ON THE COMPLEXITY OF K-STEP AND K-HOP DOMINATING SETS IN GRAPHS

Transcription

1 MATEMATICA MONTISNIRI Vol XL (2017) MATEMATICS ON TE COMPLEXITY OF K-STEP AN K-OP OMINATIN SETS IN RAPS M FARAI JALALVAN AN N JAFARI RA partmnt of Mathmatcs Shahrood Unrsty of Tchnology Shahrood Iran Emals: mfarhad438@gmalcom njafarrad@gmalcom Summary n a post ntgr 2 two rtcs n a graph ar sad to -stp domnat ach othr f thy ar at dstanc apart A st S of rtcs n a graph s a -stp domnatng st of f ry rtx s -stp domnatd by som rtx of S Th -stp domnaton numbr of s th mnmum cardnalty of a -stp domnatng st of A subst S of rtcs of s a -hop domnatng st f ry rtx outsd S s -stp domnatd by som rtx of S Th -hop domnaton numbr of s th mnmum cardnalty of a -hop domnatng st of In ths papr w show that for any ntgr 2 th dcson problms for th -stp domnatng st and -hop domnatng st problms ar NP -complt for planar bpartt graphs and planar chordal graphs 1 INTROUCTION For notaton and graph thory trmnology not gn hr w rfr to [6] Lt = ( V E) b a graph wth rtx st V = V ( ) and dg st E E( ) n ( ) = V ( ) and th sz of s m ( ) E ( ) last on dg Th opn nghborhood of a rtx s N ( ) = u V ( ) u E( ) and th closd nghborhood of s N [ ] { ( ) dg( ) s N ( ) = Th ordr of s = A graph s non-mpty f t contans at { = N Th dgr of dnotd by Th opn nghborhood of a subst S V s N ( S) = N ( ) S N S = N S S A subst S of rtcs V S has a nghbor n S Th and th closd nghborhood of S s th st [ ] ( ) of a graph s a domnatng st of f ry rtx n ( ) domnaton numbr of s th mnmum cardnalty of a domnatng st of Th dstanc u -path btwn two rtcs u and n dnotd d( u ) s th mnmum lngth of a ( ) n A chordal graph s a graph that dos not contan an nducd cycl of lngth gratr than3 A planar graph s a graph whch can b drawn n th plan wthout any dgs crossng For an ntgr 1 two rtcs n a graph ar sad to -stp domnat ach othr f thy ar at dstanc xactly apart n A st S of rtcs n s a -stp domnatng st of f ry rtx n Vs ( ) -stp domnatd by som rtx of S Th -stp domnaton numbr γ ( ) stp of s th mnmum cardnalty of a -stp domnatng st of 2010 Mathmatcs Subjct Classfcaton: 05C69 Kywords and phrass: -Stp domnatng st -Stp domnaton numbr -op domnatng st -op domnaton numbr Stp domnatng st NP-complt 36

2 M Farhad Jalal and N Jafar Rad Th concpt of 2-stp domnaton n graphs was ntroducd by Chartrand arary ossan and Schultz [3] and furthr studd for xampl n [4811] Rcntly Ayyaswamy and Natarajan [1] ntroducd a paramtr smlar to th 2-stp domnaton numbr namly th hop domnaton numbr of a graph A subst S of rtcs of s a hop domnatng st f ry rtx outsd S s 2-stp domnatd by som rtx of S Th hop domnaton numbr γ h( ) of s th mnmum cardnalty of a hop domnatng st of Th concpt of hop domnaton was furthr studd for xampl n [210] nnng t al [7] studd th complxty ssu of th 2 -stp domnaton as wll as th hop domnaton n a graph and showd that th dcson problms for th 2 -stp domnatng st and hop domnatng st problms ar NP -complt for planar bpartt graphs and planar chordal graphs In ths papr w gnralz ths rsults for any ntgr 2 For an ntgr 2 a subst S of rtcs of s calld a -hop domnatng st f ry rtx outsd S s -stp domnatd by som rtx of S Th -hop domnaton numbr γ h ( ) of s th mnmum cardnalty of a -hop domnatng st of W show that for any ntgr 2 th dcson problms for th -stp domnatng st and -hop domnatng st problms ar NP -complt for planar bpartt graphs and planar chordal graphs 2 MAIN RESULTS W wll stat th corrspondng dcson problms n th standard Instanc Quston form [5] and ndcat th polynomal-tm rducton usd to pro that t s NP -complt Lt 2 b a post ntgr Consdr th followng dcson problms: -Stp omnatng St Problm ( SP ) Instanc: A non-mpty graph and a post ntgr t Quston: os ha a -stp domnatng st of sz at most t? -op omnatng St Problm( P ) Instanc: A non-mpty graph and a post ntgr t Quston: os ha a -hop domnatng st of sz at most t? W us a transformaton of th Vrtx Cor Problm whch was on of Karp's 21 NP - complt problms [9] A rtx cor of a graph s a st of rtcs such that ach dg of th graph s ncdnt wth at last on rtx of th st Th Vrtx Cor Problm s th followng dcson problm Vrtx Cor Problm ( VCP ) Instanc: A non-mpty graph and a post ntgr Quston: os ha a rtx cor of sz at most? 37

3 M Farhad Jalal and N Jafar Rad W frst consdr th -stp domnatng st problm Thorm 1 Th SP s NP -complt for planar bpartt graphs Proof Clarly th SP s n NP snc t s asy to rfy a "ys" nstanc of th SP n polynomal tm W show how to transform th rtx cor problm to th SP so that on of thm has a soluton f and only f th othr has a soluton Lt b a connctd planar graph of ordr n = n and sz m = m 2 Lt b th graph obtand from as follows For ach dg = u E( ) w subdd th dg 2 1tms and lt x1 x2 x2 1 b th rtcs that rsultd from subddng th dg 2 1tms and add a path and jon 1 to both u and Th rsultng graph n n 2 1 m 2 m n 4 1 m m = 4+ 1 m has ordr = + ( ) + = + ( ) and sz ( ) Clarly th transformaton can b prformd n polynomal tm W not that s connctd and planar snc s connctd and planar Furthr by th constructon dosn`t contan odd contour so s bpartt Thus s a connctd planar bpartt graph W show that has a rtx cor of sz at most t f and only f has a -stp domnatng st of sz at most t+ m Assum that has a rtx cor S of sz at most t W now consdr th st S = S Snc m 2 w fnd that rtx ( ){ 1 2 E -stp domnats th rtcs S For ry dg u E( ) + x an = and 1 1 th x + and th rtx -stp domnats th rtcs 2 u and n Furthr snc s connctd and m 2 for ry two adjacnt dgs and f n for 1 1 th rtx s -stp domnatd f by th rtx Snc S s a rtx cor n and x ar -stp domnatd by th st S n Thrfor th st S s a -stp domnatng st of sz at most t + m n Fgur 1: Th graphs and n th proof of Thorm 1 38

4 M Farhad Jalal and N Jafar Rad Suppos nxt that has a -stp domnatng st = u E( ) In ordr to -stp domnat + Lt of sz at most t m + n th rtcs for blongs to th st In ordr to -stp domnat th rtx that u or or both u and blong to Furthr snc for 1 blong to Thus V ( ) for ry u E( ) m = t Thus has a rtx cor of sz at most t Thorm 2 Th SP s NP -complt for planar chordal graphs x n w not = s a rtx cor of = w not that Proof As notd n th proof of Thorm 1 th SP s n NP Now lt us show how to transform th rtx cor problm to th SP so that on of thm has a soluton f and only f th othr has a soluton Lt b a connctd planar chordal graph of ordr n and sz m 2 Lt b th = w add a path graph obtand from as follows For ach dg u E( ) P2 : and jon 1 to both u and and add a path P : a1 a2 a and jon a 1 to both u and Th rsultng graph has ordr n = n + 3m and sz m = ( 3+ 3) m Th transformaton can clarly b prformd n polynomal tm W not that snc s a connctd planar chordal graph so too s W show that has a rtx cor of sz at most t f and only f has a -stp domnatng st of sz at most t+ m Assum that has a rtx cor S of sz at most t W now consdr th st S = S Snc m 2 w fnd that rtx -stp domnats th rtcs domnats th rtcs 2 u and n ( ){ 1 2 E S For ry dg u E( ) + and a = and 1 1 th and also th rtx -stp Fgur 2: Th graphs and n th proof of Thorm 2 39

5 M Farhad Jalal and N Jafar Rad Snc s connctd and m 2 for ry two adjacnt dgs and f n for 1 1 th rtx f s -stp domnatd by th rtx Snc S s a rtx cor n and -stp domnatng st of sz at most domnat a ar -stp domnatd by th st S n Thrfor th st t+ m n Lt u E( ) n th rtcs -stp domnat th rtx Thus V ( ) S s a = In ordr to -stp 1 blongs to th st In ordr to a n w not that u or or both u and blong to blong to = s a rtx cor of Furthr snc for 1 for ry = u E( ) w not that m = t Thus has a rtx cor of sz at most t W nxt consdr th -hop domnatng st problm Thorm 3 Th P s NP -complt for planar bpartt graphs Proof Lt b a graph of ordr n and sz m and lt b th connctd planar bpartt graph constructd n th proof of Thorm 1 W show that has a rtx cor of sz at most t f and only f has a hop domnatng st of sz at most t+ m If has a rtx cor S of sz at most t thn ths s mmdat snc th -stp domnatng st S constructd n th proof of Thorm 1 s also a hop domnatng st n of sz S t+ m Suppos nxt that has a hop domnatng st of sz at most t + m If { n { { for som dg E( ) thn at last on rtx + + s not hop domnatd by a contradcton Thrfor for ry dg E( ) Lt = u b an arbtrary dg of If x thn n ordr to hop domnat th rtx both u and blong to W now consdr th st follows For ach rtx x assocatd wth an dg E( ) f x x n w not that u or or obtand from V ( ) as thn w add u or to th st Th rsultng st s a rtx cor of of sz at most m t Thus has a rtx cor of sz at most t Thorm 4 Th P s NP -complt for planar chordal graphs Proof Lt b a graph of ordr n and sz m and lt b th connctd planar chordal graph constructd n th proof of Thorm 2 W show that has a rtx cor of sz at most t f and only f has a hop domnatng st of sz at most t+ m If has a rtx cor S of sz at most t thn ths s mmdat snc th -stp domnatng st S constructd n th proof of Thorm 2 s also a hop domnatng st n of sz 40

6 M Farhad Jalal and N Jafar Rad S t m + Suppos nxt that has a hop domnatng st If { of sz at most t + m 1 for som dg E( ) thn at last on rtx of { s not hop domnatd by a contradcton Thrfor { for ry dg E( ) Lt = u b an arbtrary dg of If a both u and blong to follows For ach rtx thn n ordr to hop domnat th rtx a n w not that u or or W now consdr th st obtand from V ( ) as a assocatd wth an dg E( ) f a thn w add u or to th st Th rsultng st s a rtx cor of of sz at most m t Thus has a rtx cor of sz at most t REFERENCES [1] SK Ayyaswamy and C Natarajan op domnaton n graphs Manuscrpt [2] SK Ayyaswamy B Krshnaumar C Natarajan and YB Vnatarshnan Bounds on th hop domnaton numbr of a tr Procdngs of Mathmatcal Scncs Indan Acadmy of Scnc (2015) OI:101007/s [3] Chartrand F arary M ossan and K Schultz Exact 2-stp domnaton n graphs Mathmatca Bohmca (1995) [4] ror A L and Y Rodtty A not: som rsults n stp domnaton of trs scrt Math (2004) [5] M R ary S Johnson Computrs and Intractblty: A ud to th Thory of NPcompltnss Frman Nw Yor (1979) [6] T W ayns S T dtnm and P J Slatr Fundamntals of omnaton n raphs Marcl r Inc Nw Yor (1998) [7] M A nnng and N Jafar Rad On 2-Stp and op omnatng Sts n raphs raphs and Combnatorcs (2017) OI /s [8] P rsh On xact n-stp domnaton scrt Math (1999) [9] R M Karp Rducblty mmong combnatoral problms In R E Mllr and J W Thatchr (dtors) Complxty of Computr Computatons Plnum Nw Yor (1972) [10] C Natarajan and SK Ayyaswamy op omnaton n raphs-ii An Stt Un Odus Constanta 23(2) (2015) [11] Y Zhao L Mao and Z Lao A Lnar-Tm Algorthm for 2-Stp omnaton n Bloc raphs J Mathmatcal Rsarch wth Applcatons (2015) Rcd Sptmbr