Complementary Acyclic Weak Domination Preserving Sets
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1 International Journal of Research in Engineering and Science (IJRES) ISSN (Online): , ISSN (Print): ijresorg Volume 4 Issue 7 ǁ July 016 ǁ PP Complementary Acyclic Weak Domination Preserving Sets N Saradha a,vsaminathan b And K Angammal c A Assistant Professor, Department Of Mathematics, SCSVMV University, Enathur, Kanchipuram, Tamilnadu, India B Head & Coordinator, Ramanujan Research Center in Mathematics, Sarasathi Narayanan College, Madurai, Tamilnadu, India C Research Scholar, Department of Mathematics, SCSVMV University, Enathur, Kanchipuram, Tamilnadu, India Abstract: Let G = (V, E) be a simple graph A subset D of V(G) is called complementary acyclic eak domination preserving set of G (c-adp set of G) if < V D > is acyclic and ( D ) ( G) The minimum cardinality of a c-adp set in G is called the complementary acyclic eak domination preserving number of G and denoted by c-adpn(g) A c-adp set of G of cardinality c-adpn(g) is called a c-adpn-set of G In this paper, e introduce and discuss the concept of complementary acyclic eak domination preserving sets Keyords: Complementary acyclic eak domination preserving set,complementary acyclic eak domination preserving number I INTRODUCTION By a graph e mean, simple and undirected graph G(V, E) here V denotes its vertex set and E its edge set Degree of a vertex u is denoted by d(u) The maximum degree of a graph G is denoted by Δ(G) We denote a cycle on n vertices by C n, a path on n vertices by P n and a complete graph on n vertices by K n A graph G is connected if any to vertices of G are connected by a path A maximal connected sub graph of a graph G is called a component of G The number of components of G is denoted by ω(g) The complement G of G is the graph ith vertex set V in hich to vertices are adjacent if and only if they are not adjacent in G A graph G is said to be acyclic if it has no cycles A tree is a connected acyclic graph A bipartite graph is a graph hose vertex set can be partitioned into to disjoint non empty sets V 1 and V such that every edge has one end in V 1 and another end in V A complete bipartite graph is a bipartite graph here each vertex of V 1 is adjacent to every vertex in V The complete bipartite graph ith partitions of order V 1 = m and V = n, denoted by K m,n A star denoted by K 1,n-1 is a tree ith one root vertex and n-1 pendant vertices A bistar, denoted by D(r,s) is the graph obtained by joining the root vertices of the stars K 1,r and K 1,s A heel graph denoted by W n is a graph ith n vertices formed by joining a single vertex to all vertices of C n-1 A helm graph, denoted by H n is a graph obtained from the heel W n by attaching a pendant vertex to each vertex in the outer cycle of W n Corona of to graphs G 1 and G, denoted by G1 G is the graph obtained by taking one copy of G 1 and V(G 1 ) copies of G in hich i th vertex of G 1 is joined to every vertex in the i th copy of G If D is a subset of V, then D denoted the vertex induced sub graph of G induced by D The open neighborhood of a set D of vertices of graph G, denoted by N(D) is the set of all vertices adjacent to some vertex in D, and D D N is called the closed neighborhood of D, denoted by N[D] The diameter of a connected graph is the maximum distance beteen to vertices in G and is denoted by diam(g) A cut-vertex of a graph G is a vertex hose removal increases the number of components A vertex cut of a connected graph G is a set of vertices hose removal results in a disconnected graph The connectivity or vertex connectivity of a graph G, denoted by k(g) (here G is not complete) is the size of a smallest vertex cut A connected sub graph H of a connected graph G is called a H-cut if ω(g-h) For any real number denotes the largest integer less than or equal to x A subset D of V is called a dominating set of G if every vertex in V-D is adjacent to at least one vertex in D The domination number γ(g) of G is the minimum cardinality taken over all dominating set D in G A dominating set D of G is called a eak dominating set of G if for every v V D there exist a vertex u D such that uv V(G) and d( u) d( v) The minimum cardinality taken over all eak dominating sets is the eak domination number and is denoted by (G ) In this paper, e use this idea to develop the concept of complementary acyclic eak domination preserving number of a graph ijresorg 44 Page
2 Complementary Acyclic Weak Domination Preserving Sets N Saradha a,vsaminathan b Complementary Acyclic Weak Domination Preserving: Definition: 1 A subset D of G is called a complementary acyclic eak domination preserving set of G (c-adp set of G) if V D is acyclic and ( D ) ( G) The minimum cardinality of a c-adp set in G is called the complementary acyclic eak domination preserving number of G and is denoted by c-adpn(g) A c-adp set of G of cardinality c-adpn(g) is called a c-adpn- set of G Example: v {,v } is a complementary acyclic eak domination preserving set of G Remark: 3 Let G be a simple graph If c-adp set of G is independent, then (G) = 1 Remark: 4 Since V(G) is c-adp set, the existence of a c-adp sets is guaranteed in any graph Remark: 5 The c-adp set property is superhereditary, since any super set of a c-adp set is a c-adp set Hence a c-adp set is minimal if and only if it is 1-minimal c-adpn for standard graphs 1 c-adpn(k ) = n 1 c-adpn(k n ) = n, n 3 3 c-adpn(k 1,n ) = n 4 c-adpn(k m,n ) = max {m, n} 5 c-adpn(d m,n ) = m +n Theorem: 31 For any Path P n, n 3 II n c adpn( Pn ) if n 1(mod 3) 3 n 1if n 0 or (mod 3) 3 Theorem: 3 For any cycle C n, n 3 c adpn( C n n ) if n 1(mod 3) 3 n 1if n 0 or (mod 3) 3 Theorem: 33 For any heel W n, n 4 n c adpn( Wn ) if n is even n 1 if n is odd Fig 1 MAIN RESULT Observation 34 A c-adpn-set of a connected graph G need not induce a connected sub graph ijresorg 45 Page
3 Complementary Acyclic Weak Domination Preserving Sets N Saradha a,vsaminathan b Example: 35 v v 7 v 8 Fig 31 D = {v,,v 8 } is a c-adpn-set hich is disconnected Observation: 36 ( G) c adpn ( G) if and only if any c-adpn-set induces a complete subgraph Proof: By hypothesis c adpn ( G) ( G) Let D be a c-adpn-set of G c adpn ( G) D ( G) D ( Therefore D is a complete sub graph Conversely, let any c-adpn-set induces a complete sub graph Let D be a c-adpn-set of G D ( D ( G) That is c adpn ( G) ( G) Example: 37 v D={v,, } is a eak dominating set ( D ) 3 and also D is a c-adpn-set Remark: 38 If ( G) ( G), then dpn( G) ( G) But c-adpn(g) may be greater than (G ) Example: 39 D = {v,, }, ( G) ( G) and c-adpn(g) = 3 Fig 3 ijresorg 46 Page
4 Complementary Acyclic Weak Domination Preserving Sets N Saradha a,vsaminathan b v Fig 33 Definition: 310 A subset D of G is called a minimal c-adp set of G if D is a c-adp set of G and no subset of D is a c-adp set of G Theorem: 311 Let D be a c-adp set of G D is minimal if and only if for any u in D, either V (D {u}) contains a cycle or ( D { u} ) ( G) Proof: obvious Definition: 31 The maximum cardinality of a minimum c-adp set of G is called the upper c-adp number of G and is denoted by c-adpn(g) Remark: 313 There are graphs G ith c-adpn(g) < c-adpn(g) III FINE C-AWDP GRAPHS Definition: 41 A graph G is a fine c-adp graph if all minimal c-adp sets have the same cardinality Example: 4 (i) K n,n is a fine c-adp graph (ii) All cycles are fine c-adp graphs Theorem: 43 Let G 1 and G be graphs Suppose G 1 contains a cycle and G is acyclic and G ) ( ) Then have equal cardinality Proof: Suppose G 1 contains a cycle and G is acyclic and G ) ( ) 1 G ( 1 G G is a fine c-adp graph if and only if G is a fine c-adp graph and all minimal c-a sets of G 1 G1 G but it is not a dp set G1 G Any c-adp set of G 1 is a c-a ( 1 G set of Let D be a minimal c-adp set of G and D 1 be a minimal c-a set of G 1 Clearly, ( D ) ( G) ( G1 ) ( D1 ) Therefore D1 D is a minimal c-adp set of G1 G Let D D 1 D, here D 1 V ( G 1 ) and D V(G ) Since D is a c-a set of G1 G, D 1 is a c-a set of G 1 and D is a c-a set of G D) ( D D ) max{ ( D ), ( D } ( G G ) ( ) If D ) ( ) then ( G ( D1 ) ( G) ( G1 ) ( D1 ) ( D Therefore ( D1 ) ( D ) ( G) ( 1 D, a contradiction, since D 1 is a subset of V(G 1 ) Therefore ) Therefore D is a c-adp set of G and G is a fine c-adp graph if D 1 is c-a set of G 1 Since D is minimal, D 1 and D are also minimal Thus, and only if G is a fine c-adp graph and all minimal c-a set of G 1 have equal cardinality Similar argument can be given if G 1 contains a cycle and G is a cyclic 1 G IV CONCLUSION We found complementary acyclic eak domination preserving number for some standard graphs and general graphs REFERENCES [1] Frank Harary, Graph Theory, Narosa Publishing House, Reprint 1997 [] Gary Chartrand, Ping Zhang Chromatic Graph Theory CRC press, Taylor Dr Francis group A chapman and Hall book, 009 [3] Teresa WHaynes, Stephen THedetniemi, Peter JSlater, Fundamentals of Domination in Graphs, Marcel Dekker, Inc,ne York,Basel,Hong Kong 1998 ijresorg 47 Page
5 Complementary Acyclic Weak Domination Preserving Sets N Saradha a,vsaminathan b [4] SM Hedetniemi, STHedetniemi, DF Rall, Acyclic Domination,Discrete Mathematics (000), [5] MPoopalaranjani, On some coloring and domination parameters in graphs, PhD Thesis, Bharathidasan University, India, 006 [6] MValliammal, SPSubbiah, VSaminathan, Complementary acyclic chromatic preserving sets in graphs, Vol3, 013, No8, ijresorg 48 Page
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