Intrntionl J.Mth. Comin. Vol.1(2014), 80-86 Algorithmi n NP-Compltnss Aspts of Totl Lit Domintion Numr of Grph Girish.V.R. (PES Institut of Thnology(South Cmpus), Bnglor, Krntk Stt, Ini) P.Ush (Dprtmnt of Mthmtis, Signg Institut of Thnology, Tumkur, Krntk Stt, Ini) E-mil: girishvr1@ps.u, ushpmurthy@yhoo.om Astrt: A ominting st of grph η(g), is totl lit ominting st if th ominting st os not ontin ny isolts. Th totl lit ominting numr γ t(η(g)) of G is minimum rinlity of totl lit ominting st of G. Th urrnt ppr stuis totl lit omintion in grph from n lgorithmi point of viw. In prtiulr w h otin th lgorithm for totl lit omintion numr of ny grph. Also w h otin th tim omplxity of propos lgorithm. Furthr w isuss th NP-Compltnss of totl lit omintion numr of th split grph, iprtit grph n horl grph. Ky Wors: Smrnhly k-ominting st, totl lit ominting numr, lit grph, vrtx inpnn numr, iprtit grph, split grph, horl grph. AMS(2010): 05C69 1. Introution All grphs onsir hr r finit, onnt, unirt without loops or multipl gs n without isolt vrtis. As usul p n q nots th numr of vrtis n gs of grph G. Th onpt of omintion in grph thory is nturl mol for mny lotion prolms in oprtions rsrh. In grph G, vrtx is si to omint itslf n ll of its nighors. A st D V of G is si to Smrnhly k-ominting st if h vrtx of G is omint y t lst k vrtis of S n th Smrnhly k-omintion numr γ k (G) of G is th minimum rinlity of Smrnhly k-ominting st of G. Prtiulrly, if k = 1, suh st is ll ominting st of G n th Smrnhly 1-omintion numr of G is ll th omintion numr of G n not y γ(g) in gnrl. A ominting st D of grph G is totl ominting st if th ominting st D os not ontin ny isolts. Th totl omintion numr γ t (G) of grph G is th minimum rinlity of totl ominting st. Th lit grph η(g) of grph G is th grph whos vrtx st is th union of th st 1 Riv Sptmr 24, 2013, Apt Frury 26, 2014.
Algorithmi n NP-Compltnss Aspts of Totl Lit Domintion Numr of Grph 81 of gs n th st of ut vrtis of G in whih two vrtis r jnt if n only if th orrsponing gs r jnt or th orrsponing mmrs of G r inint. A ominting st of grph η(g), is totl lit ominting st if th ominting st os not ontin ny isolts. Th totl lit ominting numr γ t (η(g) of G is th minimum rinlity of totl lit ominting st of G. A vrtx ovr C of grph G = (V, E) is sust C V suh tht for vry g uv E, w hv u C or v C. A ut-vrtx of onnt grph G is vrtx v suh tht G {v} is isonnt. A stl st in grph G is pir-wis non-jnt vrtis sust of V (G), n liqu is pirwis jnt vrtis sust of V (G). A grph is split if its vrtx st n prtition into stl st n liqu. A grph is iprtit if its vrtx st n prtition into two stl sts. A grph is horl if vry yl of lngth t lst 4 hs t lst on hor, whih is n g joining two non-onsutiv vrtis in th yl. In this ppr, w otin th lgorithm for totl lit omintion numr of ny grph. Also, w h otin th tim omplxity of propos lgorithm. Furthr w isuss th NP-Compltnss of totl lit omintion numr of grph with rspt to split grph, iprtit grph n horl grph. 2. Algorithm To fin th lgorithm for th minimum totl lit omintion st of grph w us initilly, th DFS lgorithm to th fin th ut vrtis of givn grph [1], th VSA lgorithm [2] to fin th minimum vrtx ovr of grph n shortst pth lgorithm [3] to fin th shortst pth in grph. Th gs in th shortst pth givs totl lit omintion st of grph G. Thn w ru this to minimum st whih givs th minimum totl lit omintion st of ny grph G. Algorithm to fin th minimum totl lit omintion st of givn grph: Input: A grph G = (V, E). Output: A minimum totl lit omintion st D of grph G = (V, E). Stp 1: Initiliz D = φ. Stp 2: Ll th vrtis of grph G s {v i /i = 1, 2, 3, 4, 5,, n} n ll th gs of grph G s { j /j = 1, 2, 3, 4, 5,, m}. Stp 3: Lt A={v i /v i is ut vrtx of grph G(V, E)}. Stp 4: Comput th st C of ll miniml vrtx ovrs in G, suh tht C os not ontin vrtx of gr on. Stp 5: FOR th miniml vrtx ovr st C, DO Stp 6: IF V () = 1. GOTO Stp 7. ELSE
82 Girish.V.R. n P.Ush IF V () = 2 n thy r jnt GOTO Stp 8. ELSE GOTO Stp 9. END IF. Stp 7: D = D { ny two jnt gs of E(G)}. GOTO Stp 13. Stp 8: D = D {( i, j ), i is ommon g inint with V () n j N( i )} GOTO Stp 13. Stp 9: Lt E 1 = { q / q E(G), whr q is th st of gs in th shortst pth onnting ll th vrtis of V () n E 1 K 1,n if thr is ny othr shortst pth }. K = { l / l is n n g E 1 }. R = { j / j E(G) E 1 / j is jnt to K} FOR E 1 1 or 0 DO, Lt two gs E 2 = ( i, j ) E 1 suh tht j N( i ). IF i N( j ) n i N( k ), whr k or j is n n g. Thn E 2 = ( i,n n g) ELSE IF i N( j, k ) n j N( l, m ), ( l, m ) i Thn E 2 =( i, k ) END IF END IF D = D E 2. B = { p / p N( i, j ) in E 1 }. C 1 = { r / r N(B) E 1 (D B), r is not inint with A, r (v i, v j ), v i, v j C}. E 1 = E 1 (B C 1 ). END FOR. Stp 10: IF E(E 1 ) = 0 thn GOTO Stp 11. ELSE D = D {E 1 i, i E 1 n i N(D)}. GOTO Stp 11. END IF. Stp 11: FOR R φ DO, Lt ny g in R D = D { k, k E 1 n k N( i )}. R = R { i } { s / s N(D)}. END FOR Stp 12: END FOR (from Stp 4) Stp 13: RETURN D, minimum totl lit omintion st of grph G. Stp 14: STOP.
Algorithmi n NP-Compltnss Aspts of Totl Lit Domintion Numr of Grph 83 3. Tim Complxity Th worst s tim omplxity of fining th solution of th minimum totl lit omintion prolm of grph using th propos lgorithm n otin s follows: Assum tht thr r n vrtis n m gs in th propos lgorithm. (i) DFS lgorithm [1] to fin th ut vrtis of givn grph whih rquirs running tim of O(mn). (ii) VSA lgorithm [2] to fin th minimum vrtx ovr of givn grph whih rquirs th running tim of 0(mn 2 ). (iii) Shortst pth lgorithm [3] to fin th shortst pth onnting th vrtis of V () whih rquirs th worst s of running tim of O(m + n). (iv) For FOR loop in stp 9 rquirs th worst s running tim of 0 ( ) m 1 3. (v) For FOR loop in stp 11 rquirs th worst s running tim of 0 ( 2n 3 2). (vi) So th ovrll tim is ( ) ( ) m 1 2n O(mn) + 0(mn 2 ) + O(m + n) + 0 + 0 3 3 2 = 0(mn 2 ). 4. NP-Compltnss of totl lit omintion numr of grph This stion stlishs NP-Complt rsults for th totl lit omintion prolm in iprtit grph, split grph n in hrol grph. Th trnsformtion is from th vrtx ovr prolm, whih is known to NP-Complt. 1 1 1 x y 1 1 Fig.1 A onstrut iprtit grph G from th grph G Thorm 4.1 Th totl lit omintion numr prolm is NP-Complt for iprtit grph. Proof Th totl lit omintion numr prolm for iprtit grph is NP-Complt s w n trnsform th vrtx ovr prolm to it s follows. Givn non-trivil grph G = (V, E),
84 Girish.V.R. n P.Ush onstrut th grph G = (V, E ) with th vrtx st V onsists of two opis of V not y V n V, togthr with two spil vrtis x n y n whos gs E onsists of (i) gs uv n u v for h g uv E(G). (ii) gs of th form uu for h vrtx u V. (iii) gs of th form u x for vry vrtx u V. (iv) th on itionl g xy. W lim tht G = (V, E) hs vrtx ovr of siz k if n only if G = (V, E ) hs miniml totl lit omintion st of siz k + (p k). Lt C th vrtx ovr of G of siz k. Lt B = {u x/u V } suh tht B = k. Lt D = B R, whr R = {u x/u V C} with R = p k. Thn it is lr tht, D is totl lit ominting numr of iprtit grph with rinlity k + (p k). On th othr hn suppos D is miniml totl lit omintion st of th grph G with rinlity k + (p k). Lt A = {v i /v i V, v i is inint with i D} with A = D. Th vrtx st A in G is V (G), suh tht A onsists of opis of V n V C n whos vrtis r jnt to tlst on vrtx of C. So, th grph G hs vrtx ovr of siz k. ¾ Thorm 4.2 Th totl lit omintion numr prolm is NP-Complt for split grph. Proof Th totl lit omintion numr prolm for split grph is NP-Complt s w n trnsform th vrtx ovr prolm to it s follows. Givn non-trivil grph G = (V, E) onstrut th grph G = (V, E ) with th vrtx st V = V En E = {uv : u v, u, v V } {v : v V, E, v }. f f g g G G Fig.2 A onstrut split grph G 1 from grph G W lim tht G = (V, E) hs vrtx ovr of siz k if n only if G = (V, E ) hs totl lit omintion st of siz k + (p k) 1. Lt C th vrtx ovr of G of siz k. Lt B = { i / i E(G ) E(G), i is inint with V C n V V C in G}. Thn it is lr tht B is totl lit ominting st of split grph with rinlity k + (p k) 1. On th othr hn, suppos D is th totl lit omintion numr of th grph G with
Algorithmi n NP-Compltnss Aspts of Totl Lit Domintion Numr of Grph 85 rinlity k + (p k) 1. Lt A = {v i /v i V, v i is inint with i D E(G)} with rinlity qul to D + 1 = k + (p k). Th vrtx st A in G is V (G) suh tht A onsists opis of V n V C whos vrtis r jnt to t lst to on vrtx of C. So, th grph Ghs vrtx ovr of siz k. ¾ Thorm 4.3 Th totl lit omintion numr prolm is NP-Complt for horl grph. Proof w shll trnsform th vrtx ovr prolm in gnrl grph to th totl lit omintion in horl grph. Thrfor, th NP-Compltnss of th totl lit omintion prolm in horl grph follows from tht of th vrtx ovr prolm in gnrl grph. For ny grph G onsir th horl grph G = (V, E ) with vrtx st V = {v 1, v 2, v 3, v 4 /v V } n th g st E = {v 1 v 2, v 2 v 3, v 3 v 4 /v V } {u 3 v 4 /uv E} {u 4 v 4 /uv V, u v}. 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 G G Fig.3 A onstrut horl grph G 1 from grph G W lim tht G = (V, E) hs vrtx ovr of siz k if n only if G = (V, E ) hs miniml totl lit omintion st of siz 2(k + (p k)). Lt C th vrtx ovr of G of siz k. Lt B = {v 2 v 3, v 3 v 4 /v V }. Thn it is lr tht B is miniml totl lit ominting st of horl grph with rinlity 2(k + (p k)). On th othr hn suppos D is th miniml totl lit omintion numr of th grph G with rinlity 2(k + (p k)). Lt A = {v 3 /v 3 V, v 3 is inint with v 2 v 3, v 3 v 4 D} with A = D 2 =k + (p k). Th vrtx st A in G is V (G) suh tht A onsists opis of V n V C whos vrtis r jnt to t lst to on vrtx of C. So, th grph G hs vrtx ovr of siz k. ¾ 4. Conlusion Th min purpos of this ppr is to stlish n lgorithm for th totl lit omintion prolm in gnrl grph. NP-Complt rsults for th prolm r lso shown for split grph, horl grph n for iprtit grphs.
86 Girish.V.R. n P.Ush Rfrns [1] J.Mstr, Dpth first srh, Algorithm n Complxity, Smstr 2, 2012. [2] S.Blji, V.Swminthn n K.Knnn, Optimiztion of unwight minimum vrtx ovr, Worl Amy of Sin, Enginring n Thnology, 43 (2010), 716-729. [3] Ltur Nots: Bs on Mrk Alln Wiss, Dt Struturs n Algorithm, Anlysis in Jv (2n ition), Aison-Wsly, 2007. [4] Trs W.Hyns, Stphn T.Htnimi n Ptr J. Sltr, Funmntl of Domintion of Grphs, Mrl Dkkr, In. Nwyork, 1998. [5] G.J.Chng, Algorithmi spts of omintion in grphs, in: D.Z.Du., P.M.Prlos (Es.) Hnook of Comintoril Optimiztion, Kluwr Ami Pu., Boston Vol.3, 1998, pp.339-405. [6] Grr J. Chng, Th wight inpnnt omintion prolm is NP-Complt for horl grphs, Disrt Appli Mthmtis, 143 (2004) 351-352. [7] Anny Lvithin, Introution to th Dsign n Anlysis of Algorithms (2n ition), Prson Eution.In., 2009.