Vishnu V. Narayan. January

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1 A approimation algorithm for 2-rt-connctd spanning subgraphs on graphs with minimum dgr at last arxi: [cs.ds] 17 Jan 2017 Vishnu V. Naraan Januar W obtain a polnomial-tim 17 -approimation algorithm for th minimum-cost 2-rtconnctd spanning subgraph problm, rstrictd to graphs of minimum dgr at last. 12 Our algorithm uss th framwork of ar-dcompositions for approimating connctiit problms, which was priousl usd in algorithms for finding th smallst 2-dg-connctd spanning subgraph b Chrian, Sbő and Sigti (SIAM J.Discrt Math. 2001) who ga a 17-approimation algorithm for this problm, and b Sbő and Vgn (Combinatorica 2014), 12 who improd th approimation ratio to 4. Introduction A graph is 2-rt-connctd if th dltion of an rt, along with its incidnt dgs, dos not disconnct th rmaindr of th graph. Th problm of finding a smallst 2-rtconnctd spanning subgraph of a gin graph is NP-hard. This can b sn ia th following rduction from th Hamiltonian ccl problm: A graph G has a Hamiltonian ccl if and onl if th numbr of dgs in th smallst 2-rt-connctd spanning subgraph is qual to th numbr of rtics in th graph. Khullr and Vishkin ga a 5 -approimation algorithm for 2-rt-connctiit in [1]. This was improd b Garg, Singla and Vmpala, who obtaind an approimation ratio of in [2]. 2 Bttr approimation ratios ha bn claimd in th past, but to th bst of our knowldg, no complt proof had bn publishd for ths. Rcntl, Hgr and Vgn ga a 10-7 approimation algorithm for this problm. Our rsarch was carrid out indpndntl in th sam priod. Th rsarch was compltd on Jul 1, 2016; this draft was dlad du to othr commitmnts. 1

2 W prsnt a 17 -approimation algorithm for th 2-rt-connctiit problm rstrictd to 12 graphs with minimum dgr at last. Appndi A contains a proof that this rstrictd rsion of th problm is also NP-hard. Our algorithm uss th framwork of ar-dcompositions for approimating connctiit problms, which was priousl usd in algorithms for finding th smallst 2-dg-connctd spanning subgraphs b Chrian, Sbő and Sigti in [] who ga a 17 -approimation algorithm for this problm, and b Sbő and Vgn in [4], who 12 improd th approimation ratio to 4. Prliminaris Lt G = (V, E) b an undirctd graph. An ar of G is a path P of lngth at last 1, such that th ndpoints of P ma coincid, but r othr pair of rtics of P ar distinct. An ar P is opn if its ndpoints ar distinct and closd othrwis. P is triial if it has a singl dg, short if it has 2 or dgs, and long othrwis. P is n if it has an n numbr of dgs, and odd othrwis. Th rtics of P that ar not ndpoints of P ar calld intrnal rtics of P, thir st is dnotd b in(p ). An ar-dcomposition of G is a squnc P 0,P 1,...,P k, whr P 0 is a rt and P 1,...,P k ar ars such that P i shars actl its two ndpoints with th rtics of P 0... P i 1. W dnot b φ(g) th minimum numbr of n ars of an ar-dcomposition D, or all th ar-dcompositions D of G. An ar-dcomposition is nmin if th numbr of n ars is qual to φ(g). For an ar P, lt φ(p ) = 1 if P is n and φ(p ) = 0 othrwis. A nontriial ar P is a pndant ar if no othr nontriial ar has an ndpoint in in(p ), othrwis it is non-pndant. W rfr th radr to [4] for dfinitions and a dtaild discussion of nic ar-dcompositions, ardrums, armuffs and maimum armuffs. W also us th lowr bounds LP (G) and L µ (G, M) and thir rlatd thorms, as dfind in Sction 4 of this papr. W dnot b OP T 2V C (G) th cost of th minimum-cost 2-rt-connctd spanning subgraph of G. At tims, w abus th notation for triial ars, and writ u for th ar corrsponding to th path containing th rtics u, and th dg u. Algorithm Our algorithm consists of a fw stps, summarid as follows. 1. Construct an opn nmin ar-dcomposition D of G. 2. Modif D to gt an opn nmin ar-dcomposition with th proprt that all of its short ars ar pndant ars. 2

3 . Modif D to gt an opn nmin ar-dcomposition that is nic. 4. Dlt all dgs in triial ars. Th rsulting graph is a 2-rt-connctd spanning subgraph of G with at most OP T 2V C(G) dgs. Our analsis is dtaild in Thorms 2, and 5 blow. Th following Lmma (Lmma 1) allows us to rplac a gin ar-dcomposition on a graph with anothr ar-dcomposition on th sam graph. It is usd in th proof of Thorm 2. Th Lmma is simpl to pro, and its proof is lft to th radr. Th radr ma find Figur 1 usful for undrstanding th statmnt of th Lmma. Lmma 1. Lt D b an ar-dcomposition of a graph G. Suppos P and Q ar nontriial ars of D such that Q is th first nontriial ar of D with an ndpoint in an intrnal rt of P. Furthr, suppos that onl on of th ndpoints of Q is an intrnal rt of P, and that this ndpoint is adjacnt, b an dg of P, to an ndpoint of P. Lt and b th nd rtics of P and w and b th nd rtics of Q, such that w is th intrnal rt of P adjacnt to. Lt P b th ar with ndpoints and and consisting of all dgs of P and Q cpt w. Lt D b th ar-dcomposition constructd from D b dlting th ars P and Q, adding th ar P in th position of Q, and adding th triial ar w at th nd of th ar-dcomposition. Thn D is a alid ar-dcomposition of G. w Q w P P (a) (b) Figur 1 Thorm 2. Er 2-rt-connctd graph G with minimum dgr at last has an opn ar-dcomposition with φ(g) n ars in which all short ars ar pndant. Such an ar-dcomposition can b computd in polnomial tim. Proof. Using Proposition.2 of Chrian, Sbő and Sigti [], construct an opn ardcomposition D = (P 0, P 1,..., P k ) of G with φ(g) n ars. Suppos th closd ar P 1 is short (that is, P 1 is a -ar). Sinc r rt of G has dgr at last, G has at last 4 rtics, hnc D has at last on opn ar. Suppos u and ar th nd rtics of P 2, thn thr is a u, -path of lngth 2 in P 1. Lt P b th union of th u, -path in P 1 and th u, -path in P 2. Dlt th ars P 1 and P 2 from D, and add th ar P in th position of P 1 in D, and th triial ar u at th nd of D. Now th closd ar in

4 D is a long ar, and D is still nmin. St k := k 1 and rlabl th nw ars of D such that D = (P 0, P 1,..., P k ). P 2 u u P 1 (a) P 0 P 1 (b) P 0 Figur 2: Th ccl-ar P 1 is a short ar W procd to mak all othr short ars pndant, starting with 2-ars. As long as D has a non-pndant 2-ar, w rpat th following procdur. Choos th first non-pndant 2-ar P in D. Sinc P is non-pndant, thr ists a nontriial ar in D with on nd incidnt on th intrnal rt of P. Lt Q b th first such ar in D. Q P u P u (a) (b) Figur : P is a 2-ar Lt u and b th nd rtics of P and and b th nd rtics of Q such that u. Dlt ars P and Q from D, and construct th ar P with nds at u and, containing intrnall th intrnal rtics of both P and Q, as shown b th thick lin in Figur b. Add th ar P to D in th position of Q. Add th triial ar at th nd of D. B Lmma 1, D is still a alid ar-dcomposition of G. Sinc Q was a nontriial ar, P has lngth at last, hnc this procdur rducs th numbr of 2-ars in D b on. Furthr, P is an opn ar, thus D is still an opn ar-dcomposition. If Q was an n ar, thn this procdur rducd th numbr of n ars b 2, contradicting our assumption that D is nmin. Hnc Q was an odd ar, and th numbr of n ars rmains unchangd in D. Aftr rpating th abo procdur for all non-pndant 2-ars, all 2-ars in D ar pndant. Nt, w mak all -ars pndant. As long as D has a non-pndant -ar, w rpat th following procdur. Prior to ach itration, w rlabl th ars in D such that th i th ar is lablld P i. Lt P b th first non-pndant -ar in D. Lt and b th ndpoints of P, and lt and b th intrnal rtics of P adjacnt to and rspctil (as shown in Figur 4). Cas 1. Thr ists a nontriial ar Q with ndpoints and. 4

5 P Figur 4: P is a -ar Lt P b th ar with ndpoints and consisting of all of th dgs of Q and th dgs and (as shown b th thick dashd lin in Figur 5b). Dlt th ars P and Q from D, and add th ar P to D in th position of P, and th triial ar at th nd of D. Th rsulting ar-dcomposition D is alid for G. Sinc Q is nontriial, P has lngth at last 4 and is a long ar. Furthr, D is still an opn ar-dcomposition, and sinc th lngth of P has th sam parit as th lngth of Q, D is still nmin. Q P (a) P (b) Figur 5: Thr ists a nontriial ar Q with ndpoints and Cas 2. Thr ist ars Q 1 and Q 2 such that Q 1 has ndpoints and, Q 2 has ndpoints and, and at last on ar in {Q 1, Q 2 } is nontriial. Lt P b th ar with ndpoints and consisting of all th dgs of Q 1 and Q 2 and th dg (as shown b th thick dashd lin in Figur 6b). Dlt th ars P, Q 1 and Q 2 from D, and add th ar P to D in th position of P, and th triial ars a and b at th nd of D. Th rsulting opn ar-dcomposition D is alid for G. If both Q 1 and Q 2 ar n ars, thn this procdur rducs th numbr of n ars in D b 2, contradicting our assumption that D was nmin. If ithr ro or actl on of ths ars is n, thn D rmains nmin. Cas. Othrwis, lt Q b th first nontriial ar with an ndpoint at an intrnal rt of P (sa ). Lt w b th othr ndpoint of Q (as shown in Figur 7). Cas a. w. Lt P b th ar with ndpoints and w, consisting of th dgs of P and Q cpt. Dlt th ars P and Q from D, add th ar P to D in th position of Q, and th triial ar at th nd of D. P is both opn and long, and th rsulting ar-dcomposition D is alid for G b Lmma 1. Sinc th lngth of P has th sam parit as th lngth of Q, th numbr of n ars rmains th sam. Cas b. w =, and is th ndpoint of a triial ar u such that u X. W rfr th radr to Figur 8a for this cas. 5

6 Q 1 Q 2 P (a) P (b) Figur 6: Thr ist ars Q 1 from to and Q 2 from to, not both triial Q w P Figur 7 Lt R b th ar containing u intrnall. If R is a short ar, thn it is pndant (sinc P is th first non-pndant short ar). W ha th following cass: (i) R is a 2-ar. Choos an ndpoint a of R that dos not coincid with, and lt P b th ar au u. Dlt P and R from D and add th 4-ar P in th position of P, and th nw triial ars at th nd of D. (ii) R is a -ar. Choos th ndpoint a of R which is not adjacnt to u in R. Lt R b th ar of lngth 2 in R with ndpoints a and u. If a dos not coincid with, lt P b th ar R u Q, which has th sam parit as Q. Dlt R, P and Q from D and add P in th position of P in D and th triial ars at th nd of D. If a coincids with, lt P b th ar R u of lngth 5. Dlt R and P from D and add P in th position of P in D and th triial ars at th nd of D. In both of th abo cass, w do not crat tra n ars, and th rsulting ar-dcomposition is opn, nmin, and alid for G. If R is a long ar, lt P b th ar Q u. Dlt P and Q from D and add P in th position of P in D, and th triial ars at th nd of D. This ar has th sam parit as th ar Q, so th rsulting ar-dcomposition is opn, nmin, and alid for G. Cas c. Othrwis, sinc th graph has minimum dgr at last, is adjacnt to a rt u / X {}. Obsr that this is th onl rmaining cas. Lt R b th ar containing th dg u, and lt t b th othr ndpoint of R. In particular, if u is a triial ar, thn t is th rt u. W rfr th radr to 6

7 Q P R u Q R P (b) t L 2 L 1 (a) Figur 8 Figur 8b, which will b usful throughout th following analsis. Th following sub-procdur constructs thr sts of ars (F old, F1 nw and F0 nw ), which ar latr usd to modif th ar-dcomposition in ordr to add th intrnal rtics of P to a nw long ar. Th procdur adds som of th isting ars of D to th st F old, and constructs sts of nw ars F1 nw and F0 nw. Whn suitabl sts ar found, th ars in F old ar dltd from D and rplacd with th ars in F0 nw, along with a suitabl constructd long ar. Rpat th following sub-procdur until t is in X {, }. Initiali F old, F1 nw and F0 nw with th mpt st. Lt S b th ar that intrnall contains t, with ndpoints c and d. Add S to F old. Partition th dgs of S into th ars S1 nw (with ndpoints c and t) and S0 nw (with ndpoints t and d). If S is an n ar, ithr S1 nw and S0 nw ar both odd or th ar both n. If S is an odd ar, suppos without loss of gnralit that S1 nw is n and S0 nw is odd. Add S1 nw to F1 nw and S0 nw to F0 nw. St t = c. Th following obsrations will b usful in our analsis. If S0 nw is n, thn S is n (thus rplacing S with S0 nw in an ardcomposition will not, b itslf, incras th numbr of n ars in that ar-dcomposition). If S1 nw is odd, thn S0 nw is odd and S is n. Whn this sub-procdur trminats, w ha th following cass: (i) t / {,, }. Lt P b th ar R L 1... L k, whr F1 nw = {L 1,..., L k }. Dlt P and all ars in F old from D, and for ach ar in F old, rplac it with th corrsponding ar in F0 nw at th sam position in D (this dos not add an tra n ars, but might crat nw non-pndant short ars; obsr that ths ars occur latr in th ar-dcomposition than th nwl cratd long ar in this itration). In th position of P, add th ar P. Add th triial ar at th nd of D. Th rsulting ar dcomposition is alid bcaus th sub-procdur is trminatd whn a rt in X is ncountrd, 7

8 thus r ar in F old appard aftr P in D, hnc r ar in F0 nw appars aftr P. If F1 nw contains onl n ars, thn P has th sam parit as R and w do not introduc an tra n ars. If not, thn F1 nw contains at last on odd ar, in which cas th corrsponding ar in F old is n, and th corrsponding ar in F0 nw is odd. Sinc w ha alrad rducd th numbr of n ars b at last on, P is n and w do not introduc tra n ars. (ii) t =. Discard th prious sts F old, F1 nw and F0 nw. Choos u to b th nighbour of on th last ar that was lablld S. Lt R b this ar and lt t b its othr ndpoint. Sinc r choic of R that w mak in this mannr appars strictl arlir in th ar dcomposition than all th prious choics, th sub-procdur can onl b rpatd O(n) tims bfor w no longr ha this cas. (iii) t =. Lt F1 nw = {L 1,..., L k }, and lt P b th ar R L 1... L k. Dlt P and all ars in F old from D, and for ach ar in F old, rplac it with th corrsponding ar in F0 nw at th sam position in D. In th position of P, add th ar P. Add th triial ar at th nd of D. This ar-dcomposition is alid, as plaind arlir. If all of th ars in L ar n, thn P has th sam parit as R, and this stp dos not introduc an tra n ars. If not, thn L contains at last on odd ar, in which cas P is n and w do not introduc tra n ars (as plaind arlir). (i) t =. Lt P b th ar Q R L 1... L k, whr F1 nw = {L 1,..., L k }. Dlt P, and in th position of P, add th ar P. Dlt all ars in F old from D, and for ach ar in F old, rplac it with th corrsponding ar in F0 nw at th sam position in D. Add th triial ars and at th nd of D. As plaind arlir, this ar-dcomposition is alid. If all th ars in F1 nw ar n, w ha th following cass: (a) Q and R ar odd. In this cas, P is odd and w do not introduc tra n ars. (b) Eactl on of Q and R is n. In this cas, P is n and w do not introduc tra n ars. (c) Q and R ar n. In this cas, P is odd, contradicting th assumption that D was nmin; this cas cannot occur. If F1 nw contains an odd ar, thn th corrsponding ar in F0 nw is odd and th corrsponding ar in F old is n. Sinc w ha alrad rducd th numbr of n ars b at last on, w do not introduc tra n ars. In all of th abo cass, th intrnal rtics of P ar addd to a long ar in D. If P was a -ar, thn it is possibl that w cratd nw non-pndant short ars that appar aftr P in th nw ar-dcomposition. Ths short ars ar handld in futur itrations of th abo procdur, in th sam mannr as abo (that is, first w handl all non-pndant 2-ars, thn w handl th first non-pndant -ar). 8

9 In ach itration, th abo procdur taks tim polnomial in V (G) for th non-pndant short ar undr considration. Furthr, if this short ar is a -ar, th intrnal rtics of this short ar ar addd to long ars, and ar nr again addd to a non-pndant short ar until trmination. As a consqunc, th st X in ach itration is a strict suprst of th corrsponding st in an prious itration. Hnc th running tim for th whol procdur is polnomial. On trmination of this procdur, th ar-dcomposition D is opn and has φ(g) n ars, and all of its short ars ar pndant. Thorm. Gin a 2-rt-connctd graph G with minimum dgr at last, and an associatd nmin ar-dcomposition D in which all short ars ar pndant, an opn nmin nic ar-dcomposition of G can b computd in polnomial tim. Proof. Sinc D is opn and nmin, and all short ars of D ar pndant, it rmains to obtain th proprt that thr ar no dgs conncting an intrnal rt of on short ar to an intrnal rt of anothr short ar of D. Sinc D has φ(g) n ars, thr ar no dgs conncting th intrnal rtics of 2-ars. If not, w could rplac th 2-ars and th triial ar conncting thir intrnal rtics b a pndant -ar and two triial ars, rducing th numbr of n ars b two, contradicting th assumption that D is nmin. Sinc w ha two choics for ach nd rt of such a -ar, w can alwas choos its nd rtics such that it is opn. As long as D has two short pndant ars P and P with an dg conncting an intrnal rt of P with an intrnal rt of P, w rpat th following procdur. Cas 1. On of th ars P and P is a 2-ar. Without loss of gnralit, assum P is a 2-ar and P is a -ar. Lt a and b b th ndpoints of P and b th intrnal rt of P. Lt c and d b th ndpoints of P and and b th intrnal rtics of P such that is adjacnt to both c and (Figur 9a). Construct th ar S as shown b th thick paths in Figurs 9b and 9c, that is, S consists of th dgs a,, and d if th rtics a and d ar distinct, and th dgs b,, and d if th coincid. P P a b c d (a) a b c d (b) a and d ar distinct Figur 9: P is a 2-ar a b c d (c) a and d coincid 9

10 Rmo th ars P and P from D, and add th ar S in plac of th ar P, followd b triial ars consisting of th rmaining dgs from P and P that ar not in S. Sinc P and P ar both pndant ars, th nw ar-dcomposition is a alid ar-dcomposition of G. Sinc th nd rtics of S ar distinct, it is opn, and sinc w dltd a 2-ar from D bfor adding a 4-ar to it, th numbr of n ars in D rmains qual to φ(g). Cas 2. Both P and P ar -ars. Lt a and b b th ndpoints of P and lt and w b its intrnal rtics adjacnt to a and b rspctil. Lt c and d b th ndpoints of P and lt and b its intrnal rtics adjacnt to c and d rspctil (Figur 10). Suppos and ar adjacnt. W ha th following cass. w P P a b c d Figur 10: Both P and P ar -ars Cas 2a. Th rtics b and c ar distinct. Construct th ar S with ndpoints b and c and dgs bw, w,, and c (as shown b th thick path in Figur 11). Rmo th ars P and P from D, add th ar S in plac of th ar P, and add th triial ars consisting of th rmaining dgs from P and P that ar not in S at th nd of D. Sinc P and P ar both pndant ars, th nw ar-dcomposition is a alid ar-dcomposition of G. Sinc th nd rtics of S ar distinct, it is opn, and sinc S is an odd ar, th numbr of n ars in D rmains qual to φ(g). w a b c d Figur 11: b and c ar distinct Cas 2b. Th rtics b and c coincid, as shown in Figur 12. Sinc r rt of th graph has dgr at last, is adjacnt to som rt not in th st {b, }. Cas 2b.I. is adjacnt to an intrnal rt of P. If is adjacnt to, construct th ar S with ndpoints b and d and dgs bw, w,, and d (as shown b th thick path in Figur 1a). Othrwis, 10

11 w a b d Figur 12: b and c coincid if is adjacnt to w, construct th ar S with ndpoints a and b and dgs a,,, w and wb (as shown b th thick path in Figur 1b). In ithr cas, dlt P and P from D, add S to D in plac of P, and add all of th rmaining dgs (dashd dgs in th corrsponding figur) in triial ars at th nd of D. In both cass, th ar S is an odd long ar with distinct nd points, hnc th nw ar-dcomposition is opn and is alid for G, and th numbr of n ars rmains qual to φ(g). w w a b d (a) is adjacnt to a b d (b) is adjacnt to w Figur 1: is adjacnt to an intrnal rt of P Cas 2b.II. is adjacnt to an intrnal rt of an ar R not qual to P. Sinc th input graph is simpl and dos not ha paralll dgs, dos not coincid with b or. If R is a long ar, construct th ar S with ndpoints b and and dgs bw, w,, and (as shown b th thick path in Figur 14a). Dlt P and P from D, add S to D in plac of P, and add all of th dashd dgs in th corrsponding figur in triial ars at th nd of D. Othrwis, if R is a short ar, thn it is pndant. If it is a 2-ar (Figur 14b), construct th ar S as shown b th thick path in th figur. Obsr that w ha two choics for on nd of this ar: w choos to nd th ar at ithr g or h, so as to nsur that it is an opn ar. Th ampl in th figur shows S nding at g, with dgs g,,,, w and wb. Rmo P, P and R from D, add S to D in plac of P, and add all of th dashd dgs in triial ars at th nd of D. Othrwis, R is a -ar. Lt g and h b th ndpoints of R, and i and b its intrnal rtics adjacnt to g and h rspctil (Figurs 14c and 14d). W ha two cass: ithr g and b ar distinct, or th coincid. If th ar distinct, construct th ar S as shown b th thick path in Figur 14c, with dgs bw, w,,,, i and ig. Dlt P, P and R from D, and add S 11

12 to D in plac of P, and all of th dashd dgs in triial ars at th nd of D. Othrwis, th rtics g and b coincid. Construct th ar S as shown b th thick path in Figur 14d, with dgs gi, i,, and d. Dlt th ars P and R from D and add S to D in plac of P, and all of th dashd dgs in triial ars at th nd of D. In all cass, th numbr of n ars rmains qual to φ(g), sinc th onl cas whr S is an n ar is whn R is a 2-ar. Additionall, S is an opn pndant ar, hnc th nw ar-dcomposition is alid for G. w a b d R (a) R is a long ar w w a b g R h (b) R is a 2-ar w d a b i d a b i d g R h g R h (c) R is a -ar, and b and g ar distinct (d) R is a -ar, and b and g coincid Figur 14: is adjacnt to a rt outsid P Sinc th abo procdur taks constant tim for r pair of pndant ars with adjacnt intrnal rtics, th running tim for th whol procdur is polnomial. On trmination of this procdur, th ar-dcomposition D has φ(g) n ars, and is both opn and nic. Lmma 4. Lt D b an opn nic ar-dcomposition of a 2-rt-connctd graph G, and M b th associatd ardrum composd from th short (pndant) ars of D. Dnot b V I th st of intrnal rtics of non-pndant ars, and lt µ(g, M) b th si of th maimum armuff for th ardrum M. Thn µ(g, M) V I 1. Proof. Suppos not. Thn µ(g, M) V I. Considr th graph H on th rt st V I with dg (u, ) prsnt in E(H) if and onl if thr is a path with its ndpoints at u and in th maimum armuff for M. Sinc µ(g, M) V I, this graph has at last V I dgs, and 12

13 is hnc not a forst. Sinc r dg in this graph corrsponds to a path in th maimum armuff, an ccl in this graph must b a ccl in th maimum armuff, which contradicts th dfinition of an armuff, which stats that th union of all paths in th armuff is a forst. Hnc µ(g, M) V I 1. Thorm 5. Thr is a 17 -approimation algorithm for th minimum 2-rt-connctd 12 spanning subgraph problm on graphs with minimum dgr at last. For an 2-rtconnctd graph G whr r rt has dgr at last, it finds a 2-rt-connctd spanning subgraph with at most 17OP T 12 2V C(G) dgs in polnomial tim. Proof. Construct an opn nmin nic ar-dcomposition D for G. Lt π dnot th numbr of pndant ars and π th numbr of (pndant) -ars in this ar-dcomposition. W ha π π. Lt H b th graph obtaind b dlting from G all dgs that ar in triial ars in this ar-dcomposition. Sinc th nontriial ars of D form an opn ar-dcomposition for H, H is 2-rt-connctd (Whitn [5]), and has at most 17 LP(G) dgs, which w show 12 using th following claims. Claim 5.1. Th numbr of dgs in nontriial ars is at most 5 4 LP (G) π. Proof. For an ar P with E(P ) 5, w ha E(P ) 5 in(p ). For an 4-ar or 2-ar P 4 w ha E(P ) 5 in(p ) +. For an -ar P w ha E(P ) 5 in(p ) Lt E b th st of dgs in nontriial ars. Sinc th total numbr of 4- and 2-ars in D is at most φ(g), and π π, th total numbr of dgs in nontriial ars is at most 5 4 ( V (G) 1) + 4 φ(g) π 5 4 L φ(g) π, which is at most 5 4 LP (G) π. Claim 5.2. Th numbr of dgs in nontriial ars is at most 2 LP (G) 1 4 π. Proof. Sinc D is an opn nic ar-dcomposition, th graph inducd in G b th intrnal rtics V M of th pndant short ars of D has dgr at most 1. Lt M b th st of its componnts, thn M is an ardrum in G. Lt V D b th st of intrnal rtics of pndant long ars and lt V I = V \ (V M V D ). Dnot b φ M, φ D and φ I th numbr of n ars in th sts of pndant short ars, pndant long ars and non-pndant ars rspctil. Lt E 1 b th st of dgs in pndant short ars. For r pndant short ar P, w ha E(P ) = 2 in(p ) φ(p ). Summing or all pndant short ars, w ha E 1 = 2 V M φ M. Lt E 2 b th st of dgs in pndant long ars. For r pndant long ar P, w ha E(P ) 2 in(p ) φ(p ) 1. Summing or all pndant long ars, w ha E 2 2 V D φ D (π M ). Lt E b th st of dgs in non-pndant ars. For r non-pndant ar P cpt th singl rt ar P 0, sinc P is a long ar, w ha E(P ) 5 in(p ) + 1 φ(p ). For th rt 4 2 ar P 0, E(P 0 ) = 0 and in(p 0 ) = 1. Summing or all non-pndant ars including P 0, w ha E 5 V 4 I 1 + 1φ 2 I. 1

14 Lt E = E 1 E 2 E b th st of dgs in nontriial ars. Summing or th abo inqualitis, w gt E 2 V (G) +1 2 φ(g) π + M 1 4 V I 5 4 = [ V (G) + M µ(g, M) 1] + 1 [ V (G) +φ(g) 1] 2 π ( µ(g, M) 1 ) 4 V I = L µ (G, M) + 1 ( 1 2 L φ(g) π µ(g, M) 1 ) 4 V I ( 1 = LP (G) π µ(g, M) 1 ) 4 V I 2 LP (G) π + µ(g, M) using Lmma LP (G) π + π sinc µ(g, M) M π 4 2 LP (G) 1 4 π. (Proof of Thorm continud) If π 1LP (G), thn from Claim 1, E 5LP (G) + 1π LP (G) OP T V C(G). If π > 1LP (G), thn from Claim 2, E LP (G) 1π < LP (G) OP T V C(G). Appling Thorms 2 and to G, and dlting all dgs in triial ars, w obtain a 2-rtconnctd spanning subgraph of cardinalit at most OP T 2V C(G) in polnomial tim. 14

15 Rfrncs [1] S. Khullr and U. Vishkin. Biconnctiit approimations and graph carings. In: Journal of th ACM (JACM) 41.2 (1994), pp [2] N. Garg, A. Singla, and S. Vmpala. Improd approimation algorithms for biconnctd subgraphs ia bttr lowr bounding tchniqus. In: Proc. 4th Annual ACM-SIAM SODA (199), pp [] J. Chrian, A. Sbő and Z. Sigti. Improing on th 1.5-Approimation of a Smallst 2-Edg Connctd Spanning Subgraph. In: SIAM J. Discrt Math (2001), pp [4] A. Sbő and J. Vgn. Shortr tours b nicr ars: 7/5-approimation for th graph-tsp, /2 for th path rsion, and 4/ for two-dg-connctd subgraphs. In: J. Combinatorica 4.5 (2014), pp [5] H. Whitn. Non-sparabl and planar graphs. In: Transactions of th Amrican Mathmatical Socit 4 (192), pp

16 Appndi A A1. Th 2-rt-connctiit problm is NP-hard whn rstrictd to graphs with minimum dgr at last. Proof. Lt G b an input graph to th gnral 2-rt-connctiit problm, and dnot b n(g) th numbr of rtics with dgr 2 in G. Considr th graph G constructd as follows: rplac r rt with dgr 2 in G b an instanc of K 4 (th complt graph on 4 rtics), such that th two dgs incidnt on th dgr-2 rt in G ar incidnt on distinct rtics of th K 4 instanc in G. Thn G has minimum dgr at last, and r 2-rt-connctd spanning subgraph H of G, with E(H) dgs, corrsponds to a 2-rt-connctd spanning subgraph H of G (constructd b adding a path of lngth btwn th dgr-4 nods of r K 4 instanc cratd b rplacmnt), with E(H ) = E(H) +n(g) dgs, and ic-rsa. Hnc an algorithm that sols th 2-rt-connctiit problm in polnomial tim on graphs with minimum dgr at last can b usd to sol th unrstrictd problm in polnomial tim, which implis th statmnt of A1. 16

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