Cmjv02i03p0463

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J. Comp. & Math. Sci. Vol.2 (3), 463-469 (2011)

On the Domination Globle Alliances in Some Classes of Graphs D. B. GANGADHARAPPA1 and A. R. DESAI2 1

Senior grade Lecturer in Mathematics. Coorg Institute of Technology Ponnampet – 571216. South Kodagu. Karnataka. India 2 Professor in Mathematics S.D.M college of Engg and Tech Dharawad – 580002. Karnataka. India ABSTRACT A defensive alliance in a graph G is a set vertices D⊑V satisfying the condition that for every vertex v ∈ D, the number of neighbors v has D plus 1 is at least as large as the number of neighbors it has in V−D. The dominating set D is a global (strong) alliance if for every vertex v ∈ V−D, that is every vertex in V−D is adjacent to at least one number of alliance D. Note that global defensive alliance is a dominating set. The minimum cardinality of a global defensive(strong defensive, offensive, strong offensive) alliance is denoted by γୟ (G) (γୟො (G), γ଴ (G), γ଴෡ (G)). We study global defensive and offensive alliance in graphs.

INTRODUCTION We consider a graph G = (V(G),E(G)) with vertex set V(G), edge set E(G), order p(G) = |v G | and size q(G) = |E G |. the degree in G of a vertex v is denoted by d (v) or simply d(v), and the number of neighbors of v in a sub set D of V. For a non empty sub set D ⊑ V, We denote the sub graph of G induced by D by D . For any vertex v ∈ V, the open neighborhood of v is the set N(v) = {u: uv ∈ E}, while the closed neighborhood of v is the set N[v] = N(v) ∪ {v}. For a subset D ⊑ V, the open neighborhood N(D) = U

N(v) and the closed neighborhood N[D] = N(D) ∪ D. A set D is a dominating set if N[D] = V, and is a total dominating set or an open dominating set if N[D] = V. the minimum cardinality of a dominating set of G is the domination number γ(G) respectively for the total domination number γ (G). The concept of domination in graphs. With its many variations is now well studied in graph theory. Hedetniemi, Hedetnimi and Kristiansen introduced several types of alliances, incenting defensive alliances. A nonempty set of vertices D ⊑ V is called a

Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)

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D. B. Gangadharappa, et al., J. Comp. & Math. Sci. Vol.2 (3), 463-469 (2011)

defensive alliances if for every v Ďľ D, |N v D| ≼ | N v v D|. every vertex in D is defined from possible vertices in V−D. a defensive alliance D is called strong if for every vertex v ∈ |N v D| > | N v v D| In this case we say that every vertex in D is defined. An alliance D is called global if it effects every vertex in V−D, that is every vertex in V−D is a adjacent to at least one member of the alliance D this case D dominating set. The global alliance number Îł (G), global strong alliance number Îł (G) are the minimum cardinality of a alliance and strong alliance of G that also a dominating set the entire vertex set is a global strong alliance for any graph G. The dominating set is independent iff it is a maximal independent set and that in every graph Îł(G) ≤ i(G) ≤ β(G), where Îł(G)and i(G) are respectively the minimum cardinality of dominating set and of an independent dominating set and β(G) is the maximum cardinality of a global defensive respectively strong defensive Îł (G) offensive Îł (G), strong offensive Îł (G) clearly Îł(G) ≤ Îł (G) ≤ Îł (G) and Îł(G) ≤ Îł (G) ≤ Îł (G). For every graph G the corona of a graph is obtained by attaching a pendant edge at each vertex of G. Theorem 1: For any graph G, i) ii) iii)

1≤ γ(G) ≤ γ (G) ≤ γ (G) ≤ p, 1 ≤ a(G) ≤ γ (G) ≤ p, 1 ≤ a(G) ≤ a (G) ≤ γ (G) ≤ p.

We show the global alliance and global strong alliance number for complete graphs and complete bipartite graphs and total domination numbers with minimum degree.

Proposition 1: For the complete graph k , i) Îł (k ) =

and ii) Îł (k ) =

Proof: Let D be a Îł (k ) – set and let v ∈ D then D contains at least deg v /2 =

neighbors of v, and so γ (k ) ≼ set consisting of v and

.

The

of its neighbors

is a global alliance, and so Îł (k ) ≤ . This establish (i). Let D be a Îł (k ) – set and v∈ D. Then D contains at least deg v /2 =

neighbors of v, and so γ (k ) ≼

.The

set consisting of v and of its neighbors is a global strong alliance, and so γ (k ) ≤

. This establish ii).

Proposition (2): For the complete bipartite graph K , , (i) Îł (K , ) = + 1

(ii) γ (K , ) = + , if r, d ≼2 and (iii)

Îł (k , ) = + .

Proof: We first establish (i) the result is immediate when d=1. Suppose d≼2 and D is a γ (K , )-set, since D is a dominating set, the central vertex v say, belongs to D and therefore D contains at least deg v /2 = neighbors of v. Hence γ (K , ) ≼ + 1.

The set consisting of v and of its neighbors is a global alliance, and so γ (K , ) ≤ + 1. Given that a(k , ) = +

and a (k , ) =

we have γ (K , ) ≼

+ +

by the theorem 1,

and γ (k , )≼

Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)

D. B. Gangadharappa, et al., J. Comp. & Math. Sci. Vol.2 (3), 463-469 (2011)

+ . The set consisting of vertices in

the partite set and vertices in the other partite set is a global alliance , and so γ (K , ) ≤ + . This establish (ii).

Similarly , the set consisting of

vertices

in the one partite set and vertices in the other partite set is a global strong alliance establishing (iii). Lemma1: For any graph G with δ(G) ≼2 , Îł (G) ≤ Îł (G) , further if ∆(G)≤3, then Îł (G) =Îł (G) . Proof: For any Îł (G)- set D and vertex v∈

D, D contains at least ≼1 neighbors of v, and so D is a total dominating set. Thus Îł (G) ≤ Îł (G). Further if ∆(G)≤3, then for any Îł (G) –set D and vertex u∈ D , |N u D| ≼2≼|N v V D|. Hence D is a global alliance, and so Îł (G)≤ Îł (G). If G is a cubic graph, then Îł (G) =Îł (G). Proposition 3: For cycles C , p≼3, Îł (C )= Îł (C )= Îł (C ). Lemma 2: For any graph G with no isolated vertices, Îł (G) ≤ Îł (G). Proof: For any Îł (G) –set D and vertex v∈

D, D contains at least ≼1 neighbors of v and so D is a total dominating set. Thus γ (G) ≤ γ (G). The total domination number of paths P and cycles C is well known: for

p≼3, Îł (P )= Îł (C ) = + − . For paths, we show that the global strong alliance number equals the total domination number. However, the global alliance

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number of a path is not necessarily equal to its total domination number. Proposition 4: For p≼3, Îł (P )= Îł (P ). Proof: By lemma (2) , Îł (P )≤ Îł (P ). For any Îł (P )- set D and vertex u∈ D, |N u D|≼2> |N u V D|. Hence D is a global strong alliance, and so Îł (P )≤ Îł (P ) implies that Îł (P )= Îł (P ). Proposition 5: For p≼2, Îł (P ) = Îł (P )- 1.

Proof: Let R= P , since ∆(R) ≤ 2, every total dominating set of R is also a global alliance of R, and so Îł (R)≤ Îł (R). Suppose P ≥ 2 (mod4). If v denotes an end vertex of R , then either p=2, in which case Îł (R)≤ Îł (R)1 . On the other hand , let A be a Îł (R)- set . Then A is a dominating set of R. If the sub graph <A> induced by A contains an isolated vertex, then this vertex must be an end vertex of R. Hence <A> contains a most two isolated vertices. If <A> contains no isolated vertex, then A is a total dominating set, and so Îł (R) ≤ |A|. If <A> contains one isolated vertex v, then A - 'v( is a total dominating set of T−N[v] = P , and so Îł (P )≤ |A|−1. If now P≢2(mod4) then Îł (R)= Îł (P )+ 1≤|A| , while if p≥2(mod4), then Îł (R)= Îł (P )+2 ≤|A|+1. If <A> contains two isolated vertices u and v , then either R = P , in which case Îł (R)= 2 =|A| or |A|≼4, in which case A= 'u, v( is a total dominating set of R−N[u]−N[v]= P . Therefore, Îł (P )≤ |A|−2, and so Îł (R)= Îł (P )+ 2≤|A|. Since |A|= Îł (R), we have shown that Îł (R)≼ Îł (R) unless p≥2(mod4), in which case Îł (R)≼ Îł (R) −1.

Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)

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D. B. Gangadharappa, et al., J. Comp. & Math. Sci. Vol.2 (3), 463-469 (2011)

Lower bound and upper bound on the global alliance and global strong alliance number of a graph in terms of its order. Theorem 2: If G is a graph of order p, Then γ (G) ≼

,

and this bound is sharp.

Proof : Let Îł (G) = k. For any Îł (G) – set D and vertex v ∈ D, D contains at least Neighbors of v. Hence K = |D| ≼ |'v(| + ≼ (deg v +1)/2. Thus V−D contains

at most ≤ (deg v +1)/2 ≤ K neighbors of v. therefore each vertex in D has at most k neighbors in V− D, and so p− k = |V D| ≤ k . 0r equivalently, k + k−p

≼ 0. Hence K ≼ . Therefore this bound is sharp may be seen as , Let F = k and for k ≼ 2, Let F be the graph obtained from the disjoint union of k stars K , by adding all edges between the central vertices of K stars, Then G = F for some k ≼ 1has order P = k(k+1),and so k=(+4p . 1

−1)/2. If k=1, then Îł (G)=1= . If K≼2, then the k central vertices of the stars form a global alliance, and so, Îł (G)≤

.consequently, Îł (G)=

.

Proposition6: If G is a graph of order p,

then Îł (G) ≼ ๨ .

ŕ°Ž

Corollary 1. If G is a cubic graph of order p.

Then γ (G) ≼

Corollary 2. If G is a 4-regular graph of

order p. Then γ (G) ≼ .

Theorem 3: If G is a graph of order p. Then Îł (G) ≼ +p, and this bound is sharp. Proof : Let Îł (G) = k. For any Îł (G)-set D and vertex v ∈ D, D contains at least neighbors of v. Hence k = |D| ≼ . ≼ most ≤

Thus V−D contains at

≤ k−1 neighbors of v. Therefore ,each vertex in D has at most k−1 neighbors in V−D, and so P−K = |V D| ≤ k(k−1), or equivalently k ≼ +p i.e Îł (G) ≼ +p. That this bound is sharp, may be seen as ,Let F = K , F = P , and for K ≼ 3, Let F be the graph obtained from the disjoint union of k stars K , by adding all edges between the central vertices of the k stars. Then F = F For some K ≼ 1 has order p =k , and so k = +p . If K =1, ThenÎł ( G) = 1 =+p , while if k = 2, Then Îł (G) = 2 =+p , if k ≼ 3. Then the k central vertices of the stars from a global strong alliance and so Îł (G) ≤ +p, Thus Îł (G) = +p. Proposition 7: For any graph G with no isolated vertices and minimum degree δ, δ δ (i) Îł (G) ≤ p− and (ii) Îł (G) ≤ p− and these bounds are sharp. Proof : Let v be a vertex of minimum degree and D be the set of vertices formed δ by the removing neighbors of v from V. Then D dominates G for each u∈D,|N u δ V D | ≤ ≤ , and so |N u

D| ≼ +1 ≼|N u V D |. Thus D is a global alliance, and so γ (G) ≤ |D|. This establish (i). That this bound is sharp

Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)

D. B. Gangadharappa, et al., J. Comp. & Math. Sci. Vol.2 (3), 463-469 (2011)

from proposition 1. (take G=K with p is odd) Let D be the set vertices formed by δ removing neighbors of v from V, Then D dominates G. For each u∈D, ,|N u V δ D | ≤ ≤ , and so |N u D|

≼ +1 >|N u V D |. Thus D is aglobal strong alliance, and so Îł (G) ≤ |D|. This establish (ii). That this bound is sharp from proposition1(take G=K ). Corollary 3: For any graph G, Îł (G)= p if 1111 . and only if G=K Definitions: 1. 2 is the family graphs obtained from a clique D ~ K by attaching K=d (u)+1, leaves at each vertex u of D. 2. 2 is the family of bipartite graphs obtained from a balanced complete bipartite graph D ~ K , by attaching K+1=d (u)+1 leaves at each vertex u of D. 3. 2 is the family of trees obtained from a tree D by attaching a set L of d (u)+1 leaves at each vertex u of D. Proposition 8: (i) If G ∈ 2 Then i(G) = Îł (G)− Îł (G) +1. (ii) If G ∈ 2 Then i(G) = Îł (G)/4 + Îł (G). (iii) If G ∈ 2 Then i(G) = 2Îł (G) −1.

Proof: If G ∈ 2 with 1≤ i ≤ 3 then V(D) is a minimum dominating set and a defective alliance of a G. There fore Îł(G) ≤ Îł (G) ≤ |D| = Îł(G) and thus Îł (G)=|D|. (i) If G ∈ 2 , that is D ~ K then i(G) = 1+ = |D| − |D| + 1. (ii) If G ∈ 2 , that is D ~ K , then |D| = 2k and i(G) = k+k(k+1)= |D| /4 + |D|

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(iii) Let T ∈ 2 be constructed from a tree, D with bipartition classes X and Y. Every maximal independent set I of T can be written as I= (I∊V(D)) âˆŞ (5 / L u ). There fore |I| = |I V D | + ∑ / d u . 1 = |V D | + ∑ / d u . in the sum ∑ / d u , the edges of D between V(D)/I and I are counted once and the q(D−I) edges joining two vertices in V(D)/I are counted twice. Hence ∑ / d u = q(D)+q(D−I) ≼ q(D),and |I| ≼ |V D | +q(D) =2p(D)−1. For the particular sets I=X âˆŞ (5 L u and I =Y âˆŞ (5 ! L u , q(V(D)/I) = ∅. And |I| = 2p(D)−1. Therefore, i(T)=2p(D)−1=2 Îł (T)−1. Theorem 4: (i) Every graph G is satisfies i(G) ≤ Îł (G)− Îł (G) +1 with equality iff G∈2

(ii) Every bipartite graph G is satisfies i(G) = Îł (G)/4 + Îł (G) with equality iff G ∈ 2 (iii) Every tree G is satisfies i(G) ≤ 2Îł (G) −1 with equality iff G ∈ 2 . Proof: Let D be a Îł (G)-set, W is a maximal independent set of G[D], and B is a maximal independent set of G[N / (D)/ N / (W)]. Then WâˆŞB is a maximal independent set of G and i(G) ≤|W|+ |B|. For each v∈D, Let L(v) = N / (v). Since D is a defensive alliance, |L v | ≤ d v +1 for every v∈D, and since defensive alliance is dominating. "

|B| ≤ :N๏ ; <: ≤ ∑ ŕąš |L v | = # ŕąš

∑ /$ d v . 1

ŕą­

≤|D|−|w|+∑ /$ d v

Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)

(1)

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D. B. Gangadharappa, et al., J. Comp. & Math. Sci. Vol.2 (3), 463-469 (2011)

Therefore i(G) ≤|D|+∑ /$ d v .

(2)

(i) In every graph , d v ≤ |D| −1. Therefore i(G) ≤ |D| +(|D|−|w|) (|D| −1) with |w| ≼ 1. Hence i(G)≤ |D| − |D| + 1= Îł (G)− Îł (G) +1. If i(G)= |D| − |D| + 1 Then |W| =1 and d v = |D| −1 for every v ∈ D/W, i.e D is a clique and W consists of any vertex W of D. Moreover, For any W ∈ D, equality in (i) gives |B| =:N๏ D/'w(:, i.e :N๏ D/'w(: ŕąš

is independent and

ŕąš

:N๏ D/'w(: = ŕąš

∑ /%$&|L v | ? ∑ /%$& d v . 1) i.e all the sets L(v) for v ∈ D, are disjoint, independent and of order d v +1. Therefore G∈2 . The converse is true by proposition 8(i) (ii) Suppose now G bipartite. Let U be the set of isolated vertices of G [D]and XâˆŞY a bipartite G[D/U]. If we take W= XâˆŞU we get by (2), i(G) ≤|D|+∑ d v = |D| +q(D)‌‌.(3). Since G[D] is bipartite, q(D)≤ |D| /4 and thus i(G)≤ |D| /4+ |D| = Îł (G)/4+ Îł (G). If i(G)= |D| /4+ |D|, then q(D) = |D| /4 i.e U = ∅ and G[D] is a complete balanced bipartite graph. More over equality in (1) implies that all the sets L(v) for v ∈ Y are disjoint and of respective orders d v +1. By symmetry between X and Y, the same property holds for all v ∈ X. Then G∈2 , the converse is true by proposition 8(ii) (iii) If the bipartite graph G is a tree, G[D] is a forest, by (3), i(G) ≤ |D| + q(D) with q(D) ≤ |D|−1. Therefore i(G)≤ 2 |D| 1 =2 Îł (G) −1. If i(G)≤ 2 |D| 1, then q(D) = |D|−1, i.e G[D] is a tree the set L(v) are all disjoint for v ∈ Y and of respective order

d v +1, and the same holds for all v ∈ X by symmetry between X and Y. therefore G∈2 the converse is true by proposition 8(iii). Lemma 3: Let T be a tree constructed from a balanced tree D by attaching a set L(u) of d u leaves at each vertex u ∈ D. Let B be a maximal independent set of T and q be the number of components of the forest induced in T by V(D)/B. then |B| = 2 |D| − q −1. Proof: Every maximal independent set of T has the form B=(V(D)∊B) âˆŞ (5 /' L u ). Hence |B| = 2 |B V D | + ∑ /' d u = q(D) +q(D−B) and |B| = |B V D | + q(D) +q(D−B). Since D is a tree and D−B a forest with q components, q(D)= |D| −1 and q(D−B)=|V D /B| – q. therefore |B| = |B V D |+( |D| −1) +(|D| |B V D |−q)= 2 |D| − q −1 . Theorem 5: (i) For every tree T, Îł( (T) ≤ 2i(T)−1 (ii) For every tree T of order p, β(T)+2i(T) ≼ p+1. (iii) For every tree T of order p ≼ 2, i(T) ≤ Îł() (T) −1. REFERENCES 1. A.Cami, H.Balakrishnan, N. Deo and R.D. Dutton. On the complexcity of finding optimal global alliances, J. Combin. Math. Combin. Comput. 58, 23-31(2006). 2. M. Chellali and T.W.Haynes, Global alliances and independent in trees. Discuss, Math. Graph theory 27, 19-27 (2007). 3. T.W. Haynes, S.T.Hedetniemi and M. A. Henring, Global defensive alliances in graphs.

Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)

D. B. Gangadharappa, et al., J. Comp. & Math. Sci. Vol.2 (3), 463-469 (2011)

4. P. Kristiansen, S. A. Hedetniemi and S.T.Hedetniemi, Alliance in graphs, JCMCC 48, 157-177 (2004). 5. T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of domination in graphs. Marcel Dekker. Inc.

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Newyork (1998). 6. T.W. Haynes, S.T.Hedetniemi and M. A. Henning, A characterization of trees with equal domination and global strong alliance numbers, Utilitas Mat. 66, 3345 (2004).

Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)

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