An Optimal Algorithm to Find a Minimum 2-neighbourhood Covering Set on Cactus Graphs

Transcription

1 Annals of Pre Appled Mathematcs Vol 2 No ISSN: X (P) (onlne) Pblshed on 18 December 212 wwwresearchmathscorg Annals of An Optmal Algorthm to Fnd a Mnmm 2-neghborhood overng Set on acts Graphs Kalyan Das 1 Madhmangal Pal 2 2 Department of Appled Mathematcs wth Oceanology ompter Programmng Vdyasagar Unversty Mdnapore Inda e-mal: mmpalv@gmalcom 1 Department of Mathematcs Ramnagar ollege Depal Prba Mednpr Inda Receved 2 Semptember212; accepted 16 December 212 Absttact A cacts graph s a connected graph n whch every bloc s ether an edge or a cycle An optmal algorthm s presented here to fnd mnmm cardnalty 2-neghborhood-coverng set on cacts graphs n O (n) tme where n s the total nmber of vertces of the graph he cacts graph has many applcatons n real lfe problems specally n rado commncaton system Keywords: Desgn of algorthms analyss of algorthms 2-neghborhood-coverng set cacts graph AMS Mathematcs Sbect lassfcatons (21): Introdcton 11 acts graph Let G = ( V E) be a fnte connected ndrected smple graph of n vertces m edges where V s the set of vertces E s the set of edges A vertex v s called a ct-vertex f removal of v all edges ncdent on v dsconnect the graph A non-separable graph s a connected graph whch has no ct-vertex a bloc means a maxmm non-separable sb-graph A bloc s a cyclc bloc or smply a cycle n whch every vertex s of degree two A cacts graph s a connected graph n whch every bloc s ether an edge or a cycle acts graph has many applcatons hese graphs can be sed to model physcal settng where a tree wold be napproprate Examples of sch settng arse n telecommncatons when consderng feeder for rral sbrban lght rban regons [8] n materal hlng networ when atomated gded vehcles are sed [9] Moreover rng bs strctres are often sed n local area networs he 45

2 Kalyan Das Madhmangal Pal combnaton of local area networ forms a cacts graph Desgn a stable algorthms s a very mportant tas Several algorthms have been desgn for dfferent graphs see [16-27] 12 he -neghborhood-coverng set he -neghborhood-coverng ( -N) problem s a varant of the domnaton problem Domnaton s natral model for locaton problems n operatons research networng etc he graph consdered n ths paper are smple e fnte ndrected havng no self-loop or parallel edges In a graph G = ( V E) the length of a path s the nmber of the edges n the path he dstance d ( x y) from the vertex x to the vertex y s the mnmmn length of a path from x to y f there s no path from x to y then d ( x y) s taen as A vertex x -domnates another vertex y f d( x y) A vertex z s -N of an edge ( x y) f d( x z) d( y z) e the vertex z -domnates both x y onversely f d( x z) d( y z) then the edge ( x y) s sad to be -neghborhood covered by the vertex z A set of vertces V s a -N set f every edge n E s -N by some vertces n he -N nmber ρ ( G ) of G s the mnmm cardnalty of all -N sets 13 Revew of prevos wors For = 1 Lehel et al [1] have presented a lnear tme algorthm for comptng ρ (G1) for an nteval graph G hang et al [1] Hwang et al [7] have presented lnear tme algorthms for comptng ρ (G1) for a strongly chordal graph provded that strong elmnaton orderng s nown Hwang et al [7] also proved that -N problem s NP-complete for chordal graphs Mondal et al [12] have presented a lnear tme algorthm for comptng 2-N problem for an nterval graph Also a lnear tme algorthm for trapezod graph has presented by Ghosh et al [4] 14 Or reslt In ths paper we consder a cacts graph G Here we desgn an algorthm whch fnds the 2-neghborhood coverng set of the graph G n O (n) tme he algorthm also taes O (n) space 2 omptaton of blocs ctvertces of G As descrbed n [13] the blocs as well as ct vertces of a graph G can be determned by applyng DFS technqe Usng ths technqe we obtan all blocs ct vertces of the cacts graph G = ( V E) Let the blocs be B 1 B 2 B 3 B N the ct vertces be c 1 c 2 c 3 K cr where N s the total nmber of blocs R s the total nmber of ct vertces 46

3 An Optmal Algorthm to Fnd a Mnmm 2-neghborhood overng Set on acts Graphs wo blocs are sad to be adacent f they have at least one common vertex of the graph G Defne edge blocs as e { B : B = 2} cycle blocs as 47 = = { B : B > 2} where B s the cardnalty of B Let the nmber of cycle blocs be N nmber of edge blocs be N Nmber of vertces of each cycle s denoted by = 12 K N 3 onstrcton of tree blocs the tree Sppose the set S = { e1 e2 e3 K en } A tree bloc s a maxmal sbgraph of G sch that s a tree Let 1 2 K L be the tree blocs of G he tree blocs 's are formed by the members of S e S = 12 K L Now we have n a poston to constrct the tree sng tree blocs 's = 12 K L 's = 12 K N Before constrcton of the tree we defne an ntermedate graph G whose vertces are the blocs of G hs G = ( V E ) where the vertces are blocs of the graph G e V = { 1 2 K L K N } If two blocs are adacent they are connected by an edges hs E {( ) or ) or ( ) : = / ; = 12 K N = = / l; l = 12 K L ( l l are adacent blocs } Now the tree s constrcted from G as follows: We dscard some stable edges from G n sch a way that the resltant graph becomes a tree he procedre for sch redcton s gven below: Let s tae any arbtrary vertex of G contanng at least two ct-vertces of G as root of the tree mar t All the adacent vertces of ths root are taen as chldren of level one mar them If there are edges between the vertces of ths level then dscard these edges Each vertces of level one s consdered one by one to fnd the vertces whch are adacent to them bt nmared hese vertces are taen as chldren of the correspondng vertces of level one pt them at level two hese chldren at level two are mared f there be any edge between them then remove them hs process s contned ntl all the vertces are mared hs the tree = ( V E ) where { K K } V = 1 2 L N E E s obtaned For convenence we refer the vertces of as nodes We note that each node of ths tree s a bloc (cycle bloc or tree bloc) of the graph G = ( V E) he parent of the node n the tree wll be denoted by Parent( ) 4 Eler or Eler tor prodces an array of nodes he tor proceeds wth a vst to the

4 Kalyan Das Madhmangal Pal root there after vsts to the chldren of the root one by one from left to rght retrnng each tme to the root sng tree edges n both drectons Algorthm GEN-OMP-NEX of hen et al [2] mplements ths Eler tor on a tree startng from the root he npt to the algorthm s the tree represented by a `parent of' relaton wth explct orderng of the chldren he otpt of the algorthm s the tor startng from the root of the tree endng also at the root he tor s represented by an array S (1: 2( N + L) 1) that stores nformaton connected to the vsts drng the tor he element S () of the array S s a record consstng of two felds one of whch denoted by S ( ) node s the node vsted drng the th vst whle the other denoted by S ( ) sbscrpt s the nmber of tmes the node S ( ) node s vsted d sng the frst vsts of the tor wo felds of an element of S are wrtten together sng the notaton ( node) 48 sbscrpt Also we consder an array f ( ) f () whch stores the total nmber of occrrence of the bloc = 123 K N = 123 K L n the array S( ) = 123 K 2( N + L) 1 hs f ( ) f () represents the total nmber of vsts of the bloc n the Eler tor e f ( ) f () are the maxmm sbscrpt of n the array S () For each =123N =123L ( ) f() ( ) f () occrs only once n th array S() before ( ) f() ( ) f () all of ( ) 1 ( ) 2 ( ) f()-1 ( ) 1 ( ) 2 ( ) f ()-1 occr respectvely n order of ncreasng sbscrpts of he followng mportant lemma s proved n [11] Lemma 1 If S ( ) sbscrpt = 1 S ( + 1) sbscrpt = / 1 then S ( ) node s a leaf node of the tree 5 Determnaton of 2-N set from cycles paths Lemma 2 For 2-N problem a vertex n a cacts graph can cover at least 4 edges Proof: Let G = ( V E) be a cacts graph V Now degree of may be two or more ase 1: Let the degree of be two hen there exst two vertces v w so that d ( r) = 1 r = v w Now f v w are of degree two then there also exst two vertces x y so that d ( x) d ( y) = 2 hs covers for edges ( v)( w)( w x)( v y) where v w are adacent to x y are adacent to v w respectvely Bt f any one of v w are of degree more than two then cover all edges ncdent on v or w or both as d ( r) = 1 r = v w In ths case covers more than for edges ase 2: Sppose the degree of s more than two In ths case s adacent to more than two vertces so that the neqalty d ( v) 2 s satsfed for more than for vertces e covers more than for edges

5 An Optmal Algorthm to Fnd a Mnmm 2-neghborhood overng Set on acts Graphs hs t s evdent from the above cases that covers at least for edges Hence the proof Lemma 3 A cycle of 4 m vertces contrbte at least m vertces n the 2-N set X Proof: he degree of all vertces of a cycle are two Let v w x y be the fve consectve vertces of a cycle Hence from Lemma 2 t s evdent that a vertex can cover for edges ( v)( w)( w x)( v y) where v w are adacent to x y are adacent to w v respectvely as d ( r) 2 r = v w x y Now a cycle of 4 m vertces contan 4 m edges hs to cover 4 m edges at least m vertces are needed Lemma 4 A cycle contanng 4 m m m + 3 vertces contrbte at least m + 1 vertces n the 2-N set X Lemma 5 A path contanng 4 m edges e contanng 4 m + 1 vertces contrbte at least m vertces n the 2-N set X Proof: In a path every vertex except the end vertces are of degree two So f s a vertex on the path whch s not an end vertex or ts adacent then t mst cover for edges becase end vertces of a path cover at most two edges where as the adacent vertex of an end vertex can cover at most three edges one from the sde where end vertex les two edges from the other sde Hence to cover all the edges on ths path at least m vertces are needed Lemma 6 A path contanng 4 m + 14m + 2 or 4 m + 3 edges contrbte at least m +1 vertces n 2-N set X Proof: As a path contanng 4 m edges contrbte at least m vertces n X then for the rest 1 or 2 or 3 edges one vertex s reqred to cover them hs m + 1 vertces are necessary to cover all the edges of these paths Lemma 7 Between two vertces X also (or path) there exst at most 4 r + 4 edges e 4 r + 3 vertces r beng the nmber of vertces between nclded n both X the cycle (respectvely path) Proof: Sppose (path) Let there exsts r vertces n X between edges of X cover are any two vertces of X belong to the same cycle covers two edges between the edges of Now covers two Also r vertces 4 r edges Hence there are 4 r + 4 edges between 4 r + 3 vertces hs f r = then there exsts 4 edges between e e 3 49

6 Kalyan Das Madhmangal Pal vertces f r = 1 then there exsts 8 edges between f r = 2 then there exsts 12 edges between on Hence the proof 6 Determnaton of 2-N set from the tree blocs For a tree bloc consder the vertex common to 5 e 7 vertces e 11 vertces so the adacent cycle bloc of as the root say Here the adacent cycle bloc of s the node whch s consder after n the Eler seqence hen the adacent vertces of the root are placed at level 1 the adacent vertces of the vertces of level 1 are placed at level 2 so on hs the heght of the tree h s defned as h = max { d ( v ) beng the root v } he vertces for whch maxmm level s obtaned one of that s denoted by h the path between h on whch h occrs s treated as the man path of the tree he vertces on the man path are denoted as K where sbscrpts denote the level of vertces at 1 2 h For every vertex = 12 K h there exsts one or more sbtrees rooted hese are denoted by B ( ) B ( ) B ( ) K B ( ) M beng the total M nmber of sbtrees rooted at learly the heght of B ) s less than or eqal to h for all ( = 12 K M Some of the sbtrees are paths rooted at hs for each sch path B ( ) h for some as the maxmm nmber of edges vertces excldng on that path s Lemma 8 he vertex h 2 h s the frst member of X n d( h 2 v) 2 for all v B ( h ) = 12 Proof: Snce the heght of the tree s h there s no sbtree (path) rooted at h If there s a sbtree (path) rooted at h 1 heght of that sbtree s one Smlarly h 2 may has sbtrees of heght 2 Otherwse they all exceed the heght of the tree Also d( h 2 h) = 2 Hence for all v B ( h ) = 12 d ( v ) 2 Lemma 9 If the path rooted at the vertex h = 12 K h 2 wth B ( ) = 4m + 2 for at least one then s a member of X Proof: From Lemma 2 we have seen that one vertex cover at least for edges For any path rooted at f we start from ts leaf m vertces are nclded n X to cover 4 m edges For the rest two edges ether adacent to on that path or s the

7 An Optmal Algorthm to Fnd a Mnmm 2-neghborhood overng Set on acts Graphs vertex to cover them Bt s the most stable vertex to cover them becase t also covers more edges on the man path as well as on the other sbtrees rooted at Hence the proof Lemma 1 he path rooted at for some ad( ) B ( ) s a member of X = 12 K h 2 wth B ( ) = 4m + 3 Proof: A path wth B ( ) = 4m + 3 contans 4 m + 3 vertces 4 m + 3 edges Here also m vertces covered 4 m edges f we start from the leaf of the path hree edges are left ncovered below hs f the vertex ad( ) B ( ) s selected then t covers those edges Also ad ) can cover all the edges ncdent on Lemma 11 he path rooted at for some B ) ( ( = 12 K h 2 wth B ( ) = 4m + 1 or ad ) s a member of X where ad ) s not a member of ( Lemma 12 he path rooted at wth B ( ) = 4 m ( or 4 m + 1 or 4 m + 2 for some contrbte m vertces wth B ( ) = 4m + 3 contrbte m + 1 vertces Proof: he path contanng 4m 4 m m + 2 vertces contan 4 m 4 m m + 2 edges By Lemma 5 m vertces on the path are selected to cover 4 m edges Also by Lemma 9 Lemma 1 m vertces on the path are selected or ad( ) B ( ) are selected to cover the 4 m m + 2 edges Hence B ) wth 4 m 4m + 14m + 2 vertces contrbte m vertces n X ( From Lemma 1 for B ( ) = 4m + 3 m vertces are selected for 4 m edges for the other three edges ad( ) B ( ) s selected Hence B ( ) contrbte m + 1 vertces n X Procedre to determne the 2-N set from the tree Usng the above lemmas the procedre for selectng coverng vertces from the tree s descrbed below Step-1: Start from Step-2: Go to the vertex each h he frst member of X s h h 2 = 34 K h one by one wo cases arse here for 51

8 ase 1: here exst no sbtrees rooted at Kalyan Das Madhmangal Pal In ths case we proceed to the vertex h 1 h apply ase 1 ase 2 of Step 2 ase 2: here exst some sbtrees rooted at h Here also two cases arse Sbcase 21: If some sbtrees are paths then consder the leaf vertex of the path at the frst poston consder the followng statons () If the paths are of length 4 m + 2 then select h the vertces at (4 1) th poston = 12 K m are selected n X () If the paths are of length 4 m + 3 then select the vertces at (4 1) th poston = 12 K m + 1 n X () If the paths are of length 4 m then select the vertes at (4 1) th poston = 123 K m n X (v) If the paths are of length 4 m + 1 then select the vertces at (4 1) th poston = 12 K m n X vertex In ths case one edge ncdent on h 52 h s left ncovered It s covered by the or ad ) whch s ether belongs to the man path or another ( h sbtrees (paths) rooted at the vertex e they are stated at level h + 1 or h 1 Now f the vertex ad ) of level h + 1 or h ( h s not already selected for X then the vertex h 1 mst be a member of X Sbcase 22: If the sbtrees are another trees then fnd ther heght followng the same procedre as descrbed n Step 1 Step 2 fnd the coverng vertces from those sbtrees hs applyng the Step 1 Step 2 repeatedly for the vertces one by one on the man path from level h to the level the 2-N set of the tree wll be obtaned At the tme of consderaton of h f some edges ncdent on the vertex e the root of the tree or tree bloc s ncovered then select or ad ) from the tree or from cycle contanng the vertex If ad ( ) from the tree or tree bloc s not selected then t s selected from the cycle when we consder the cycle 7 Determnaton of 2-N set from the nodes of the tree Here we consder the nodes of the tree one by one from the seqence obtaned from Eler tor he nodes or for whch f ( ) or f () s 1 are leaf nodes Otherwse the nodes are nteror nodes 71 Fndng coverng vertces from a leaf node of If the leaf node s a cycle we apply the followng procedre Sppose v s the (

9 An Optmal Algorthm to Fnd a Mnmm 2-neghborhood overng Set on acts Graphs ctvertex of the leaf node cycle Now (1) For the leaf node consder the ctvertex v n the frst poston of the havng 4 m vertces mar the ctvertex v frst thereafter mar the vertces at (4 + 1) th poston = 12 K m (2) For the leaf node havng 4 m + 1 vertces one of the ncdent edges on the ctvertex v s left ncovered mared the vertces at (4 1) th poston = 12 K m (consderng the ctvertex v n the frst poston) (3) For the leaf node havng 4 m + 2 vertces both the edges ncdent on the ctvertex v are left ncovered mared the vertces at 4 th poston = 12 K m (4) For the leaf node havng 4 m + 3 vertces mar the ctvertex v frst then mar vertces at (4 + 1) th poston = 12 K m If the leaf node be a tree bloc then fnd the coverng vertces from for X by applyng the procedre descrbed n Secton 6 After selectng the vertces for the coverng set from leaf nodes mar all edges whch are covered by those vertces 72 Fndng coverng vertces from an nteror node of After marng the covered edges from leaves or chldren nodes respectve Parent( ) or Parent( ) whch s another cycle 53 or tree ( or ) the have the followng statons (1) None of the edges of the Parent ) or Parent ) s covered Also there may some ncovered edges ncdent on the some ctvertces wth ts chldren nodes (2) One or more edges of the Parent ) or Parent ) are covered from ts chldren node Here also may arse some ncovered edges ncdent on the ctvertces wth ts chldren nodes ase 1: Here () f the node s a tree bloc ( ) then the ncovered edges of ts chldren nodes ncrease ts heght as well as length of some sbtrees of that tree bloc herefore applyng the method descrbed n Secton 6 we fnd the coverng vertces from that mproved tree bloc () If the node be a cycle bloc ctvertces of of the chldren nodes of ( ( frst we have to mar ether the s or any one of the adacent vertces of the sad ctvertces wth ( ( It depends on the nmber of vertces les between these ctvertces Sppose are the ctvertces whch have branch of length one l be the path between + 1 l be the nmber of vertces n l Now we select

10 Kalyan Das Madhmangal Pal or ad ) by sng the followng procedre ( From Lemma 7 we see that between two members of X of a cycle or a path there are 4 r + 3 vertces r = 12 K Usng ths lemma the followng cases may arse Sbcase 1: Let there be 4 m + 1 vertces between + 1 In ths case select the vertces ad( ad( + 1 or ad ( 1 ad ( For the frst par there are 4m 1 vertces between ad ) ad ) for ( ( + 1 ( + 1 the second par there are 4 m + 3 vertces between ad ( ) ad ) whch satsfy Lemma 7 Sbcase 2: Let there be 4 m + 2 vertces between + 1 In ths case select the vertces ad( + 1 or ad ( For the frst for the second par there are 4 m + 3 vertces between ad ( + 1) between ad ( ) + 1 whch satsfy Lemma 7 Sbcase 3: Let there be 4 m + 3 vertces between + 1 In ths case select the vertces ad( ad ( or ad ( 1 ad( + 1 or + 1 here are 4 m + 3 vertces between the vertces of each par whch satsfy Lemma 7 Sbcase 4: Let there be 4 m vertces between + 1 In ths case select the vertces ad( + 1 or ad( + 1 here are 4m 1 vertces between the vertces of each par whch satsfy Lemma 7 Lemma 13 For l = 4m or 4 m + 2 m = 12 K () + 1 s the frst member of X from f l 1 s of length 4 m + 1 () s the frst member of X from f l + 1 s of length 4 m + 1 () or + 1 s the frst member of X from f both l + 1 l 1 are of length 4 m + 1 Proof: (a) For l = 4m select ether the vertces ad( + 1 or ad( + 1 (b) For l = 4m + 2 select ether the vertces ad ( or ad ( (c) For l 1 = 4m + 1 select ether the vertces ad ( 1 ad ( 1) l 1 or ad( ad ( 1) l 2 (d) For l + 1 = 4m + 1 select ether the vertces ad ( ad ( or ad( + 1 ad (

11 An Optmal Algorthm to Fnd a Mnmm 2-neghborhood overng Set on acts Graphs ase 1: hs from (a) (c) t s evdent that f l s of length 4 m l 1 s of length 4 m + 1 then select + 1 ad( ad( 1 for X (1) If l s of length 4 m + 2 l + 1 s of length 4 m + 1 (b) (d) gve the coverng vertces as + 1 ad( 1) l 1 ad( ) l 1 n X (2) hs (1) (2) shows that + 1 mst be member of X So t s the frst member of X from that cycle ase 2: Now f l s of length 4 m l + 1 s of length 4 m + 1 then (a) (d) gve ad( + 1 ad ( (3) f l s of length 4 m + 2 l + 1 s of length 4 m + 1 (b) (d) gve ad( ad ( (4) hs (3) (4) shows that mst be a member of X So t s the frst member of X from the cycle ase 3: If both l + 1 l 1 are of length 4 m + 1 l s of length 4 m or 4 m + 2 select vertces from any one of (1) or (2) or (3) or (4) whch shows that any one of + 1 s the frst member of X from Lemma 14 For l l + 1 ( l 1) are ether both of length 4 m or 4 m + 2 or one s 4 m other s 4 m ( ) s the frst member of X from that cycle Proof: (a) If l s of length 4 m select ether the vertces ad( + 1 or +1 ad( (b) For l + 1 s of length 4 m select ether + 1 ad ( or + 2 ad ( (c) If l s of length 4 m + 2 select ether the vertces ad ( or + 1 ad ( 1 (d) For l + 1 s of length 4 m + 2 select ether the vertces + 1 ad ( or + 2 ad( + 1 Now f l l + 1 both are of length 4 m or 4 m + 2 then from (a) (b) (c) (d) the vertex + 1 s the common member to be selected Hence + 1 mst be a member of X So t s the frst member of X from If l l + 1 one of whch s 4 m other s the length 4 m + 2 then from (a) (c) (b) (d) + 1 s the common member to be selected Hence + 1 mst be a member of X So t s the frst member of X from 55

12 Kalyan Das Madhmangal Pal For l 1 nstead of l + 1 we can prove smlarly that s the frst member of X Hence the proof Lemma 15 For all l = 12 K r 1 of length 4 m or 4 m + 2 any one of = 12 K r s the frst member of X from the cycle Proof: (a) If l s of length 4 m select ether 56 ad ( + 1 or + 1 ad( (b) If l s of length 4 m + 2 select ether ad ( or + 1 ad ) l ( 1 From (a) (b) t s evdent that for each l = 123 K r 1 of length 4 m or 4 m + 2 or + 1 s a common member of X hs for all l of length 4 m or 4 m + 2 any one of = 12 K r s the frst member of X from the cycle Lemma 16 For all l = 12 K r 1 of length 4 m + 1 or 4 m + 3 any one of ad ( ) = 12 K r s the frst member of X from the cycle Proof: (a) If l s of length 4 m + 1 select ether the vertces ad( ad( + 1 or ad ( 1 ad ( (b) If l s of length 4 m + 3 select ether the vertces ad( ad ( or ad ( 1 ad( + 1 or + 1 From (a) (b) t s evdent that for each l = 123 K r 1 of length 4 m +1 or 4 m + 3 ad ) s a common member for both the cases So t s taen ( as the frst member of X hs for all l of length 4 m + 1 or 4 m + 3 any one of ad ( ) = 12 K r s the frst member of X from the cycle ase 2: In ths case also f the node be a tree bloc then heght of the tree or length of some paths be decreased Also f there be some ncovered edges ncdent on some vertex of the tree then the length of some path or heght of the tree be ncreased If the node be a cycle there occr two or more than two trees Hence the steps to fnd the coverng vertces from the nteror node are: Step 1: For the cycle arses n ase 1 Select the frst member of X from mar the edges covered by ths vertex hen the nmared edges of the cycle ths form two trees rooted at the ctvertex of Parent( ) For every tree we apply the procedre as descrbed n Secton 6 to select the coverng vertces from that tree Step 2: For the cycle arses n the ase 2 Here the nmared vertces occrs as two or more than two trees For every tree we apply the procedre as descrbed n Secton 6 to select the coverng vertces

13 An Optmal Algorthm to Fnd a Mnmm 2-neghborhood overng Set on acts Graphs Step 3: For the tree bloc arses n ase 1 or ase 2 we apply the procedre as descrbed n Secton 6 to select the coverng vertces from that tree bloc 8 Algorthm ts complexty In ths secton we present an algorthm 2NOV to compte the 2-neghborhood coverng set on cacts graphs he tme space complextes are also compted here he proof of correctness of the algorthm s also presented n ths secton Algorthm 2NOV Inpt: he cacts graph G Otpt: he 2-neghborhood covered set X Step 1: ompte the blocs ctvertces of G as descrbed n Secton 3 //Let S be the set of edge blocs form the tree blocs = 12 K L Also denote the cycle blocs as Step 2: onstrct a tree descrbed n Secton 4 Step 3: Apply Eler tor on 57 = 12 K N // whose nodes are the tree blocs cycle blocs as store the otpt n the array = 12 K = 12 K L N + L S (1: 2( N + L) 1) N + L s the total nmber of nodes of Step 4: ompte f ( ) f () N whch stores total nmber of occrrences of the node n the array S Step 5: Note the order n whch ( ) f ( ) ( ) f ( ) = 12 K N = 12 K L occrs n the array S Step 6: For each node of the resltng seqence f () f ( ) = 1 or f () = 1 then fnd the vertces of G sng rle descrbed n Secton 71 pt them n the set X () f ( ) 1 or f ( ) 1 then fnd the vertces of G sng the rle descrbed n Secton 72 pt them n the set X end 2NOV Lemma 17 he set X obtaned from the algorthm 2NOV s a 2-neghborhood coverng set Proof: Here the problem s to fnd 2-N set he set X s constrcted n sch a way that for every vertex X we fnd v S so that d ( v) 2 Now t s seen that U S X = V herefore all the edges connected wth the vertces of X vertces of S are covered by the vertces of X e E s covered by the vertces of the set X hs X s the 2-N coverng set of the graph G Lemma 18 he set X obtaned from 2NOV s mnmm among all the 2-N coverng set of the cacts graph G Proof: From the Lemmas t s evdent that the selecton of coverng vertces from cycles paths tree blocs s made n sch a way that these

14 Kalyan Das Madhmangal Pal contrbte least nmber of vertces n the coverng set Also drng consderaton of the nodes of the tree we mnmze the nmber of coverng vertces for the cases where leaf nodes contan 4 m m + 2 vertces Sometmes these contrbte m vertces n X nstead of m + 1 vertces as n Lemma 4 Smlarly for the tree bloc the paths contanng 4 m m + 2 edges also contrbte m vertces nstead of m + 1 vertces as n Lemma 12 Hence the lemmas procedre are so desgned that they fnd mnmm nmber of vertces to 2-N set X hs the set X s the mnmm cardnalty 2-N set for the cacts graph G heorem 1 he mnmm 2-neghborhood coverng set X obtaned from the algorthm 2NOV can be compted n O (n) tme Proof: he blocs ctvertces of any graph can be compted n O(m+n) tme [13] For the cacts graph m=o(n) hence step 1 of Algorthm 2NOV taes O(n) tme Also formaton of tree blocs sng the edge blocs of G taes O(n) tme Hence step 2 can be compted n O(n) tme In step 3 the constrcton of the tree sng tree blocs cycles fndng f() f () for each node fndng seqence of nodes sng Eler or tae O(n) tme Hence steps tae O(n) tme Step 7 can be performed by comparng f() f () wth 1 for = 12 N = 1 2 L So ths step taes only O(n) tme Hence the algorthm 2NOV compted the 2-N set n O(n) tme REFERENES 1 hang G J Farber M za Z Algorthmc aspects of neghborhood nmbers SIAM J Dscrete Math 6 (1993) hen Y Das S K Al S G A nfed approach to parallel depth-frst traversals of general trees Informaton Processng Letters 41 (1991) Golmbc M Algorthmc Graph heory Prefect Graphs Academc Press New Yor Ghosh P K Pal M An Optmal algorthm to solve 2-neghborhood coverng problem on trapezod graph Advanced Modelng Optmzaton 9 (1) (27) Garey M R Johnson D S ompter Intractablty: A Gde to the theory of NP-ompleteness Freeman San Francsco A Gamst A Some lower bonds for a class of freqency assgnment problems IEEE ransactons of Vehclar echnology 35(1) (1986) Hwang S F hang G J -neghborhood- coverng ndependence problems for chordal graphs SIAM J Dscrete Math 11 (4) (1998) Koontz W L G Economc evalaton of loop feeder relef alternatves Bell System echncal J 59 (198) Karv O Ham S L An algorthmc approach to networ locaton problems Part 1: he p-center SIAM J Appl Math 37 (1979) Lehel J za Z Neghborhood perfect graphs Dscrete Math 61 (1986)

15 An Optmal Algorthm to Fnd a Mnmm 2-neghborhood overng Set on acts Graphs 11 Pal M Bhattacharee G P An optmal parallel algorthm for all-pars shortest paths on nweghted nterval graphs Nordc Jornal of omptng 4 (1997) Mondal S Pal M Pal K An optmal algorthm to solve 2-neghborhood coverng problem on nterval graphs Intern J ompter Math 79 (2) (22) Rengold E M Nvergent J Deo N ombnatoral Algorthms : heory Practce (Prentce Hall Inc Englewood hffs New Jersy 1977) 14 Sarrafgadeh M Lee D A new approach to topologcal va mnmzaton IEEE rans ompter Aded Desgn 8 (1989) Schaerf A A srvey of atomated tmetablng Artfcal Intellgence Revew 13 (1999) Pal M Bhattacharee GP Seqental parallel algorthms for comptng the center the dameter of an nterval graph Intern J ompter Mathematcs 59(1+2) (1995) Pal M Bhattacharee GP Parallel algorthms for determnng edge-pacng effcent edge domnaton sets n an nterval graph Parallel Algorthms Applcatons 7 (1995) Pal M Bhattacharee GP A seqental algorthm for fndng a maxmm weght -ndependent set on nterval graphs Intern J ompter Mathematcs 6 (1996) Pal M Bhattacharee GP A data strctre on nterval graphs ts applcatons Jornal of rcts System ompters 7(3) (1997) Saha A Pal M An algorthm to fnd a mnmm feedbac vertex set of an nterval graph Advanced Modelng Optmzaton 7(1) (25) Pal M Effcent algorthms to compte all artclaton ponts of a permtaton graph he Korean J ompt Appl Math 5(1) (1998) Bera D Pal M Pal K An optmal parallel algorthm for comptng ct vertces blocs on permtaton graphs Intern J ompter Mathematcs 72(4) (1999) Hota M Pal M Pal K An effcent algorthm for fndng a maxmm weght -ndependent set on trapezod graphs omptatonal Optmzaton Applcatons 18 (21) Bera D Pal M Pal K An effcent algorthm for generate all maxmal clqes on trapezod graphs Intern J ompter Mathematcs 79 (1) (22) Bera D Pal M Pal K An optmal PRAM algorthm for a spannng tree on trapezod graphs J Appled Mathematcs omptng 12(1-2) (23) Barman S Mondal S Pal M An effcent algorthm to fnd next-to-shortest path on trapezodal graph Advances n Appled Mathematcal Analyss 2(2) (27) Das K Pal M An optmal algorthm to fnd maxmm mnmm heght spannng trees on cacts graphs Advanced Modelng Optmzaton 1 (1) (28)