Bounds on double Roman domination number of graphs a

Transcription

1 Proceedigs o the 2 d Iteratioal Coerece o Combiatorics, Cryptography ad Computatio (I4C217) Bouds o double Roma domiatio umber o graphs a 1 Doost Ali Mojdeh a, Ali Parsia b, Ima Masoumi Departmet o Mathematics, Uiersity o Mazadara, Babolsar, Ira b Departmet o Mathematics, Uiersity o aresh, aresh, Ira * i_masoumi@yahoocom b Let (, E) be a simple graph A double Roma domiatig uctio o a graph is a uctio : {,1,2, haig the property that i (), the the ertex must hae at least two eighbors, such that ( w1 ) ( w2 ) 2 or oe eighbor such that (w) ; ad i () 1, the the ertex must hae at least oe eighbor w such that ( w) 2 he weight o a double Roma domiatig uctio is the sum w ( ), ad the miimum weight o ( ) w 1 w 2 w or eery double Roma domiatig uctio o is called double Roma domiatio umber o We deote this umber with ) I this paper; we obtai some ew lower ad upper bouds o double ( Roma domiatio umber o graphs 21 Mathematical ubject Classiicatio: 5C69 Keywords ad phrases: Double Roma domiatio, bouds o double Roma graph, domiat dieretial 1 Itroductio raph domiatio applies aturally too may tasks, icludig acility locatio ad etwork costructio, or example, i costructig a cellular phoe etwork, oe eeds to w 1 Correspodig author 429

2 choose locatios or the towers to coer a large regio as cheaply as possible May ariats o domiatio hae bee studied extesiely ad hae applicatios such as costructios o errorcorrectig codes or digital commuicatio ad eiciet data routig i wireless etworks he origial study o Roma domiatio was motiated by the deese strategies used to deed the Roma Empire durig the reig o Emperor Costatie the reat, AD He decreed that or all cities i the Roma Empire, at most two legios should be statioed Further, i a locatio haig o legios was attacked, the it must be withi the iciity o at least oe city at which two legios were statioed, so that oe o the two legios could be set to deed the attacked city his part o history o the Roma Empire gae rise to the mathematical cocept o Roma domiatio, as origially deied ad discussed by tewart [7] i 1999, ad Reelle ad Rosig [6] i 2 Let (, E) be a graph o order We cosider the ollowig game You are allowed to buy as may tokes rom a bak as you like, at a cost o 1 dollar each For example, uppose you buy tokes You the place the tokes o some subset o ertices o For each ertex o which has o toke o it, But is adjacet to a ertex with a toke o it, you receied 1 dollar rom the bak Your objectie is to maximize your proit, that is, the total alue receied rom the bak mius the cost o the tokes bought Let be the set o ertices i X that hae a eighbour i the set X For a oempty subset X we write C( X ) ( X bd( X )) Based o this game, We deie the dieretial o a set to be ( X ) bd( X ) X [5], ad the dieretial o a graph to be equal to ( ) max{ ( X ) : X Oe o ariatios o a dieretial o graphs, is B-dieretial o graphs We deote this parameter o graph with () ad we deie ( ) max{ bd( X ) : X [5] A graph is said to be domiat dieretial [] i it cotais a -set which is also a domiatig set ome examples o domiat dieretial graphs are complete graphs, star graphs, wheel graphs, path graphs ad cycle graphs with k ( ) P k bd(x ) k or k 2 A graph is said to be double Roma graph i dr ( ) ( ) bhh We deote miimum degree o ertices o graph with () ad maximum degree o ertices o graph with () he ope eighborhood o a ertex () is the set N( ) { u : u E( ) he ope eighborhood o a set is the set N( ) { N( ) : he closed eighborhood o a set is the set N[ ] N( ) Let be the set o edges icidet with i that is, E { u E( ) : u N ( ) We E deote the degree o by d ( ) E A ertex o degree zero is called a isolated ertex ie a set the priate eighborhood p [, ] o is deied by p[, ] N[ ] N[ { ], equialetly, p[, ] { u : N[ u] { Each ertex i p [, ] is called a priate eighbor o he exteral priate eighborhood ep (, ) o with respect to cosists o those priate eighbors o i hus ep(, ) p[, ] ( ) l A set is a domiatig set i N [ ] A domiatio umber () is the miimum cardiality a () -et A graph has property EPN i or eery () -set ad or eery, ep (, ) We call a tree with property EPN, a EPN -tree [5] C X 4

3 graph For a graph (, E) let i { ( ) : ( ) i A Roma domiatig uctio o is a uctio : {,1,2 such that i or some, the there exists w N() such that w 2 he weight o a Roma domiatig uctio is the sum w ( ), ad the miimum weight o or eery Roma domiatig uctio ( ) o is called Roma domiatio umber o We deote this umber with ( R ) We say that graph is Roma graph i We hae R ( ) 2 ( ) A double Roma domiatig uctio o graph is a uctio : {,1,2, such that the ollowig coditios are met: (a) i, the ertex must hae at least two eighbors i or oe eighbor i () (b) i () 1, the ertex w must hae at least oe eighbor i 2 2 he weight o a double Roma domiatig uctio is the sum w ( ), ad the miimum weight o w double Roma domiatio umber o or eery double Roma domiatig uctio We deote this umber with Beeler et al, i [1] hae bee studied the double Roma graph 2 Kow results ( ) o dr () he ollowig results are importat or our iestigatios heorem A [5] For ay graph o order, ( ) ( ) heorem B [] A graph is domiat dieretial i ad oly i ( ) 2 ( ) heorem C [2] I is a graph o order, the R ( ) ( ) heorem D [2] I is a graph o order, the ( ) ( )( ( ) 1) R ( ) 2 is called Robert A Upper ad lower bouds heorem 1 Let be a simple coected graph o order I -uctio o exists such that (,,, ) ad, the ( ) 2 1 2( ) 1 2 Proo For eery double Roma domiatio uctio g ( W, W, W, ) o graph 1 2 W we cosider -uctio h ( U, U1, U) o such that U bd( W ), U, U C( W ), U W Clearly we hae g( W) h( U) hereore, 1 2 ( ) mi{ h( U) : U ( U, U1, U) mi{ h( U) : U ( U, U1, U), U 41

4 mi {2 U U : U mi{2 C( W ) W : W, W 2 mi{2 C( W ) W 2 bd( W ) 2 bd( W ) : W mi{2 C( W ) 2 W 2 bd( W ) W 2 bd( W ) : W mi{2 W 2 bd( W ) : W 2 max{2 bd( W ) W : W Now sice W 1, we hae, ( ) 2 max{2 bd( W ) 1: W 2 1 2max{ bd( W ) : W 2 1 2( ) heorem 2 Let be a simple coected graph o order exists such that (,,, ) ad, the ( ) 2 ( ) i -uctio o Proo By heorem 1 ad heorem A, we hae: ( ) 2 ( ) 1 heorem Let be a otriial tree o order he ( ) 2 ( ) 1 Proo uppose o the cotrary that or eery -uctio (, 1, 2, ) o, we hae hereore 2 Now we cosider two cases as ollows: Case1 I at least two ertices i set are adjacet, the by deiitio o double 1, 2 Roma domiatio uctio we must hae distict ertices, 4, 5, 6 i set o 2 such that is adjacet with, ad 4 2 is adjacet with, 5 6 ice, i the ertices were ot distict, the there was exist a cycle Now we assig label to ertices, 1 2 ad label to ertices, 4, 5, 6 hereore we achiee a double Roma domiatio uctio with less weight Case2 I eery two ertices, i 1 are ot adjacet, the there is ot exist ay cycle i a tree, thereore by deiitio o double Roma domiatio uctio we must hae two ertices, 1 with at least three ertices o label 2 are adjacet Now we assig label to ertices, 1 ad assig label to three ertices o label 2 which are adjacet hereore we achiee a double Roma domiatio uctio at most with equal weight Hece accordig to 1 42

5 upper two cases we achiee cotradictio hus accordig to heorem 2 we hae, ( ) 2 ( ) 1 heorem 4 I a graph is domiat dieretial ad without isolated ertices, the ( ) 2 ( ) ( ) But sice a graph Proo By proo o heorem 1 we hae, ( ) mi{ h( U) : U ( U, U1, U) hereore we hae 2 max{2 bd( W ) W : W 2 max{ ( W ) ( W ) : W is domiat dieretial ad without isolated ertices thus we hae, ( ) /2, ( ) 2 ( ) hus, ( ) 2 max{ ( ) ( ) 2 ( ) ( ) ( ) hus, heorem 5 I a graph is domiated dieretial ad without isolated ertices, the is a double Roma graph Proo By heorems A, B ad 4 we hae, ( ) 2 ( ) ( ) 2 ( ( )) ( 2 ( )) But or eery graph that ( ) ( ) we hae ( ) ( ) thereore ( ) ( ) heorem 6 I a tree is a double Roma, the it is EPN -tree Proo We proe, i a tree is double Roma, the is EPN -tree We suppose is ot EPN -tree hus we hae or some -set ad or some ertex () ep (, ) We ca suppose that {,,, ad 2 1 ( ) 1, hereore p[ ( ) hus p[ Now clearly, we ca say that a domiatig set is a miimal domiatig set i ad oly i eery ertex i has at least oe priate eighbor ice is a ( ) -set, the or eery i {,1,, ( ) 1 we hae p[ i hus there exists a ertex w such that N w] { But we hae p[, hece w Now [ i we hae w, the we should hae w i or some i O the other had, eery eighbor o ertex i set o is adjacet to at least oe aother ertex i set o because the ertex o does ot hae priate eighbor i set o Now we ca delete the ertex o rom the set o such that this ew set will be a ( ) -set with smaller 4

6 cardiality hereore we hae tree thus But we hae w Hece is a isolated ertex i set o is coected hereore there exists a ertex w ( ) such that p[ thus we must hae a ertex 1 such that But w N ( 1 ) is a w N( ) Now i there exists aother ertex w ( ) such that w N ], the we should hae a ertex [ or example such that w 1 N[ 2 ] because a ertex does ot hae ay priate eighbour i set o Now we ca deie uctio : {,1,2, such that ) 2, ( ),, ( ) ad or eery w ( ) Clearly is a ( 1 ( ) 1 double Roma domiatio uctio o with beig double Roma tree (w) such that w < ( ) his iequality cotradicts Reereces [1] RA Beeler, W Hayesa, Hedetiemi, Double Roma domiatio, Discrete Applied Mathematics, ol 211 (216) 2-29 [2] Bermudo, H Ferau, J M igarreta, he dieretial ad the Roma domiatio umber o a graph, Appl Aal Discrete math 8(214) [] Bermudo, JC Herádez-ómez, JM Rodríguez, JM igarreta, Relatios betwee the dieretial ad parameters i graphs, Amsterdam: Elseier Electroic Notes i Discrete Mathematics 46 (214) [4] W Hayes, Hedetiemi, PJ later, Fudametals o Domiatio i raphs, Marcel Dekker, New York, 1998 [5] JR Lewis, Dieretials o graphs, Master s hesis, East eessee tate Uiersity, 24 [6] C Reelle, KE Rosig, Deedes imperium romaum: a classical problem i military strategy, Amer Math Mothly 17 (7) (2) [7] I tewart, Deed the Roma empire!, ci Amer 281 (6) (1999) [8] DB West, Itroductio to raph theory, Book, ecod Editio, 21 44