The connected monopoly in graphs

Transcription

1 Iteratioal Joural of Multidisipliary Researh ad Developmet Volume :2, Issue :, April 2015 wwwallsubjetjouralom e-issn: p-issn: Impat Fator: 372 Ahmed Mohammed Naji Departmet of Studies i Mathematis Uiversity of Mysore, Maasagagotri Idia N D Soer Departmet of Studies i Mathematis Uiversity of Mysore, Maasagagotri Idia The oeted moopoly i graphs Ahmed Mohammed Naji, N D Soer Abstrat I a graph G(V, E), a set d V( is said to be a moopoly set of G if ay vertex v V - D has at d ( v ) least 2 eighbors i D, where d ( is a degree of v i G A moopoly set D of G is alled a oeted moopoly set of G if the subgraph D idued by D is oeted The miimum ardialities of oeted moopolies sets of G, deoted by mo( is alled the oeted moopoly size of G I this paper, we ivestigate the relatioship betwee mo( ad some other parameters of graphs Bouds for mo( ad its exat values for some stadard graphs are foud Keywords: Coeted moopoly set Coeted moopoly size Mathematis Subjet Classifiatio: 05C9, 05C99 1 Itrodutio I this paper, we are oeder with a simple graph G( V, E), that oempty, fiite, have o loops o multiple, direted edges Let G be suh a graph ad let ad m be the umber of its verties ad edges, respetively A graph H is a subgraph of G if V ( H ) V ( ad E( H ) E( For subset V ( S, the subgraph S of G is alled the subgraph idued by S if E( S) = { uv E( u, v S A graph G is said to be oeted if for every pair of verties there is a path joiig them The maximal oeted subgraphs are alled ompoets The oetivity umber ( is defied as the miimum umber of verties whose removal from G results i a disoeted graph or i the trivial graph (a sigle vertex) A graph G is said to be k -oeted if ( k We refer to [] for graph theory otatio ad termiology ot desribed here A set D of verties i a graph G is a domiatig set of G if every vertex i V D is adjaet to some vertex i D The domiatio umber ( of G is the miimum ardiality of a domiatig set i G The oept of oeted domiatio umber was itrodued by E Sampathkumar ad H Walikar [8] A domiatig set D of a graph G is oeted domiatig set if a subgraph idued by D is oeted The oeted domiatio umber ( of G is the miimum ardiality of a oeted domiatig set i G for more details i domiatio theory of graphs we refer to [5] Correspodee: Ahmed Mohammed Naji Departmet of Studies i Mathematis Uiversity of Mysore, Maasagagotri Idia A subset D of verties set of a graph G is alled a moopoly set if for every vertex v V ( D has at least 2 eighbors i D The moopoly size of G is the smallest ardiality of a moopoly set i G, deoted by mo( A moopoly set D of a graph G is miimum if for ay other moopoly set D of G, D D Ay moopoly set D of a graph G with miimum ardiality is alled a miimum moopoly set I partiular, moopolies are a dyami moopoly (dyamos) that, whe olored blak at a ertai time ~ 273 ~

2 Iteratioal Joural of Multidisipliary Researh ad Developmet step, will ause the etire graph to be olored blak i the ext time step uder a irreversible majority oversio proess Dyamos were first itrodued by Peleg [9] For more details i dyamos i graphs we refer to [1, 2, 3, 7, 10] I [], the author defied a moopoly set of a graph G, proved that the mo( for geeral graph is at least 2, disussed the relatioship betwee mathigs ad moopolies ad he showed that ay graph G admits a moopoly with at most ( verties A moopoly set D of a graph G is alled a oeted moopoly set of G if the subgraph D idued by D is oeted The miimum ardialities of oeted moopolies sets of G, deoted by mo(, is alled the oeted moopoly size of G I this paper, we itrodue ad study the oeted moopoly size of graphs ad we ivestigate the relatioship betwee mo (G ) ad some other parameters of a graph Bouds for mo (G ) ad its exat values for some stadard graphs are foud It is lear that, a oeted moopoly size of a graph G is exists if ad oly if G is oeted The we osider all graphs i this paper is oeted, uless refer to otherwise To illustrate this oept, osider the followig graph G i Figure 1 Fig 1: A graph G with = 2, = 3, mo = 3, mo = The set { v is a domiatig set of a graph G with miimum ardiality, the (G ) = 2 { v,,, the set v5 v is a oeted domiatig set of G with miimum (G ) = 3 ardiality, the, the set { v 7 is a moopoly set of G with miimum ardiality, the mo (G ) = 3 ad the set { v 5 7 is a oeted moopoly set of G with miimum ardiality, the mo (G ) = 2 Exat Values of Coeted Moopoly Size of Some Stadard Graphs The oeted moopoly size of some stadard graphs a be easily foud ad are give as follows: Observatio 21 mo ( K ) = 1 2, 2 mo( P ) = mo( C ) = 2 2, 3 mo( K1, ) = 1 3, 2 mo( K s, r ) = s 1, 2 s r, if 1 (mod mo ( W 3 ) = 1, otherwise 5 3 W Where is a wheel graph of order 3); ~ 27 ~

3 Iteratioal Joural of Multidisipliary Researh ad Developmet Theorem 22 Let T be a tree with verties The mo( T ) = l( T ) Where l (T ) is the umber of pedet verties i T Proof Let T be a tree with verties, L( T ) = { v T : = 1 ad let l( T ) = L( T ) The sie for every ( T L( T )) = 1 v L(T ), 2, it follows that T L(T ) is a moopoly set of T Furthermore, T L(T ) is a oeted idue subgraph of T Therefore, mo( T ) T L( T ) = l( T ) (1) Coversely, sie for every v T L(T ) is a ut-vertex of a tree T, it follows that mo( T ) l( T ) (2) From equatios (1) ad (2) we get mo ( T ) = l ( T ) 3 Bouds o Coeted Moopoly Size of Graphs Relatioships betwee a oeted moopoly size mo (G ) ad some other parameters of a graph G as a moopoly size mo (G ), a domiatio umber (G ), a ( oeted domiatio umber, a idepedet umber (G ) ad a oetivity umber (G ) may be get its as followig: Sie a oeted moopoly set of graphs is eessarily a moopoly set, it follows that the followig result is obvious Propositio 31 For ay oeted graphg, mo( mo( It is immediate observatio, from defiitios, that a oeted moopoly set of a graph G is a oeted domiatig set The the followig results proof is ot hard to get it Propositio 32 For ay oeted graphg, ( mo( The followig results is immediate osequees of Propositio 32 Corollary 33 For ay oeted graph G, ( mo( Theorem 3 Let G be a oeted graph of order ad let p a iteger umber suh that 0 p ( 2 If ( ( p, the ( p 1 Proof Let G be a oeted graph of order ad let 0 p ( 2 If ( ( p, the the subgraph V I idued by subset V I is oeted, where I is a idepedet set of G with ardiality equals to ( V I ) ( p 1 Sie I is a idepedet set, it follows that 2 for every v I Hee, V I is a oeted moopoly set of G Therefore, V I ( p 1 If p = 0 the 1 r r, this situatio, we a observe it for example, i, where ( K r, r ) = ( K r, r ) = r mo( K r, r ) = r 1 = 0 1 ad Theorem 35 Let G be a oeted graph with miimum degree The ( mo( 2 Proof Let G be a oeted graph with miimum degree ad let D be a oeted moopoly set of G Sie D is a oeted moopoly set, it follows that D N ( D 2 2 for every v V D But ( for every oeted graph Hee, ( mo( = D 2 K, ~ 275 ~

4 Iteratioal Joural of Multidisipliary Researh ad Developmet Theorem 3 Let D be a miimum moopoly set of a oeted graph G If V D ( 1, the = mo( Proof Let D be a miimum moopoly set of G ad let V D ( 1 The the subgraph D D = mo( mo( mo( oeted Hee, graph Ad this ompletes the proof But from Propositio 31 we have The followig results is immediate osequees of Theorem 3 Corollary 37 For ay oeted graph G, if mo( ( 1, the = mo( Lemma 38 For a oeted graph G of order 3 p, Where p is the umber of pedet verties of G ( Verties of degree equal to 1) Theorem 39 Let G be a oeted graph of order 3 The 2 P The boud is sharp, ad C attaitig it Proof Let G be a oeted graph of order 3, ad let L G defie as Lemma 38 we have L L 2 L < 2 If, the the theorem is hold Otherwise, if Case 1: L = 1, amely L = { v, Choose a vertex that u N ( Hee, the subgraph V { v, u Therefore, V { v, u = 2 Case 2: L = 0, the 2 for every uw E(, the we osider the followig ases: u V ( idued by D is for ay oeted L = { v V ( : = 1 The by u suh that u is ot a ut-vertex i G It is lear V { v, u is oeted Furthermore, v V ( Choose subset ad either u or w is a ut-vertex i G Hee, the subgraph u) u) S = mo( 2 We have from Theorem 39 that the lower boud is a moopoly set i G S V ( ad S = { u, w suh that V S is oeted Furthermore, Similarly for w Hee, V S is a oeted moopoly set i G Therefore, I the followig result we haraterize all graphs whih attaitig Theorem 310 Let G be a oeted graph of order The if G has oly oe uiversal vertex ( vertex with degree equal mo( = 1 to -1) ad all its other verties have degrees at most 2, the Proof Let G V = { v, v,,, be a oeted graph with vertex set 1 v2 v 1 suh that = 1 v ) 2 ad i for vi ) vi ) D = 1 every 1 i 1 ad let D = { v The 2, for every 1 i 1 Hee, the set D is a oeted moopoly set i G, ad mo( = 1 Theorem 311 Let G be a oeted graph with maximum degree 2 = ( ( mo( Proof By Propositio 32 we have Coversely, Let D be a oeted domiatig set of G The D 1 from defiitio we have that D is oeted, ad 2 2 for every v V D Hee, D is a oeted moopoly set of G ( Therefore, This ompletes the proof The ~ 27 ~

5 Iteratioal Joural of Multidisipliary Researh ad Developmet The overse of Theorem 311 ot true For example, mo( K ) = ( K1, 1, ) = 1, while ( K1, ) = for > 2 Theorem 312 Let G be a oeted graph The 3mo( 2 Proof Let D be a moopoly set i G ad let ( be the umber of ompoets i the subgraph D idued by D It is lear that mo( ( Sie every moopoly set i G is a domiatig set, it follows that there exists two ompoets of D ( amely, i ( ad j (, i j (, ( ) 3 ) suh that i j By ( addig the verties i the path betwee i ( ad j to set D dereases the umber of ompoets i D by oe This proedure a be repeated it util remai oly oe ompoet i D Thus resultig i a oeted moopoly set i G it is lear that there exists at most 2( ( 1) verties added to D to form a oeted moopoly set Hee D 2( ( 1) mo( 2( mo( 1) 3mo( 2 8 E Sampathkumar ad H B Walikar, The oeted domiatio umber of a graph, joural of mathematis ad physis siee, 13(1979), D Peleg, Loal majorities; oalitios ad moopolies i graphs; a review, Theoretial Computer Siee, 282(2002), M Zaker, O dyami moopolies of graphs with geeral thresholds, Disrete Mathematis, 312(2012), The boud is sharp, P ad C attaitig this boud Theorem 313 Let G be a oeted graph of order ad size m The om( 2m P The boud is sharp, ahieves it Proof Sie for ay oeted graph 1 m it follows by Theorem 39 that 2 2( 1) 2m Referees 1 E Berger, Dyami moopolies of ostat size, Joural of Combiatorial Theory, Series B, 83(2001), J Bermod, J Bod, D Peleg ad S Perees, The power of small oalitios i graphs, Disrete Applied Mathematis, 127(2003), P Flohii, R Kralovi, A Roato, P Ruzika ad N Satoro, O time versus size for mootoe dyami moopolies i regular topologies Joural of Disrete Algorithms, 1(2003), F Harary, Graph Theory, Addiso Wesley, Massahusetts, T W Hayes, S T Hedetiemi ad P J Slater, Fudametals of Domiatio i Graphs, Marel Dekker, New York, 1998 K Khoshkhak, M Nemati, H Soltai, ad M Zaker, A study of moopoly i graphs, Graph ad Combiatorial Mathematis, 29(2013), A Mishra ad S B Rao, Miimum moopoly i regular ad tree graphs, Disrete Mathematis, 30(1)(200), ~ 277 ~