CHAPTER 2 NEIGHBORHOOD CONNECTED PERFECT DOMINATION IN GRAPHS

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1 22 CHAPTER 2 NEIGHBORHOOD CONNECTED PERFECT DOMINATION IN GRAPHS 2.1 INTRODUCTION Various types of domiatio have bee studied by several authors ad more tha 75 models of domiatio are listed i the appedix of Hayes et al. (1998). The followig are some of the fudametal types of domiatio. (i) Domiatio (Oyesti Ore 1962) (ii) Idepedet domiatio (Cockaye et al., 1978) (iii) Coected domiatio (Sampathkumar ad Walikar 1979) (iv) Total domiatio (Cockaye et al., 1980) (v) Paired domiatio (Hayes ad Slater 1998) Weichsel [See Hayes et al. (1998)] itroduced the cocept of perfect domiatio i graphs. Arumugam ad Sivagaam (2010) itroduced the cocept of eighborhood coected domiatio i graphs. Motivated by these two cocepts i this chapter we itroduce the cocept of eighborhood coected perfect domiatio ad iitiate a study of the correspodig parameter. We cosider oly coected graphs i this chapter.

2 23 I sectio 2.2 we itroduce the cocept of eighborhood coected perfect domiatio ad eighborhood coected perfect domiatio umber γ cp (G). We determie γ cp for stadard graphs such as paths ad cycles. We also discuss the relatio betwee γ cp ad other domiatio parameters such as γ, γ p ad γ c. Also we obtai a characterizatio of miimal eighborhood coected perfect domiatig sets. I sectio 2.3 we preset some bouds for eighborhood coected perfect domiatio umber γ cp. I sectio 2.4 we fid the upper boud for the sum γ cp +χ where χ is the chromatic umber of G ad characterize the correspodig extremal graphs. I sectio 2.5 we preset some of the ope problems that arise aturally ad idicate directios for further research. 2.2 NEIGHBORHOOD CONNECTED PERFECT DOM- INATION IN GRAPHS Defiitio A perfect domiatig set S of a graph G is called the eighborhood coected perfect domiatig set (cpdset) if the iduced subgraph N(S) is coected. The miimum cardiality of a cpd-set of G is called the eighborhood coected perfect domiatio umber of G ad is deoted by γ cp (G). Example γ cp value for well kow graphs. (i) γ cp (K ) = 1

3 24 (ii) γ cp (K r,s ) = 2, r 1 ad s 2 (iii) γ cp (W ) = 1. Example (i) For the graph G = C 6 we have γ = 2, γ p = 2, γ c = 3 ad γ cp = 4. (ii) For the graph G i figure 2.1 we have γ(g) = 5, γ p (G) = 6, γ c (G) = 7 ad γ cp (G) = 8. Figure 2.1: Graph for which γ, γ p, γ c ad γ cp are distict Remark (i) Clearly γ cp (G) γ c (G) γ(g). (ii) For ay coected graph G, γ cp (G) = 1 if ad oly if there exists a o cut vertex v such that degv = 1. Thus γ cp (G) = 1 if ad oly if G = H +K 1 for some coected graph H. (iii) For a tree T with 3, γ cp (T) 2.

4 25 Theorem For ay path P, γ cp (P ) = 2. Proof. Let P = (v 1,v 2,,v ). If 1(mod 4) the S = {v i : i = 2k,2k+1adk is odd} is a cpd-set of P ad if 1(mod 4) the S 1 = S {v 1 } is a cpd-set of P. Hece γ cp (P ) 2. Sice γ c (P ) = 2 ad γ c(g) γ cp (G), we have 2 γ cp(p ). Thus γ cp (P ) = 2. Corollary For ay o trivial path P, (i) γ cp (P ) = γ(p ) if ad oly if = 2 or 4 (ii) γ cp (P ) = γ p (P ) if ad oly if = 2 or 4. Proof. Sice γ(p ) = γ p (P ) = 3 the corollary follows. Theorem For a cycle C, 2 if 0, 1(mod 4) γ cp (C ) = 2 +1 if 2(mod 4) 2 if 3(mod 4). Proof. Let C = (v 1,v 2,,v,v 1 ) ad = 4k + r, where 0 r 3. Let S = {v i : i = 2j, 2j +1, j is odd ad 1 j 2k 1}

5 26 Let S 1 = S if 0(mod 4) S {v 1 } if 1, 3(mod 4) S {v 2,v 1 } if 2(mod 4) Clearly S 1 is a cpd-set of C ad hece 2 if 0, 1(mod 4) γ cp (C ) 2 +1 if 2(mod 4) 2 if 3(mod 4). Sice γ cp (C ) γ c (C )ad γ c (C ) = 2 2 if 3(mod 4) if 3(mod 4), it follows that values give for γ cp (C ) are correct uless 2 (mod 4). If 2(mod4), the for ay γ c -set S of C, there exists a vertex v V S adjacet to two vertices i S ad hece γ cp (C ) Hece the result follows. Corollary For ay cycle C,

6 27 (i) γ cp (C ) = γ(c ) if ad oly if = 3, 4, or7. (ii) γ cp (C ) = γ p (C ) if ad oly if = 3, 4, 5, 7or8 (iii) γ cp (C ) = γ c (C ) if 2(mod 4). Proof. Sice γ(c ) = 3, γ p (C ) = if 2(mod 3) otherwise ad γ c (C ) = 2 2 if 3(mod 4) if 3(mod 4) the result follows. Theorem Let S be a miimal cpd-set of a graph G. The for every u S, oe of the followig holds (i) p[u, S] φ (ii) N(u) (S {u}) 2 (iii) < N(S {u}) > is discoected. Proof. Let S be a miimal cpd-set of G. Let u S ad let S 1 = S {u}. The ay oe of the followig is true. (a) S 1 is ot a domiatig set

7 28 (b) < N(S 1 ) > is discoected (c) there exists a vertex v V S 1 such that N(v) S 1 2 If < N(S 1 ) > is discoected the (iii) is true. If S 1 is ot a domiatig set of G, the p[u, S] φ. Suppose a vertex v V S 1, such that N(v) S 1 2. If v u the there exist two vertices x, y S 1 such that x, y are adjacet to v ad hece S is ot a cpd-set. Thus v = u which gives (ii) of the theorem. 2.3 BOUNDS FOR γ cp I the followig theorems we obtai the bouds for γ cp. Theorem Let G be a graph with = 1 ad let v V(G) with degv =. The γ cp (G) 1 + V(H) where H is a compoet of G v with V(H) is miimum. Proof. Let v V(G) with degv = 1. If G v is coected the {v} is a cpd-set of G ad hece γ cp (G) = 1. Suppose G v is discoected, the S = {v} is ot a cpd-set of G. Let H be a compoet of G v with miimum vertices. Hece S V(H) is a cpd-set of G. Thus γ cp (G) 1+ V(H). Remark The boud give i theorem is sharp. The graph G = K 1, 1, γ cp (G) = 2 = 1+ V(H).

8 29 Corollary LetGbeagraphwith = 1. The γ cp (G) = 2 if ad oly if there exists a support vertex v such that degv = 1. Example For the graph G i figure 2.2 γ cp (G) = 1 + mi V(H) = 3 where H is ay compoet of G v. Figure 2.2: A graph with γ cp = 1+mi V(H) Theorem Let G be ay graph ad H be a coected spaig subgraph of G with γ cp (G) > γ cp (H). The γ cp (G) > γ c (G). Proof. Suppose γ cp (G) = γ c (G). Sice γ c (G) γ c (H) we have γ cp (G) γ cp (H) which is a cotradictio. This proves the result. Theorem For ay graph G, γ cp (G). Further, if G is a ( 2)-regular graph, 6, the γ cp (G) =.

9 30 Proof. First part is obvious. Suppose G is ( 2)-regular ad let S be ay γ cp -set of G. Clearly S cotais at least two vertices. Suppose γ cp (G) <. Case 1. γ cp (G) = 2 The there exists a vertex x V S which is adjacet to vertices of S which is a cotradictio. Case 2. 3 γ cp (G) 1 The w V S is adjacet to at least two vertices of S which is a cotradictio. Hece γ cp (G) =. Example For the graph G i figure 2.3, γ cp (G) = 6 =. Figure 2.3: A graph with γ cp = Theorem Let G be a graph with k pedat vertices. The γ cp (G) k +1 ad equality holds if ad oly if G is a star.

10 31 Proof. Let X be the set of all pedat vertices of a graph G ad let X = k. Let u X. The (V X) {u} is a cpd-set of G. Heceγ cp (G) k+1.letgbeagraphwithγ cp (G) = k+1 ad let X be the set of all pedat vertices of G with X = k. If V X > 1 the V X is a cpd-set of G with V X = k which is a cotradictio. Hece V X = 1. Thus G is a star. Example For the graph G give i figure 2.4 γ cp = k where k is the umber of pedat vertices. Figure 2.4: A graph with γ cp = k 2.4 NEIGHBORHOOD CONNECTED PERFECT DOM- INATION NUMBER AND CHROMATIC NUMBER Several authors have studied the problem of obtaiig a upper boud for the sum of a domiatio parameter ad a graph theoretic parameter ad characterized the correspodig extremal graphs. Paulraj Joseph ad Arumugam (1999) proved that γ(g)+χ(g) +1. They also characterized the class of graphs for which the upper boud is attaied. I the ext two theorems we fid a upper boud for sum of the eighborhood coected

11 32 perfect domiatio umber ad chromatic umber ad characterize the correspodig extremal graphs. Theorem For ay otrivial graph G, γ cp (G)+χ(G) 2 1 ad equality holds if ad oly if G is isomorphic to K 2. Proof. Suppose γ cp (G) + χ(g) = 2 the γ cp (G) = ad χ(g) =. The G is a complete graph with γ cp (G) = which gives G is trivial ad hece γ cp (G)+χ(G) 2 1. Let G be a graph with γ cp (G)+χ(G) = 2 1. The either (i)γ cp (G) = 1, χ(g) = or (ii)γ cp (G) =, χ(g) = 1. Suppose(i)holds. TheGisacompletegraphwithγ cp (G) = 1 which gives = 2. Hece G is isomorphic to K 2. Suppose (ii) holds. The G is a isomorphic to K X where X is a o empty subset of set of edges icidet with a vertex v of K with X 2 which implies γ cp (G) = 1 or 2. The = 2 ad hece G is discoected which is a cotradictio. The coverse is obvious. Theorem Let G be a graph. The γ cp (G)+χ(G) = 2 2 if ad oly if G is isomorphic to K 3 or P 3 or the graph obtaied from K H where K = K 2 ad H is either K 2 or K 2 with V(H) = {u,v} by addig 1 edges betwee u ad K ad addig 2 edges betwee v ad K, 2 i 5, i = 1 or 2, such that [N(u) N(v)] {u,v} = φ ad < 2.

12 33 Proof. Let γ cp (G)+χ(G) = 2 2. The oe of the followig is true(i)γ cp (G) = 2, χ(g) = (ii)γ cp (G) = 1, χ(g) = 1 (iii)γ cp (G) =, χ(g) = 2. Suppose (i) holds. The G is a complete graph with γ cp (G) = 2 this implies = 3. Hece G is isomorphic to K 3. Suppose (ii) holds. The G is isomorphic to K X, where X is a o empty subset of set of edges icidet with a vertex of K with X 2 which implies γ cp (G) = 1 or 2. The = 2 or 3 ad hece G is isomorphic to P 3. Suppose (iii) holds. Because χ(g) = 2, either G has a complete subgraph of order 2 or > 4 ad G is the joi of K 5 with C 5. (I case = 5, by the joi of K 5 ad C 5 we mea C 5.) If G is the joi of K 5 with C 5 the γ cp (G)+χ(G) = 6, if = 5, or 1, if > 5. I either case, γ cp (G)+χ(G) 2 2. Thus G has a complete subgraph G 1 of order 2. Let Y = V(G) V(G 1 ) = {u,v}. The Y = K 2 or K 2. Case 1. Y = K 2 Sice G is a coected graph each u ad v are adjacet to at least oe vertex of G 1. If either u or v is a pedat vertex, the γ cp (G) <. Hece each u ad v are adjacet to at least two vertices i G 1. If u ad v have a commo eighbor w i G 1, the γ cp (G) = 1 which gives a cotradictio. Hece N(u) N(v) = φ. If N(u) N(v) = V(G 1 ) the γ cp (G) = 2 which is a cotradictio. The the graph is isomorphic to the graph give i theorem. Case 2. Y = K 2.

13 34 Sice G is coected ad γ cp (G) = we have each u ad v are adjacet to at least oe vertex of G 1. If u ad v have a commo eighbor w i G 1, the γ cp (G) = 1 or 3 which gives a cotradictio. Hece N(u) N(v) = φ. Suppose N(u) V(G 1 ) = {x} the {u,x} is a γ cp -set G which is a cotradictio. Hece each u ad v are adjacet to more tha oe vertex i G 1. If [N(u) N(v)] {u,v} = V(G 1 ) the γ cp (G) = 2 which is a cotradictio. The the graph is isomorphic to the graph give i theorem. The coverse is obvious. 2.5 CONCLUSION AND SCOPE I this chapter we have itroduced a ew type of domiatio, amely, eighborhood coected perfect domiatio ad preseted several results o the correspodig domiatio parameter. The followig are some iterestig problem for further ivestigatio. Problem Characterize the class of graphs for which γ cp (G) =. Problem Characterize the class of graphs for which γ cp (G) = k where k is the umber of pedat vertices i G.