International J.Math. Combin. Vol.2(2018), 87-96 Accurate Independent Domination in Graphs B.Basavanagoud (DepartmentofMathematics,KarnatakUniversity,Dharwad-580003,India) Sujata Timmanaikar (DepartmentofMathematics,GovernmentEngineeringCollege,Haveri-581110,India) E-mail: b.basavanagoud@gmail.com,sujata123rk@gmail.com Abstract: Adominating set D of agraph G=(V,E)is an independent dominating set, if theinducedsubgraphhDihasnoedges. AnindependentdominatingsetDofGisanaccurate independent dominating set if V −D has noindependent dominating set of cardinality |D|. The accurate independent domination number i (G) of G is the minimum cardinality of an a accurate independent dominating set of G. In this paper, we initiate a study of this new parameter and obtain some results concerning this parameter. Key Words: Domination, independent domination number, accurate independent domination number,Smarandache H-dominatingset. AMS(2010): 05C69. §1. Introduction All graphs considered here are ﬁnite, nontrivial, undirected with no loops and multiple edges. For graph theoretic terminology we refer to Harary [1]. LetG=(V,E)beagraphwith V =pand E =q. Let∆(G)(δ(G))denotethemaximum | | | | (minimum)degreeand x ( x )theleast(greatest)integergreater(less)thanorequaltox. The ⌈ ⌉ ⌊ ⌋ neighborhood of a vertex u is the set N(u) consisting of all vertices v which are adjacent with u. The closed neighborhood is N[u] = N(u) u . A set of vertices in G is independent if ∪{ } no two of them are adjacent. The largest number of vertices in such a set is called the vertex independence number ofGandis denotedbyβ (G). ForanysetS ofverticesofG,the induced o subgraph S is maximal subgraph of G with vertex set S. h i The corona of two graphs G and G is the graph G = G G formed from one copy 1 2 1 2 ◦ of G and V(G ) copies of G where the ith vertex of G is adjacent to every vertex in the 1 1 2 1 | | ith copy of G . A wounded spider is the graph formed by subdividing at most n 1 of the 2 − edges of a star K for n 0. Let Ω(G) be the set of all pendant vertices of G, that is the 1,n ≥ set of vertices of degree 1. A vertex v is called a support vertex if v is neighbor of a pendant vertex and d (v) > 1. Denote by X(G) the set of all support vertices in G, M(G) be the set G 1Supported by University Grant Commission(UGC), New Delhi, India through UGC-SAP- DRS-III, 2016- 2021: F.510/3/DRS-III/2016 (SAP-I). 2ReceivedJanuary11,2018,Accepted May25,2018. 88 B.BasavanagoudandSujataTimmanaikar of vertices which are adjacent to support vertex and J(G) be the set of vertices which are not adjacent to a support vertex. The diameter diam(G) of a connected graph G is the maximum distance between two vertices of G, that is diam(G) = max d (u,v). A set B V is u,v∈V(G) G ⊆ a 2-packing if for each pair of vertices u,v B,N [u] N [v]=φ G G ∈ ∩ Aproper coloring ofagraphG=(V(G),E(G))isafunctionfromtheverticesofthegraph to a set of colors such that any two adjacent vertices have diﬀerent colors. The chromatic number χ(G) is the minimum number of colors needed in a proper coloring of a graph. A dominatorcoloring ofagraphGisapropercoloringinwhicheachvertexofthegraphdominates every vertex of some color class. The dominator chromatic number χ (G) is the minimum d number of color classes in a dominator coloring of a graph G. This concept was introduced by R. Gera at.al [3]. AsetD ofverticesinagraphG=(V,E)isadominating set ofG,ifeveryvertexinV D − isadjacenttosomevertexinD. Thedomination number γ(G)ofGistheminimumcardinality of a dominating set. For a comprehensive survey of domination in graphs, see [4, 5, 7]. Generally, if D H, such a dominating set D is called a Smarandache H-dominating h i ≃ set. A dominating set D of a graph G = (V,E) is an independent dominating set, if the inducedsubgraph D hasnoedges,i.e.,aSmarandacheH-dominatingsetwithE(H)= . The h i ∅ independent domination number i(G)istheminimumcardinalityofanindependentdominating set. AdominatingsetDofG=(V,E)isanaccuratedominatingset ifV Dhasnodominating − setofcardinality D . Theaccurate domination number γ (G) ofGisthe minimumcardinality a | | of an accurate dominating set. This concept was introduced by Kulli and Kattimani [6, 9]. AnindependentdominatingsetD ofGisanaccurate independent dominating set ifV D − has no independent dominating set of cardinality D . The accurate independent domination | | number i (G) of G is the minimum cardinality of an accurate independent dominating set of a G. This concept was introduced by Kulli [8]. For example, we consider the graph G in Figure 1. The accurate independent dominating sets are 1,2,6,7 and 1,3,6,7 . Therefore i (G)=4. a { } { } 2 1 6 7 b b bb b G: b b b 3 4 5 Figure 1 §2. Results Observation 2.1 1. Everyaccurateindependent dominating setis independent anddominating. Hence itis a minimal dominating set. AccurateIndependentDominationinGraphs 89 2. Every minimal accurate independent dominating set is a maximal independent dominating set. Proposition 2.1 For any nontrivial connected graph G, γ(G) i (G). a ≤ Proof Clearly, every accurate independent dominating set of G is a dominating set of G. Thus result holds. 2 Proposition 2.2 If G contains an isolated vertex, then every accurate dominating set is an accurate independent dominating set. Now we obtain the exact values of i (G) for some standard class of graphs. a Proposition 2.3 For graphs P ,W and K , there are p p m,n (1) i (P )= p/3 if p 3; a p ⌈ ⌉ ≥ (2) i (W )=1 if p 5; a p ≥ (3) i (K )=m for 1 mm =i (G), a contradiction. a 1 a ⌊ ⌋ 2 Corollary 2.1 For any graph G, i (G)=γ(G) if diam(G)=2. a Proposition 2.5 For any graph G without isolated vertices i(G) i (G). Furthermore, the a ≤ equality holds if G=P (p 3), W (p 5)or K for 1 m

Accurate Independent Domination in Graphs

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