Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 745 https://internationalpubls.com The Non Split Eccentric Domination Number of Corona Product and Join of Some Standard Graphs Sudhasenthil1 and N.Anbarasi2 1S.D.N.B. Vaishnav College for Women (Autonomous) Chennai - 600 044, India e-mail: drsudhasenthilmaths@gmail.com 2S.D.N.B. Vaishnav College for Women (Autonomous) Chennai - 600 044, India e-mail: anbarasimohan22@gmail.com Article History: Received: 12-08-2024 Revised: 15-09-2024 Accepted: 25-10-2024 Abstract: A subset D of the vertex set V(G) of a graph G is said to be a dominating set if every vertex not in D is adjacent to at least one vertex in D. A dominating set D is said to be an eccentric dominating set if for every , there exists at least one eccentric point of v in D. An eccentric dominating set D of G is a non split eccentric dominating set if the induced sub graph < V- D > is connected. The minimum of the cardinalities of the non split eccentric dominating sets of G is called the non split eccentric domination number of G. This paper evaluates the non split eccentric domination number of Corona product and join of some standard graphs. Keywords: Domination, Eccentric Domination, Non Split Eccentric Domination, Corona product, Join. 1. Introduction Let G be a finite, simple undirected graph on p vertices and q edges with vertex set V(G) and edge set E(G). For graph theoretic terminology refer Harary [8] Buckley and Harary [5]. In 2010 T.N. Janakiraman M. Bhanumathi and S. Muthammai defined an eccentric domination in graph [9]. V.R. Kulli and Janakiram introduced the concept of split and nonsplit domination number of a graph in 1997 [11] and in 2000 [12] M. Bhanumathi and Sudhasenthil introduced the cocept of split and nonsplit eccentric domination number of a graphs in 2014 [4]. Motivated by these, we have defined Nonsplit eccentric domination number of Corona product and Join of some standard graphs. Let G be a connected graph and v be a vertex of G. The eccentricity e(v) of v is the distance to a vertex farthest from v. Thus, e(v) = max{d(u, v): u  V}. The radius r(G) is the minimum eccentricity of the vertices whereas the diameter diam(G) is the maximum eccentricity. For any connected graph G, r(G)  diam(G)  2r(G). v is a central vertex if e(v) = r(G). The center C(G) is the set of all central vertices. The central subgraph of a graph G is the subgraph induced by the center v is a peripheral vertex if e(v) = d(G). The periphery P(G) is the set of all peripheral vertices. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 746 https://internationalpubls.com For a vertex v, each vertex at a distance e(v) from v is an eccentric vertex of v. Eccentric set of a vertex v is defined as E(v) = {u  V(G) / d(u, v) = e(v)}. The open neighbourhood N(v) of a vertex v is the set of all vertices adjacent to v in V. N[v] = N(v)  {v} is called the closed neighbourhood of v. For a v  V(G). Ni(v) = {v  V(G); d(u, v) = i} is defined to be the ith neigborhood of v in G. A dominating set D of a graph G is a nonsplit dominating set if the induced subgraph is connected. The nonsplit domination number ns(G) of a graph G is the minimum cardinality of a nonsplit dominating set. A set D  V(G) is an eccentric dominating set if D is a dominating set of G and for every v  V − D, there exists atleast one eccentric point of v in D. The eccentric domination number ed(G) of a graph G is the minimum cardinality of an eccentric dominating set. An eccentric dominating set with cardinality ed(G) is known as ed-set. Let S  V(G). Then S is known as an eccentric point set of G if for every v  V − S, S has atleast one vertex u such that u  E(v). An eccentric point set S of G is a minimal eccentric point set if no proper subset S of S is an eccentric point set of G. S is known as a minimum eccentric point set if S is an eccentric point set with minimum cardinality. The minimum cardinality of an eccentric point set of G denoted as e(G) is known as eccentric number of G. 2. Prior Results Theorem 2.1. [4] (i) nsed(K1,n) = n, n  2 (ii) nsed(Wn) = 3, for n  4 (iii) nsed(Pn) = n-2, for n  4 (iv) nsed(Cn) = n-2, for n  3 (v) nsed(Kn) = 1, for n  3 (vi) nsed(Km,n) = 2, for n  2 Observation 2.1. 1. For any connected graph, (G)  ns(G)  nsed(G). 2. For any connected graph G, (G)  ed(G) ≤nsed(G). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 747 https://internationalpubls.com 3. There are graphs with ns(G) = ed(G) and nsed(G) = ns(G). 4. There are graphs with (G) = ed(G) = nsed(G). Theorem 2.2. [3] For a connected graph G with even number of vertices p and ( ) 2 p Ged = if and only if G is for some connected graph H. 3. Main Results In this section, we determine the exact values of nonsplit eccentric domination number of corona product of graph Definition 3.1. An eccentric dominating set D of G is a nonsplit eccentric dominating set if the induced subgraph is connected. The nonsplit eccentric domination number nsed(G) of a graph G equals the minimum cardinality of a nonsplit eccentric dominating set. That is nsed(G) = min |D|, where the minimum is taken over D in D, where D is the set of all minimal nonsplit eccentric dominating sets of G. V(G) is a nonsplit eccentric dominating set for any graph G. Hence nsed(G) is a well defined parameter. Example:3.1 Here D = {v1, v4} is a dominating set. D = {v1, v5, v6} is an eccentric dominating set and also nonsplit eccentric dominating set. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 748 https://internationalpubls.com nsed(G) = 3, ns(G) = 3, ed(G) = 3, (G) = 2 )()()( GGG nseded   . Definition 3.2 Let G and H be two graphs on n and m vertices respectively. The corona of the graphs G and H denoted by and is defined as the graph obtained by taking one copy of G and n copies of H and then joining the ith vertex of G to every vertex in the ith copy of H. Example:3.2 Theorem 3.1: For ( ) nWCmn mnnsed = ,4,3 , where 11 −+= mm CKW . Proof: Let ( )   ( )  mjniuwWVvvvvCV ijimnn == 1,1/,,,....,,, 321 and ( )    mjniuwvvvvWCV ijinmn = 1,1/,,....,,, 321  . Let  niWD i = 1 where Wi are the central vertices of Wm. Then D is a minimum eccentric dominating set and is connected. Therefore D is a minimum nonsplit eccentric dominating set and nD = .Therefore ( ) nWC mnnsed = . Theorem 3.2: For ( ) ( ),,2 11 GVKGm mnsed =  where G1 be any connected graph with n vertices. Proof: Let ( )  nvvvvGV ,....,,, 3211 = and let  imiii uuuu ,....,,, 321 be the ith copy of Km adjacent to vi. Then ( )  nmnnmmnm uuuuuuuuuvvvvKGV ,...,,,....,.....,,,,....,,,,....,,, 2122221112113211 = . Let  mjuuuuD njjjjj = 1/,....,,, 321 are some nonsplit dominating set. Further every vertex of < V-Dj> has an eccentric vertex in Dj. Therefore Dj is a nonsplit eccentric dominating set HG 34 KP  Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 749 https://internationalpubls.com and nD j = . Hence each Dj is a minimum nonsplit eccentric dominating set of mKG 1 . Hence ( ) nDKG jmnsed ==1 . Theorem 3.3: If H is any self centred unique eccentric point graph with m vertices and 12KHG = then mGnsed 2)( = . Proof: If H is any self-centred unique eccentric point graph. Then every vertex of H is an eccentric vertex. Hence m is even and G has 3m vertices. Let mvvvv ,....,,, 321 be the vertices of H and   ii VV , for i = 1, 2, ...,m be the vertices of m copies of 2k, then in G,  ii VV , are adjacent to vi and if vj is the eccentric vertex of vi in H. Then  ii VV , are the eccentric vertices of vj in G and  jj VV , are the eccentric vertices of vi. It is clear that     = mm VVVVVVD ,...,,,...,, 2121  is a minimum eccentric dominating set of G. Further is connected and mD 2= . Therefore D is a nonsplit eccentric dominating set of G. Hence ( ) mGnsed 2= . Theorem 3.4: For ( ) nKCmn mnnsed = ,1,2,3  . Proof: Let  nn vvvCV ,...,,)( 21= and  mm uuuwKV ,...,,,)( 211,1 = , then    mjniuwnivKCV ijiimn = 1,1/,1/)( ,1  . By choosing a vertex set  nwwwD ,...,, 21= which dominates all the vertices of mn KC ,1 and is connected. Further every vertex of has an eccentric vertex in D and nD = . Therefore D is a nonsplit eccentric dominating set of mn KC ,1 . Hence ( ) nKC mnnsed =,1 . Theorem 3.5: For ( ) ( )mmnnsed PnPCmn  = ,4,3 . Proof: Let  nn vvvCV ,...,,)( 21= and the set imii uuu ,...,, 21 be the ith copy of Pn adjacent to the vertex vi then    mjniunivPCV ijimn = 1,1/1/)(  . Let D be the n copies of dominating sets of Pm which dominates all the vertices of mn PC  . Further every vertex of has an eccentric Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 750 https://internationalpubls.com vertex in D and is connected. Therefore D is a nonsplit eccentric dominating set and ( )mPnD = . Hence ( ) ( )mmnnsed PnPC  = . Theorem 3.6: For a connected graph G with even number of vertices p, ( ) 2 p Gnsed = if and only if G is 1KH  for some connected graph H. Proof: Let 1KHG = , where H is a connected graph on 2 p vertices. V(H) is a set− of G and D is the set of all pendent vertices in G is a minimum eccentric dominating set . Further is connected. Therefore the set of all pendent vertices in G is a minimum nonsplit eccentric dominating set. Hence ( ) 2 p Gnsed = . Conversely assume that ( ) 2 p Gnsed = . Since G is a graph with even number of vertices p. By theorem 2.2, we get G is 1KH  for some connected graph H. 4. Non Split Eccentric Domination in join of graphs In this section we determine the exact values of non split eccentric domination number of Join graph G + H Definition 4.1. The join G + H of two graphs G and H is the the graph with vertex set and the edge set . . Theorem 4.1: For ( ) 3,1,4 =+ mnnsed KPmn  . ( ) ( ) ( )HVGVHGV =+  )(),(/)()()( HVvGVuuvHEGEHGE =+  34 KC + Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 751 https://internationalpubls.com Proof: Let Let ( )  nn vvvPV ,...,, 21= and ( )  mjuKV jm = 1/ then ( )  mjniuvKPV jimn =+ 1,1/ . Let  uvvD n ,,1= , where v1 and vn are the end vertices of Pn and u is any vertex of Km. Then D is a minimum nonsplit dominating set. Further every vertex of has an eccentric vertex in D. Therefore D is a minimum nonsplit eccentric dominating set of Pn+Km and 3=D . Therefore ( ) 3=+ mnnsed KP . Theorem 4.2: For ( ) 3,2,4 ,1 =+ mnnsed KPmn  . Proof: Let ( )  nn vvvPV ,...,, 21= and ( )  mjuwKV jm = 1/,,1 then ( )    mjuwnivKPV jimn =+ 1/,1/,1  . Let  nvvwD ,, 1= , where w is the root vertex of K1,n, v1 and vn are the end vertices of Pn. Then D is a minimum nonsplit dominating set and is connected. Further every vertex of has an eccentric vertex in D. Therefore D is a nonsplit eccentric dominating set and 3=D . Hence ( ) 3,1 =+ mnnsed KP . Theorem 4.3: For ( ) 4,4,4 =+ mnnsed WPmn  . Proof: Let ( )  nn vvvPV ,...,, 21= and ( )  mjuwWV jm = 1/, then ( )    mjuwnivWPV jimn =+ 1/,1/  . Let  211 ,,, uuvvD n= , where v1, vn are the vertices of Pn and u1, u2 are adjacent vertices of Wn. Every vertex of has an eccentric vertex in D. Therefore D is an eccentric dominating set. Further is connected. Hence D is a minimum eccentric dominating set and 4=D . Hence ( ) 4=+ mnnsed WP . Conclusion Here we have evaluated the results on non split eccentric domination number of corona product and join of some standard graphs and also studied some bounds for non split eccentric domination number of a graph. References 1. Bhanumanthi M and Muthammai S, Eccentric domination in trees, International Journal of Engineering Science, Advanced Computing and Bio-technology, Vol. 2, No. 1, pp. 38−46, (2011) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 752 https://internationalpubls.com 2. Bhanumathi M and Muthammai S, Further results on eccentric domination in graphs, International Journal of Engineering Science, Advanced Computing and Bio- technology, Vol. 3, Issue 4, pp. 185−190, (2012) 3. 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