Graph Theory in which he used for the first time the names dominating sets and domination number and used the notation d G) for the domination number. In 977, Cockayne and Hedetniemi [7] published a survey of few results known at that time about dominating sets in graphs and they were the first to use the notation G) for the domination number of a graph which has now became the accepted notation. In 979, H.B. Walikar, B.D. Acharya and E. Sampathkumar [8] brought a monogram on Recent developments in the theory of domination and Graphs and its Applications. Recently in 998, T.W. Haynes, S.T. Hedetniemi and P.J. Slater [9 and 0] published a survey of the results in Domination theory in two volumes Fundamentals of domination in graphs and Advanced Topics in Domination..4 Chess Queen problem What is the minimum number of chess pieces of a given type which are necessary to cover / attack / dominate every square of a n n) board? This is an example of problem of finding a dominating set of minimum cardinality. These type of problems were studied in detail around 964 by Yaglom and Yaglom []. These two brothers produced elegant solutions to some of these problems for the rooks, nights, kings and bishops chess pieces. In 850s, Chess 5

6 enthusiasts in Europe considered the problem of determining the minimum number of queens that can be placed on a chess board so that all the squares are either attacked by a queen or are occupied by a queen. It was found that five queens are sufficient to dominate all the squares of 8 8) chess board. Two of the arrangements of five queens such that all the squares are dominated is shown in the Fig..4. a) and b). Q Q Q Q Q Q Q Q Q Q a) b) Fig..4. Two arrangements of queens in 8X8 chess board such that each square is attacked dominated) by one of the queens Definition.4.2 A set S V is a dominating set if for every vertex v V S there exists a vertex u S such that u is adjacent to v. The domination number = G) is the minimum cardinality of a dominating set of G Fig..4.3

7 For the graph G in Fig..4.3, {, 3, 5} is a minimal dominating set of cardinality three, {3, 6, 7, 8} is a minimal dominating set of cardinality four and {2, 4, 6, 7, 8} is a minimal dominating set of cardinality five and G) = 3..5 Applications of Domination The concept of dominance in nature is quite fundamental it is rather a rule for natural stability. In order to have a stable organizational set up in which things are required to go smooth, one cannot ignore imposition of the hierarchy of dominance in its structure. In many instances, dominance is nature s way of maintaining itself as, for instance, one species preying upon another so maintaining a sort of ecological balance. To quote a few, we may consider a whole range of natural and as well as artificial phenomena where dominance is inherent or is invoked hereditary dominance of parental traits in the off springs, the influence of forces of attraction among celestial bodies, surveillance of strategic position or units in a defense network, power hierarchy among individuals working in an organization..6 Bounds on the domination Number The first result gives the domination number of some standard graphs. Proposition.6. For any path P n and cycle C n with n vertices ; For any complete graph K n with n vertices and for any complete bipartite graph K m, n, ) where m, n > Proposition.6.2 For any graph G of order p without isolates, 7

8 Proposition.6.3 For an isolate free bipartite graph, if and only if One may be interested in graphs attaining the upper bound. These graphs are characterized in the next proposition. Definition.6.4 For a graph G we denote by the graph obtained from G by the adjunction of a vertex for every vertex v in G and making the adjacencies for every v Theorem.6.5 Let G be a connected graph of order p. Then if and only if or for some connected graph H. B.A. Reed [2] in 996 obtained the bound for graphs with Theorem.6.6 If G is a connected graph of order p with minimum degree 3, then In 997, L.A. Sanchis [3] obtained bounds related to graphs with minimum degree two. In 2009, Moo Young Sohn and Yuan Xudong [4], published an article in Journal of Korean Mathematical Science, in which a bound for graphs with minimum degree 4 is established. Theorem.6.7 If G is a connected graph of order p with minimum degree 4, then In 2006, Xing Hua-Ming, Sun Lian, Chen Xue-Gang [4] found a bound for graphs with minimum degree five. 8

9 Theorem.6.8 If G is a connected graph of order p with minimum degree 5, then The above bounds are summarized in the following table. Table..6.9 Bound for domination number decreases as the minimum degree increases. Sl.No Lower bound for Upper bound for 0 p The next bound is given in terms of maximum degree. Proposition.6.0 Let G be any p, q) graph and be the maximum degree of G. Then The next bound is given in terms of independence number and vertex covering number. 9

10 Proposition.6. For any graph G with independence number and vertex covering number, Similarly, there are bounds in terms of Matching and edge covering numbers. Proposition.6.2 For any graph G with edge independence number, Corollary.6.2. For any graph G with edge covering number, Proof. The result immediately follows as number. Vizing [6] obtained an upper bound for the size of the graph given the domination Theorem.6.3 Let G be graph of order p and size q with Then Definition.6.4 The vertex connectivity is the minimum number of vertices in such that G - S is disconnected or trivial. If G is disconnected then Its edge analogue the edge connectivity is the minimum number of edges in L such that G - L is disconnected or trivial. The vertex connectivity and edge connectivity are related by the well known Whitney s inequality [7] given in the next proposition. 0

11 Proposition.6.5 For any graph G of order p,. The following result establishes a relation between vertex connectivity and domination number of a graph. Proposition.6.6 For any graph G of order Proposition.6.7 If G is a connected graph of order p such that then G is regular. But the converse of the above result is not true. For example, C 6 is regular graph, does not hold..7 Varieties of Domination A large varieties of domination are studied in literature by imposing the method by which vertices in V D are dominated. Here we mention a few varieties of domination. Definition.7. Independent Dominating set. A dominating set is said to be independent dominating set if the graph induced by the set D has no edges. The independent domination number exists for any given graph and it is observed that The solution of Queens problem in Fig..4. a) is an example of an independent dominating set. Here all the queens are independent no queen attack each other) but they together attack dominate) all the 64 squares.

12 Definition.7.2 Connected Dominating sets Sampathkumar and Walikar [8] defined a connected dominating set to be a dominating set S such that the subgraph S, induced by the set S is connected. The minimum cardinality of a connected dominating set is the connected domination number G). It is immediate c c that G) G) and if G), then G) G) i G). Hedetneimi and Laskar [9] c further studied the connected domination. For an application, consider a network design in which it is desirable that a smallest collection S of processors manage the system s resources and that at least one processor in S be directly accessible by each processor not in S. If it is important that each pair of processors in S can communicate privately, keeping message among the vertices in S, then in the corresponding graph G, we look for G). c c Definition.7.3 Total Dominating sets A solution to the famous Five Queens Problem inspired Cockayne, Dawes and Hedetniemi [20] to introduce total domination. They observed that in the solution of five queens problem in Fig..4. b) not only the squares without queens are dominated by queens but each queen is dominated by another queen. A dominating set S is a total dominating set if S has no isolated vertices. Any graph G without isolated vertices, has a total dominating set and hence G) the total domination number. For an application, we consider a computer network t in which a core group of file servers has the ability to communicate directly with every computer outside the core group. In addition, each file server is directly linked to at least one other backup fileserver where duplicate information is stored. A smallest core group with this property is a t -set for the graph representing the network. 2

13 Definition.7.4 Distance Domination Henning [2] generalized the concept of domination as distance domination. Slater [22] termed a distance k-dominating set as a k-basis. Much of the motivation for the study of distance domination in graphs stems from problems involving the placement of a minimum number of objects hospitals, fire stations, post offices, police stations, warehouses, service centers and the like) within acceptable distances of a given population and the placement of undesirable objects like toxic wastes, nuclear reactors, airports etc.) from maximum distances from a given population. A set S is a distance k -dominating set if for every vertex v V S there is a vertex u S such that d u, v) k. The distance k-domination number G) is the k k minimum cardinality of a distance k-dominating set. It is immediate that G) G ) k where k G is the k th power graph of G..8 Strong Weak) Domination in Graphs The concepts of weak and strong domination were introduced by Sampathkumar and Pushpa Latha in [23]. Definition.8. A set D V is a weak dominating set, denoted wd -set, if every vertex u not in D is adjacent to a vertex v in D where degv) degu). The weak domination number w G) is the order of a minimum wd set of G. Definition.8.2 A set D V is a strong dominating set, denoted sd-set, if every vertex u not in D is adjacent to a vertex v in D where degv) degu). The strong domination number s G) is the order of minimum sd-set of G. Later this theory of strong domination is studied by Swaminathan, R.Laskar, Domkey, Hattingh, Rautenbach et.al in [24-30]. The independent weak domination number i w G) independent 3

14 strong domination number i s G)) is the order of minimum independent wd-set independent sdset of G. Example.8.3 G G 2 Fig. 2.6 G 3 G 4 Fig.8.4 Examples of graphs for which strong and weak domination numbers are incomparable For the graphs G and G 2 in Fig..8.4, and. Hence and are not comparable. Note that if D is a dominating set of a connected graph then V- D is also a dominating set of G. But this result need not be true for strong or weak domination. If D is a sd set then V-D need not be a wd set. For G 3 the non pendant vertices form a sd set but the pendant vertices do not form a wd set. For the graph in Fig.8.4,..9 Motivation and Remarks Degree conditions can be used to modify the definition of the domination concepts which was essentially the starting point of the new concepts. Many real life applications and illustrations have been given for ordinary domination concept. Before to proceed to study 4

15 mathematical properties of these concepts, some motivating remarks are given. Sampathkumar and Pushpa Latha [23] offers the following motivation for strong and weak domination is. Consider a network of roads connecting a number of locations. In such a network, the degree of a vertex v is the number of roads meeting at v. Suppose degu) degv). Naturally, the traffic at u is heavier than that at v. If we consider the traffic between u and v, preference should be given to the vehicles going from u to v. Thus, in some sense, u strongly dominates v and v weakly dominates u. As suggested in the concluding remark [23], the idea of strong weak) domination has been generalized by Routenbach [24] as f-strong domination and in particular made an extensive study on strong weak) domination. Let G V, X ) be a graph and f : V N be a mapping on the vertex set with values in N, the set of positive integers. A vertex u V is an f-strong neighbour of v V if u Nv) and f u) f v). A set D V is a f-strong dominating set of G if every vertex in V D has a f-strong neighbour in D. The f-strong domination number G) is the minimum cardinality of a f-strong dominating set. In the above mapping f if we take f v) d v) we get the strong weak) domination as originally defined in [23]. The application of domination can also be used for f - strong domination concept if, as an additional condition, not all objects can dominate all their neighbors. Hence the mapping f can be seen as some kind of power or force which decides who dominates whom. One further illustration of this kind for f- strong domination is illustrated here. Consider a collection of countries v,v 2,..,v n. We define a graph G with VG) = { v,v 2,..,v n }. The vertices v i and v j for i j are joined in G by an edge if and only if the countries v i and v j have diplomatic) relations. Let fv i ) measures the military, economical or ethical force of the country 5

16 v i. The country v i can economically, militarily or ethically) control the country v j if and only if they have diplomatic relations and fv i ) fv j ). In this context f G) is the minimum number of countries which have to unite in order to control all countries..0 Full sets They also defined the complementary aspect of domination, strong domination and weak domination called full number, s-full number, and w-full number. Definition.0. A set D V is full s-full, w-full respectively) if every u D dominates strongly dominates, weakly dominates respectively) some v V D. The full number s-full number, w-full number, respectively) f f G) f f G), f f G) respectively) is the maximum cardinality of a full set s-full set, w-full set respectively) of G. s s w w Theorem.0.2 For any graph G with p vertices, f p w f s p and s f w p. Mixed Domination in Graphs In the beginning domination was studied between vertices. Later, varieties dominations are studied modifying the concept that a vertex dominate an edge and edge dominating vertex. This lead to study of mixed domination concepts. Mixed domination was first studied by R. Laskar and Ken Peters [3]..2 Neighbourhood Sets in Graphs In 986, Sampathkumar and P.S.Neeralagi [32 and 33] initiated another concept similar to vertex covering, called neighbourhood sets. 6

17 Definition.2. A set S V is a neighbourhood set n- set) if G N v vs. A set L X is a Line neighbourhood set ln- set) if G Nx xl. The neighbourhood number n n ) [line neighbourhood number n n ) ] is the order of a minimum n- 0 0 G 0 0 G set [ln-set] of G. They have proved that G) n G) ) and also observed that if G is a 0 0 G triangle free graph then n G) ) but not conversely. Jayaram et al. [34] proved the 0 0 G existence problem and shown that given integers r, s and t with 2 r s t there exists a graph G such that G) r, n G) s and G) t. R.C. Brigham et.al [35] studied the 0 0 same concept by calling it as full domination number. Proposition.2.2 For any graph G, G) n 0 G) 0 G) Proposition.2.3 For any triangle free graph, n 0 G) = 0 G) But the converse of the above proposition is not true. That is if n 0 G) = 0 G) then we cannot conclude that G is triangle free. For example for the graph in Fig..2.4, n 0 G) = 0 G) = 2. But G is not triangle free. For every odd cycle n 0 G) = 0 G). v v 3 v 6 v 2 v 5 v 4 Fig

18 Proposition.2.5 For any graph G G) n 0 G) p - Proposition.2.6 For any bipartite graph G having a factor n 0 G) = 0 G) = G) = G) = p/2.3 Edge Vertex Dominating sets In 992, Sampathkumar and S.S. Kamath [36] independently studied the same concept of mixed domination with a different but simpler approach. Definition.3. An edge x, m- dominates a vertex v if v N[x]. A set L X is an Edge Vertex Dominating set EVD-set) if every vertex in G is m-dominated by an edge in L. The edge vertex domination number G) is the minimum cardinality of an EVD-set. ev Definition.3.2 A vertex v V, m-dominates an edge X x if N v x. A set S V is a Vertex Edge Dominating set VED-set) if every edge in G is m-dominated by a vertex in S. The vertex edge domination number G) is the minimum cardinality of a VED-set. ve They observed that and ev n0. In [36] it is proved that and n 0 where is the edge domination number of a graph. Theorem.3.3 For any graph G, 2 ev In the next Proposition, we get a simple but sharp upper bound for ev 8

19 Proposition.3.4 For any p, q) graph G without isolate edges p q γ ev min, 2 2 The bound in the above proposition is sharp for any spider T as 2n p 2n q ev T) n and ev T) n Later in 2004, S.S.Kamath and R.S.Bhat [37] extended it to strong weak) neighbourhood sets. Same authors also studied strong weak) independent sets and vertex coverings of a graph in [38]..4 Inverse Domination The concept of inverse domination is introduced by V.R.Kulli and S.C. Sigarakanti [39]. Definition.4. Let D be a - set of G. If D V-D is a dominating set, then D is called the inverse dominating set of G with respect to D. The inverse domination number G) is the order of a smallest inverse dominating set. If D is a minimal dominating set of G, then V-D is also a dominating set of an isolate free graph G. Therefore every isolate free graph has an inverse dominating set. Henceforth, by a graph G, it means an isolate free and simple graph. It is observed that G) G) and G)+ G) p. Domke et.al [40] characterized the graphs for which G)+ G) = p. Tamizchelvam and G.S Grace Prema [4], characterized the graphs for which domination and inverse domination numbers are equal. Another parameter called disjoint domination number G) defined by Hedetnimi et.al [40] as min{s +S 2 ; S, S 2 are disjoint dominating sets of G}. They call G is -minimum if G ) 2 G) and G is -maximum if G) p. Ameenal Bibi and R.Selvakumar [43] studied the split and 9

20 nonsplit inverse domination. They also extended inverse domination to semi-total block graph in [44]. Example.4.2 For the graph in Fig..4.3 and. And Also For the graph in Fig..4.3, and. And. Also For the graph in Fig..4.3, and. And. Also For the graph in Fig..4.3, and. And. Also. For further results in Domination reader is referred to [45-65] G G 2 Fig..4.3 G 3 20 Examples for strong and weak domination G 4

21 .5 Semi graphs At the outset we recapitulate some terminologies used in semigraphs. Sampathkumar [66] introduced a new approach in Graph Theory called Semigraphs. Definition.5. A Semigraph G is a pair V, X ) where V is a nonempty set whose elements are called vertices of G and X is a set of n-tuples, called edges of G of distinct vertices for various n 2 satisfying the following conditions. ) Any two edges have at most one vertex in common. 2) Two edges u, u,..., u ) and v, v,..., v ) are considered to be equal 2 n 2 n if, and only if, i) m n and ii) ui vi for all i, i n or u i v n i for all i, i n. Thus, edge u, u,..., u ) is same as u, u,..., ) 2 n n n u Example.5.2 By definition every graph is a semigraph but not conversely. The Fig.5.3 depicts the example of a semigraph. The edges of a semigraph unlike in graphs can contain more than two vertices. The white vertices are called middle vertices and the black vertices are called end vertices of an edge. If an edge ends at a middle vertex of another edge then we draw a small line to the end of that edge. Thus, a middle vertex of an edge can be an end vertex of another edge. In the semigraph G shown in Fig..5.3, x v, v2, v3, v4 ) is an edge of the semigraph. v 2 and v 3 are the middle vertices v and v 4 are the end vertices of the edge x. Another edge in G is y v0, v, v3). Observe that v 3 is a middle vertex of the edge x which is also the end vertex of the edge y. x v, v, ), x v, v, ) are subedges of the edge x. The subedges 3 v4 2 2 v4 2

22 v, v, ), v, ), v, ) are partial edges of the edge x. Observe that v, ) is an edge 2 3 v4 v2 v 3 4 v 3 5 with two vertices like in graphs. For further discussion on Semigraphs the reader is referred to [64]. v 7 v 0 v 9 v 6 v 8 v v 5 v v 2 v 3 v 4 Fig.5.3 An example of semigraph. The definition of semigraphs inspired to define block trees in Chapter 2. Motivated by inverse domination in graphs, several new inverse domination and inverse block domination parameters in the remaining chapters are introduced and studied. 22