A study on Interior Domination in Graphs

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1 IOSR Joural of Mathematics (IOSR-JM) e-issn: , p-issn: X. Volume 12, Issue 2 Ver. VI (Mar. - Apr. 2016), PP A study o Iterior Domiatio i Graphs A. Ato Kisley 1, C. Carolie Selvaraj 2 1, 2 Departmet of Mathematics, St. Xavier s College, (Autoomous) Palayamkottai, Idia. Abstract : The cocept of iterior set of vertices of a graph G has applicatios i locatig domiatig set of a graph. A domiatig set D is said to be a iterior domiatig set if every vertex v of D is a iterior vertex of G. The miimum of the cardiality of the iterior domiatig sets of G is called as iterior domiatio umber of G. The relatio betwee the domiatio umber ad the iterior domiatio umber has bee foud. Some bouds for iterior domiatio umber have bee foud ad its exact values for some particular classes of graphs are determied. Keywords: Graphs, complete, Wheel, Star graphs, Domiatig set, Iterior vertices I. Itroductio The various domiatio parameters have bee itroduced ad used i may applicatios i graphs by takig differet sets with each of which has some specified property. These cocepts are helpful to fid cetrally located sets to cover the etire graph i which they are defied. Let G be a fiite, simple, udirected (p, q) graph with vertex set V (G) ad edge set E(G). We refer the basic defiitios ad theorems used i the book [4]. Let G be a coected graph ad u be a vertex of G. The eccetricity e(v) of v is the distace to a vertex farthest from v. Thus e(v) = max{ d(u, v): u V}. The radius r(g) is the miimum eccetricity the vertices, whereas the diameter diam (G) is the maximum eccetricity. For ay coected graph G, r(g) diam(g) 2 r(g). The vertex v is a cetral vertex if e(v) = r(g). The cetre C(G) is the set of all cetral vertices. The cetral subgraph <C(G)> of a graph G is the subgraph iduced by the cetre. The vertex v is a peripheral vertex if e(v) = d(g). The periphery P(G) is the set of all peripheral vertices. For a vertex v, each vertex at a distace e(v)from v is a eccetric vertex. Eccetric set of a vertex v is defied as E(v) = { u V /d(u, v) = e(v)}. The ope eighbourhood N(v) of a vertex v is the set of all vertices adjacet to v i G. N[v] is called the closed eighbourhood of v. A set D V(G) is a domiatig set of G, if every vertex i V D is adjacet to some vertex i D. The domiatig set D is a miimal domiatig set if o proper subset D of D is a domiatig set. The miimal domiatig set with miimum cardiality is kow as a miimum domiatig set. The cardiality of miimum domiatig set is kow as the domiatio umber ad is deoted by (G). Let x ad z be two distict vertices i G. A vertex y distict from x ad z is said to lie betwee x ad z if d(x, z) = d(x, y) + d(y, z). A vertex v is a iterior vertex of G if for every vertex u distict from v, there exists a vertex w such that v lies betwee u ad w. A vertex v is a boudary vertex of u if d(u, w) d(u, v) for all w N(v). A vertex u has more tha oe boudary vertex at differet distace levels. I this paper, we iitiate the study of ew domiatio. II. Iterior Domiatig Sets Defiitio 2.1 A set D V(G) is a iterior domiatig set if D is a domiatig set of G ad every vertex v D is a iterior vertex of G. Ay ed vertex will ot be a member i iterior set of a graph. I a tree every vertex which is ot a ed vertex is a iterior vertex. Defiitio 2.2 The iterior domiatio umber (G) of a graph G is defied as the cardiality of the miimum iterior domiatig set. Example 2. Figure 1: A graph G DOI: / Page

2 A Study O Iterior Domiatio I Graphs D= {v 2, v 4, v 6, v 9 }, D 1 = {v 2, v, v 7, v 8, v 9 }, D 2 = {v 2, v 4, v 6, v 8 }. D is a domiatig set of G but it is ot a iterior domiatig set of G. D 1 is a iterior domiatig set of G but ot miimum. D 2 is the miimum iterior domiatig set of G. Hece (G) = 4. Theorem 2.4 A domiatig set D V(G) is a iterior domiatig set if ad oly if for all v D, N(v) 2 ad for all x N(v) there exist y N(v) such that d(x, y) = d(x, v) + d(v, y) Proof: Let D be a iterior domiatig set of G. Let v be a arbitrary vertex i D. If x is the oly eighbourhood of v the for ay y V - N(v), d(x, y) d(x, v) + d(v, y). Hece v is ot a iterior vertex of G. Sice v D is a arbitrary vertex, o vertex i D is a iterior vertex of G. The D is ot a iterior domiatig set of G which is a cotradictio. Coversely, suppose that D is a domiatig set of G. The for all v D, N(v) 2 ad for all x N(v) there exist y N(v) such that d(x, y) = d(x, v) + d(v, y). Clearly v is a iterior vertex of G. Hece D is a iterior domiatig set of G. Result 2.5 A domiatig set D is a iterior domiatig set of G if ad oly if it is ot a boudary domiatig set of G. Theorem 2.6 For ay path of order, (P ) = Proof: Let v 1, v 2, v, represets a path P. Let D = {v 2, v 5, v 8 v -1 } be the miimum domiatig set of P. Sice o ed vertex is a iterior vertex ad i D the domiatig vertices start from v 2 ad eds i v -1. No vertex i D is a ed vertex of P.. Hece D is the miimum iterior domiatig set ad D =. Thereforeγ (P )=. Theorem 2.7 For ay cycle of order 4, γ (C )= or + 1. Proof: Let G be a cycle v 1, v 2, v, v 1. Sice C is a cycle, ay vertex of D is a iterior vertex of G. Case 1: If = k, where k = 2,,., the γ (C )=. Case 2: If = k -1, where k = 2,,.. Cosider D = { v 1, v 4, v 7, v k v k+,.. v k-1 } be a miimum domiatig set of G. Sice C is a cycle, ay vertex i D is a iterior vertex of G. Hece D is a iterior domiatig set of G ad also D is a miimum iterior domiatig set of G. Sub case 1:If is odd, the D =. Sub case 2:If is eve, the D = + 1. Case : If = k -2, where k = 2,,.. the it is similar to the above case. Hece γ (C ) = or + 1. Theorem 2.8 (i) γ K m, = 2, (ii) γ W = 1 ad (iii). γ K 1, = 1 Proof: (i) Whe G = K m,, V G = V 1 V 2, V 1 = m ad V 2 = such that each elemet of V 1 is adjacet to every vertex of V 2 ad vice versa. Let D = u, v, u V 1, v V 2. The vertex u domiates all the vertices of V 2 ad it is the iterior domiatig vertex. Similarly v domiates all the vertices of V 1 ad it is the iterior domiatig vertex. Therefore D is a miimum iterior domiatig set ad hece γ K m, = 2. (ii) Whe G = W,. Let D = u where u is the cetral vertex of G ad also the iterior vertex of G. The cetral vertex domiates every vertex of G. The D is a miimum iterior domiatig set of G. Thus γ W = 1. (iii) Whe G = k 1,. Let D = u where u is the cetral vertex of G ad also the iterior vertex of G. The cetral vertex domiates every vertex of G. The D is a miimum iterior domiatig set of G. Thus γ K 1, =1. Note that for ay complete graph K, there is o iterior domiatig set, sice there is o iterior vertex. We observe the idetities: (i). γ bd K m, = γ ed K m, = γ (K m, ) Theorem 2.9 For ay coected graph G with 4, γ 2e + 1 DOI: / Page

3 A Study O Iterior Domiatio I Graphs Proof: Let x G be a vertex of degree. Therefore x domiates itself ad other of eighbourig vertices. Therefore x domiates + 1 vertices of G ad γ (G) DOI: / Page. If G is a regular graph, the every vertex has degree. Hece the cardiality of every domiatig set is greater tha. For a arbitrary graph, the cardiality of every domiatig set is also greater tha. The value of i this case is greater tha the value i the first case. Hece i all cases umber of miimum domiatig set is greater tha. For ay graph, γ G 1 = e + 1. Therefore γ 2e + 1. Theorem 2.10 Let G be a graph. If G is of radius 2 with a uique cetral vertex u, the (G) deg (u) Proof: Let G be a graph with rad (G) =2. Let u be a uique cetral vertex, the u has a eccetric vertex v with distace 2. Hece N[u] G ad v G N[u], hece there are deg(u) - 1 vertices i G N[u]. Therefore every domiatig set has a cardiality deg(u). Theorem 2.11 For a coected graph G, γ G + G = if G K 1, 1 or W. Proof: If G K 1, 1 or W the γ G = 1 ad G = 1. Hece γ G + G =. Note 2.12 Let T be tree of order with 1 pedat vertices. The γ T < γ T + 1 Theorem 2.1 For a tree T, γ T (T). Proof: Let G be tree which is ot isomorphic tok 1, 1. Let D be a γ set of T. Let v be a vertex of maximum degree (G). The v domiates N[v] ad the vertices i V N[v] domiate themselves. Hece V N(v) is a domiatig set of cardiality (T) ad so γ T (T). Theorem 2.14 For ay triagle free graph coected graph G with 4 the Iterior domiatio umber lies betwee γ (G) (G). Proof: Let D be a γ set of G. First we cosider the lower boud each vertex ca domiate at most itself ad (G) other vertices. Hece γ (G). For the upper boud, let v be a vertex of maximum degree G, the v domiates N[v] ad the vertices i V N[v] domiates themselves. Hece V N(v) is a domiatig set of cardiality (G) ad so γ G (G). Hece γ (G). Theorem 2.15 γ(g) γ (G) Proof: Let D be a miimum domiatig set. The γ G = D. Let D be a miimum domiatig set i which all vertices are iterior vertices of G. If the members of D are iterior vertices of G, the D = D. Otherwise D < D. Hece D D ad so γ(g) γ (G). Theorem 2.16 For ay coected graph with 5 vertices, γ (G) + κ(g) 2. Proof: Let G be a coected graph. For every graph G of order, 0 κ G 1 by [ 4] ad also the upper boud follows from the defiitio of iterior domiatio, that is γ G 2. Hece γ G + κ G Therefore, γ G + κ G 2. Theorem 2.17 For ay coected graph G with 5 vertices, γ G + χ(g) 2 2. Proof: Let G be a coected graph with 5vertices. Suppose G is a complete graph we eed colours for colourig so χ G =. For other graphs χ G <. I geeral for ay coected graph G, χ G. From the above theorem we haveγ G 2Hece γ G + χ G Theorem 2.18 For ay coected graph G with 5, γ G + G 2 Proof: Let G be a coected graph with 5, vertices. For ay coected graph G, G 1.From the above theorem we have γ G 2. Hece γ G + G 2. Theorem 2.19 If T is a tree with l leaves, the γ (T) l+2 Proof: Delavia [4] proved that γ(t) l+2 for a tree T with vertices ad leaves. But γ (G) γ(g), therefore γ (T) l+2.

4 A Study O Iterior Domiatio I Graphs Theorem 2.20 For ay coected graph G with x cut verticesγ (G) x+2. Proof: Delavia [4] proved that γ(g) x+2 for x cut vertices. Sice γ (G) γ(g), we have γ G x+2. Next we study the iterior domiatio properties o some special graphs. III. Exact value for some special graphs Example.1: If G is a caterpillar such that each o pedet vertex is of degree three the γ (G) = ( 2 ) 1 Sice degree of each o pedet vertex to three the G is of the followig form Figure 2: A Caterpillar Figure : A Hoffma tree Every o-pedet vertex set form a iterior domiatig set D ad it is also a miimum set. Hece γ (G) = 1. If G is a Hoffma tree, the γ (G) =. 2 Example.2 The Mobius-Kator graph is a symmetric bipartite cubic graph with 16 vertices ad 24 edges as show i the figure: Figure 4: The Mobius-Kator graph Figure 5: The Desargues graph For the Mobius-Kator graph G, γ (G) = 8. Here the followig set S is a iterior domiatig set of G. Here S= {v 9, v 10, v 11, v 12, v 1, v 14, v 15, v 16 } Example. The Desargues graph is a distace-trasitive cubic graph with 20 vertices ad 0 edges as show i the figure For ay Desargues graph G, γ (G) = 10. Here the followig set S is the iterior domiatig set. Here S = {v 11, v 12, v 1, v 14, v 15, v 16, v 17, v 18,, v 19, v 20 } Example.4 The Chvatal graph is a udirected graph with 12 vertices ad 24 edges as show i the figure: Figure 6: The Chvatal graph Figure 7: The Durer graph For the Chvatal graph G, γ (G) = 4. Here S = {v 1, v 2, v, v 4 } is a iterior domiatig set. DOI: / Page

5 A Study O Iterior Domiatio I Graphs Example.5 The Durer graph is a udirected cubic graph with 12 vertices ad 18 edges as show i the figure. For the Durer graph G, γ (G) = 6. Here S = {v 7, v 8, v 9, v 10, v 11, v 12 } is a iterior domiatig set. IV. Coclusio May researchers are cocetratig the domiatig cocept i graphs. I this paper we have defied the iterior domiatio i graphs for differet types of graphs ad ivestigated iterior domiatio umber for them. May results have bee foud ad compared for some graphs. Ackowledgemet The authors would like to thak gratitude to the referee for readig the paper carefully ad givig valuable commets. Refereces [1] Bhaumathi.M ad Kavitha.M, BD-Domiatio i Graphs, Iter. J. Fuzzy Mathematical Archive, Vol. 7, No. 1, 75-80, 2015 [2] Buckley.F, Harary.F, Distace i graphs, Addiso-Wesley Publishig Compay (1990) [] Cockaye.E.J., Hedetiemi.S.T, Towards a theory of domiatio i graphs. Networks, 7: [4] Ermelida Delavia, Rya pepper, ad Bill Waller,Lower bouds for the domiatio umber, Discussioes Mathematicae Graph Theory, Jauary (2010). [5] Gary Chartad, Pig Zhag, Itroductio to Graph theory, McGraw Hill Educatio (Idia) Private Limited, New Delhi (2006) [6] Harary.F, Graph theory, Addiso-Wesley Publishig Compay Readig, Mass (1972). [7] Jaakirama T.N., Bhaumathi.M ad Muthammai.S, Eccetric Domiatio i Graphs, Iteratioal Joural of Egieerig [8] Sciece, Advaced Computig ad Bio-Techology Vol. 1, No. 2, April-Jue 2010, pp [9] Teresa W.Hayes,Stephe T.Hedetiemi, Peter J.Slater, Fudametals of Domiatio i graphs, Marcel Dekker, Ic(1998). DOI: / Page