Volume 2, Issue 5, May – 2017 International Journal of Innovative Science and Research Technology ISSN No: - 2456 - 2165 Domination of Cartesian Product of Two Graphs P Bhaskarudu Department of Mathematics S. V. Arts College,Tirupathi Andhra Pradesh India 517502 Email: [email protected]

Abstract—The paper concentrates on the theory of Acharya [22], Neeralgi [23], Nagaraja Rao [15] and many domination in graphs. In this paper we define a new others have done very interesting and significant work in the parameter on domination called matching domination set, domination numbers and other related topics. Cockayne [17] matching domination number and we have investigated and Hedetniemi [10] gave an exhaustive survey of research on some properties on matching domination of cartesian the theory of dominating sets in 1975 and it was updated in 1978 product of two graphs. The following are the results: by Cockayne [17]. A survey on the topics on domination was also done by Hedetniemi and Laskar recently.A domination number is defined to be the minimum cardinality of all dominating sets in  If ui1 and uj2 then the graph G and a set S ⊆ V is said to be a dominating set in a degG1(C)G2 (ui, vj ) = degG1 (ui).degG2 (vj ) graph, if every vertex in V/S is adjacent to some vertex in S.

In this paper, we have defined two new domination parameters  If G1, G2 are simple finite graphs without isolated viz., matching domination set and matching domination number. vertices. G1(C)G2 is a simple finite graph without isolated vertices. The matching domination is defined as follows:

 The cartisean product graph of two simple graphs is Let G :< V, E > be a finite graph without isolated vertices. not a complete graph. Let S ⊆ V . A dominating set S or G is called a matching  If G1 and G2 are bipartite graphs them G1(C)G2 is a dominating set if the < S > admits a perfect . matching. The cardinality of a minimum matching dominating set is called the matching domination number.  If G1 and G2 are any two graphs without isolated vertices then We have obtained the matching domination of the product of two graphs G1 and G2 in cartesian product graphs and obtained γm[G1(C)G2] ≤ | V1 | .γmd(G2); γmd(G1). | V2 | an expression for this number in terms of matching domination number of G1and G2. While obtaining these results, we have Keywords - Cartesian product of graphs, Domination Set, obtained several other interesting results on matching domination Domination number, Complete graphs, Isolated vertices, on cartesian product of two graphs . degree, regular graphs, bipartite graphs. II. CARTESIAN PRODUCT OF GRAPHS I. INTRODUCTION A. Definition 2.1 The study on dominating sets was initiated as a problem in the game of chess in 1850. It is about the placement of the minimum number of Queens/rooks/horses, in the game lf G1, G2 are two simple graphs with their vertex sets of chess so as to cover every square in the chess board. V1 : {u1,u2, ...... } and V2 : {v1, v2 ...... } respectively then However a precise notion of a dominating set is said to be the Cartesian product of these two graphs is defined to be a given by Konig [12], Berge [13] and Ore [7], Vizing [14] graph with its vertex set as V1xV2 : {w1, w2,.....} and if w1 = were the first to derive some interesting results on dominating (u1,v1),w2 = (u2,v2) then w1,w2 is an edge in this product sets. Since then a number of graph theorists Konig [15], Ore graph if and only if either u1 = u2 and v1, v2ϵE(G2) or u1, [7], Bauer Harary [16], Laskar [5], Berge [13], Cockayne u2ϵE(G1) and v1 = v2. This product graph is denoted by [l7], Hedetniemi [10], Alavi[18], Allan [19], Chartrand [18], G1(C)G2. Illustration follows Kulli [3], Sampathkumar [3], Walikar[20], Arumugam [21],

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Then G1(C)G2 is a simple finite graph follows from the definition 2.1.

Further G1, G2 being graphs without isolated vertices. degG1 (ui) = 0 for any i, degG2 (vj ) = 0 for any j.

u v Hence from the Theorem 2.2, degG1 (C)G2 ( i, j ) = 0 for Fig. 1. . any i,j.

Thus G1(C)G2 does not have any isolated vertices.

It is interesting to see that the cartesian product of two simple graphs is not a comlete graph and if G1, G2 are bipartite then G1(C)G2 is also a bipartite graph.

D. Theorem 2.4

The cartisean product graph of two simple graphs is not a complete graph.

Proof : If V1 = {u1,u2,...... ,um} and V2 = {v1,v2,...... ,vn}

then in G1(C)G2 the vertex (ui, vi) is not adjacent with (uj , vj Fig. 2. . ) even if u1 and uj are adjacent and/or vi and vj are adjacent (By definition 2.1). It can be proved that in this product also that if G1, G2 are finite graphs without isolated vertices then G1(C)G2 is a finite E. Theorem 2.5 graph without isolated vertices. If G1 and G2 are bipartite graphs them G1(C)G2 is a For this, we first prove the following theorem bipartite graph.

B. Theorem 2.2 Proof :

If ui1 and uj2 then Suppose G1 is a bipartite graph with bipartite X1, Y

u v and G2 a bipartite graph with bipartition (X2, Y2) where degG1 (C)G2 ( i, j ) = degG1 (ui).degG2 (vj )

X1 = {x1,x2, ...... ,xr} Proof : Y1 = {y1,y2, ...... ,ys} By definition 2.1, (ui, vj ) is adjacent with all the vertices in X2 = {u1,u2, ...... ,um} uixNG2(vj) and NG1 (ui)xvj

Y2 = {v1,v2, ...... ,vn} Further, | NG1 (ui) |= degG1 ui and | NG2 (vj ) |= degG2 vj We have the following result as an immediate consequence. We know that X11 = V1 and X22 = V2 and also

C. Theorem 2.3 X11 = ɸ = X22.

If G1, G2 are simple finite graphs without isolated vertices. Now G (C)G is a simple finite graph without isolated vertices.\ 1 2 V1xV2 = {X11}x{X22}

Proof : = {X1xX2} ∪ {X1xY2} ∪ {X2xY1} ∪ {X2Y2}

This vertex set can partitioned as

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III. MATCHING DOMINATION NUMBER X = {X1xX2}∪ {Y1xY2} A. Definition 3.1 and Y = {X1xY2}∪ {X2xY1} A set S ⊆ V is said to be a dominating set in a graph G if every vertex in V/S is adjacent to some vertex in S and the domination number ′γ′ of G is defined to be the minimum It can easily seen that no two vertices in X are adjacent in cardinality of all dominating sets in G. G1(C)G2 for if t1, t2 are two any vertices in X, then t1t2ϵ{X1xX2}∪ {Y1xY2} We have introduced a new parameter called the matching domination set of a graph. =⇒ t1, t2ϵ{X2xX2}or{Y1xY2} Case(i) It is defined as follows: if B. Definition 3.2 t ϵ{X xX },t ϵ{X xX } 1 1 2 2 1 2 A dominating set of a graph G is said to be matching t1ϵ{X1xX2} dominating set if the induced subgraph < D > admits a =⇒ t1 = (xi, uj ) say and t2 = (xk , ul) perfect matching.

If xi = xk;ui,ukϵX2 and no two vertices of X2 are The cardinality of the smallest matching dominating set is adjacent by hypothesis; on the other hand if uj = ul and if xi, called matching domination number and is denoted by γm xk ϵX1 as no two vertices of X1 are adjacent (by hypothesis),t1, Illustration t2 are not adjacent. case (ii) if t1ϵ{X1xX2} t2ϵ{Y1xY2}

From the definition 2.1, it is evident that t , t are not Fig. 3. 1 2 adjacent since no xi is equal to any yk and so also no uj is equal to any ul as X1 ∩ Y1 = ɸ, further uj and ul are in two In this graph {a, b, c, f, e, g } is a matching domination set, different partitions of G2. So t1, t2 are not adjacent in this since this is a dominating set and the induced subgraph {a, case also. b, c, e, f, g } has perfect matching formed by the edges af, case (iii) bc, eg, {a,b,e,f } is also matching dominating set. Similarly t1ϵ{Y1xY2} {a,b,c,g} is a matching dominating set where the induced subgraph of this set admits a perfect matching given by the t2ϵ{X1xX2} edges be,ag.

This case is similar to the case (ii) and thus t1, t2 are not adjacent in this case also. However there are no matching dominating sets of lower cardinality and it follows that the matching domination case (iv) number of the graph in figure 3 is 4. Thus a graph can t1ϵ{Y1xY2} have many matching dominating sets of minimal cardinality. t2ϵ{X1xX2} We make the following observations as an immediate consequence. This case is similar to case (i) discussed above and so t1, t2 are not (a) Not all dominating sets are matching domination sets. For adjacent in this case too. example in figure 3, {a,c,e } is a dominating set but it is not a Thus we have proved that no two vertices in X are adjacent. matching dominating set. Similarly it can he proved that no two vertices in Y are (b) The cardinality of matching dominating set is always adjacent. even. The matching dominating set D of a graph requires the Hence the Theorem. admission of a perfect matching by the induced subgraph < D >. Thus it is necessary that D has even number of vertices for admitting a perfect matching.

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(c) Not all dominating sets with even number of vertices are will be both matching dominating sets of G1(C)G2. Hence, it matching dominating sets. For example in figure 3,{b,d,g,f } is follows that the minimal matching dominating set will be ≤ a dominating set containing even number of vertices, but min | D |, | D′ |. induced subgraph formed by these four vertices does not have a perfect matching. | D |= p.2s =| v1 | .γm(G2) (d) The necessary condition for a graph G to have matching ′ dominating set is that G is a graph without isolated vertices. The | D |= 2r.q = γm(G1). | v2 | matching domination number of the graph G (figure 3) is 4, Hence where as the domination number is 2 ; {a,d } being a minimal dominating set. If G is a graph with isolated vertices then any γm(G1(C)G2) ≤ min.{2ps,2qr} dominating set should include these isolated vertices and ≤ min.γm(G1). | V2 |, | V1 | γm(G2) consequently the induced subgraph of this set containing isolated vertices will not admit a perfect matching. Illustrate follows:

C. Theorem 3.3

If G1 and G2 are any two graphs without isolated vertices then γm[G1(C)G2] ≤ | V1 | .γmd(G2); γmd(G1). | V2 |

Proof : Fig. 4. matching domination set {u1, u2}, p = 4, γm(G1) = 2

Let V (G1) : {u1, u2 ...... ,up} = V1 V (G2) : {v1, v2 ...... ,vq}=V2

Let D1 : {ud1 ; ud2 ;.....ud2r} be the matching dominating set of minimum cardinality of G,. Let the induced sub graph < D1 > Fig. 5. matching domination set {v1, v2; v4, v5}, q = 6, of G1 have a perfect matching in < D1 > constitute by the γm(G1) = 4 edges ud1 ; ud2 ; ud3 ; ud4 ; ...... ud2r-1 ; ud2r .

Similarly let D2 : {vd1 ; vd2 ; ...... vd2r} be the matching dominating set of minimum cardinality of G2. Let the induced subgraph < D2 > of G2 have a perfect matching in < D2 > constituted by the edges vd1 ; vd2 ; vd3 ; vd4 ; vd2r-11 ; vd2r.

Then it can easily seen that the sets

{ (u1; vd1 ) (u1; vd2 ) : : : (u1; vd2r ) D : (u2; vd1 ) (u2; vd2 ) : : : (u2; vd2r ) ...... (u ; v ) (u ; v ) : : : (u ; v ) p d1 p d2 p d2r

and the set Fig. 6. G1(C)G2

D' : (ud1 ; v1) (ud2 ; v1) : : : (ud2r ; v1) matching domination set (ud1 ; v2) (ud2 ; v2) : : : (ud2r ; v2) ... D = {u1,u2}x | V2 | ...... = {(u1, v1), (u1, v2), (u1, v3), (u1, v4), (u1, v5)(u1, v6), ... (u2, v1), (u2, v2), (u2, v3), (u2, v4), (u2, v5), (u2, v6)}

(ud1 ; vp) (ud2 ; vp) : : : (ud2r ; vp) | D |= 12

IJISRT17MY38 www.ijisrt.com 168 Volume 2, Issue 5, May – 2017 International Journal of Innovative Science and Research Technology ISSN No: - 2456 - 2165 [10]. Hedetniemi, S.T,and Laskar.R., ” Connected matching domination set domination in graphs, in and ′ Combinatorics” (Ed. Bela Bollobas), Academic Press, D =| V1 | x{v1,v2,v4,v5} London, 1984.

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