Embeddings of Graphs of Fixed Treewidth and Bounded Degree

Transcription

1 DMFA Loo AMC Loo Embddins of Graphs of Fixd Trwidth and Boundd Dr Jonathan L. Gross Columbia Univrsity, Dpartmnt of Computr Scinc NY USA, Nw York, USA Rcivd xxx 2012, accptd xxx 2013, publishd onlin xxx 2013 Abstract Lt F b any family of raphs of fixd trwidth and boundd dr. W construct a quadratic-tim alorithm for calculatin th nus distribution of th raphs in F. Within a post-ordr travrsal of th dcomposition tr, th alorithm involvs a full-powrd upradin of production ruls and root-poppin. This alorithm for calculatin nus distributions in quadratic tim complmnts an alorithm of Kawarabayashi, Mohar, and Rd for calculatin th minimum nus of a raph of boundd trwidth in linar tim. Kywords: nus distribution, partial nus distribution, trwidth, tr dcomposition Math. Subj. Class.: 05C10 Vrsion: 13:34 Auust 22, 2013 This papr was prsntd at th AMS Mtin of January, 2012 in Boston, MA. addrss: ross@cs.columbia.du (Jonathan L. Gross) Copyriht c xxxx DMFA Slovnij

2 2 Ars Math. Contmp. x (xxxx) 1 x 1 Introduction For i = 0, 1, 2,..., lt i (G) b th numbr of topoloically distinct cllular mbddins of th raph G in th orintabl surfac S i of nus i. Th nus distribution of th raph G is th squnc of numbrs i (G) : i = 0, 1,... Th smallst and larst numbrs i such that i (G) is positiv ar calld th minimum nus and th maximum nus, rspctivly, of th raph G. It is asily provd and wllknown that thr ar cllular mbddins of a raph G in vry surfac whos nus lis btwn th minimum and th maximum. Th st of numbrs btwn (and includin) th minimum and maximum nus is calld th nus ran of G. Th main objctiv of this papr is to driv a quadratic-tim alorithm to calculat th ntir nus distribution for any family of simpl raphs with fixd trwidth and boundd dr. Sinc th scond barycntric subdivision of a nral raph is a simpl raph with th sam nus distribution as th nral raph to which it is homomorphic, it follows that this can b xtndd to nral raphs by subdividin ds. This papr also introducs a nral form of partial nus distribution for arbitrarily lar dr and for arbitrary root-subraphs, i.., byond vrtics and ds, a rlativizd form of partial nus distribution modulo a fixd rotation systm for its root subraph, and a nral form of production ruls for itrativ rassmbly of a ivn raph from on or mor small subraphs. BASIC RESULTS ON GENUS DISTRIBUTION Fiv fundamntal paprs [10, 7, 18, 15, 20] of th prsnt author and his co-authors Khan and Poshni hav stablishd mthods for calculatin th nus distribution of a raph that is constructd by various kinds of amalamation of two raphs of known nus distribution. Ths paprs also stablish ways to calculat th nus distributions of chains and cycls of copis of raphs of known nus distribution. Th mthods dvlopd in ths paprs includ rcombinant strands, partitiond nus distributions, and production ruls. Mor rcntly, combinin ths calculation mthods with th alorithmic tchniqus of post-ordr travrsal and root-poppin has facilitatd th calculation of nus distributions for 3-rular outrplanar raphs [8], for 4-rular outrplanar raphs [19], and for 3-rular Halin raphs [9]. Combinin ths sam calculation mthods with d-addition [16] has ld to th nus distribution of msh raphs of th form P 3 P n. CONNECTIONS OF TREEWIDTH TO EMBEDDING PROBLEM Sinc th introduction of th concpt of trwidth by Robrtson and Symour, boundin th trwidth has bn widly usd to obtain polynomial-tim alorithms for problms that ar othrwis NP-hard. In particular, dcidin whthr an arbitrarily slctd raph can b mbddd in a ivn surfac is NP-complt [25]; howvr, for any class of raphs

3 Ars Math. Contmp. x (xxxx) 1 x 3 of boundd trwidth, Kawarabayashi, Mohar, and Rd [14] hav drivd a linar-tim alorithm for calculatin th minimum nus. Althouh outrplanar raphs hav trwidth 2, and althouh Halin raphs and P 3 P n mshs hav trwidth 3 (s [3]), trwidth plays no xplicit rol in th calculation of nus distributions in any of th paprs just mntiond. Indd, th fiv fundamntal paprs on raph amalamations citd abov includ applications to raphs of arbitrarily hih trwidth and arbitrarily hih dr. Nonthlss, th pastins in thos paprs occur in localitis of th amalamand raphs in which th trwidth and dr ar boundd. In th prsnt papr, various ky idas from th arlir paprs ar abstractd, nralizd, and combind with trwidth to yild a quadratic-tim alorithm for th nus distribution of th raphs in any family of raphs of boundd dr and boundd trwidth. It will b apparnt that as trwidth and dr incras, th multiplicativ constant of th quadratic trm rows rapidly. Accordinly, on anticipats continud intrst in th drivation of spcial mthods for calculatin th nus distributions of raph familis of spcial intrst. TERMINOLOGY In what follows, a raph is takn to b connctd and simpl, unlss somthin ls can b infrrd from th immdiat contxt. W us V G and E G to dnot th vrtx st and d st of a raph G. Th mbddins ar in orintd surfacs. Trminoloy usd hr is prdominantly consistnt with [13] and [1]. S also [17] and [28]. W abbrviat fac-boundary walk as fb-walk. DEF. In this papr, a subraph-rootd raph is a tripl (G, H, u), whr H is a subraph of G and u V H. Th third paramtr u is calld th pivot. Somtims, th form (G, H) with no pivot is usd. If H = K 1 or H = K 2, thn th raph is vrtx-rootd or drootd, rspctivly. Whn th contxt clarifis th manin, th raph may simply b calld rootd. REMARK. Pivot vrtics ar usd hr to chan th root subraph as a squnc of raphs is formd in th procss of rassmbly of a ivn raph. In [8], w achivd a chan of root with what w calld root-splittin. In [9], a chan of root was accomplishd by pi-mrs. OUTLINE OF THIS PAPER Sction 2 of this papr dscribs trwidth from a prspctiv that is rlvant to its us in th alorithm. Sction 3 dscribs how a dcomposition tr is usd to analyz a raph into framnts to b amalamatd. Sction 4 introducs a hihly nral way of partitionin th nus distribution of a raph; it shows that th numbr of clls of th partition dpnds only on th trwidth and th maximum dr, and not on th numbr of vrtics of that raph. Sction 5 shows that th numbr of mbddins of an amalamatd

4 4 Ars Math. Contmp. x (xxxx) 1 x raph dpnds on th drs of th vrtics of th subraph of amalamation, and not on th numbr of vrtics of th raph. Sction 6 dscribs production ruls, as thy occur in th alorithm. Sction 7 dscribs and analyzs th nus distribution alorithm. Sction 8 offrs som conclusions about th alorithm and opportunitis for futur rsarch. This papr is larly slf-containd, xcpt for som dtails of th wll-stablishd mthods of constructin productions (which is quit ncssary for th alorithm), as in [10] and [18]. Prior xprinc with calculatin nus distributions of raph amalamations, spcially as in [8] and [9], is likly to b quit hlpful. 2 Trwidth Th usual dfinition of trwidth is basd on th concpt of tr dcomposition. Ths ar both du to Robrtson and Symour [21]. An xcllnt xposition is ivn by [3]. For applications of trwidth to topoloical raph thory, s [17]. DEF. Lt G = (V, E) b a raph, and T a tr with nods 1, 2,..., s. Lt X = {X i 1 i s} b a family of substs of V (associatd with th rspctiv nods 1, 2,..., s) whos union is V such that th inducd raph on th st of imas in T of ach vrtx of V is a subtr of T ; for vry d uv in th raph G, thr is a nod i in th tr T such that both u and v ar mmbrs of X i. Thn th pair (X, T ) is a tr dcomposition of G, and th tr T is calld a dcomposition tr for G. TERMINOLOGY. For th sak of clarity, w will rfr to vrtics and ds in th raph G and to nods and lins in th tr T. ABUSE OF NOTATION. Throuhout this papr, w rfr to th sts X i of a dcomposition tr as nods. DEF. Th width of a tr dcomposition (X, T ) quals { max X i } 1 i V T 1 DEF. Th trwidth of a raph G is th smallst k such that G has a tr dcomposition of width k. Proposition 2.1. A connctd raph has trwidth 1 if and only if it is a tr. Proposition 2.2 ([27]). A connctd raph has trwidth 2 if and only if it contains a cycl and dos not contain a K 4 -minor.

5 Ars Math. Contmp. x (xxxx) 1 x 5 Proposition 2.3. Evry raph of trwidth 2 is planar. Proof. A non-planar raph has ithr K 5 or K 3,3 as a minor. Both ths Kuratowski raphs hav K 4 as a minor. By Proposition 2.2, a raph with a K 4 -minor cannot hav trwidth 2. TREEWIDTH CHARACTERIZATION WITH k-trees An altrnativ charactrization of trwidth, in trms of k-trs (s, for instanc, [4]), is th startin point of our prsnt approach to nus distributions: DEF. A k-tr is dfind rcursivly: Th complt raph K k+1 is a k-tr. If G is a k-tr and C is a k-cliqu in G, thn th raph obtaind by joinin a nw vrtx to th vrtics of C is a k-tr. Proposition 2.4. Th trwidth of a raph G is th last numbr k such that G is a subraph of a k-tr. Proof. Th proof is a dirct consqunc of th dfinitions. FULL DECOMPOSITION TREES DEF. A full dcomposition tr of width k for a raph G is a dcomposition tr T in which vry nod has k + 1 vrtics, and vry pair of adjacnt nods intrscts in k vrtics. W obsrv that ach lin of a full dcomposition tr corrsponds to th k vrtics shard by th two nods whos adjacncy is rprsntd by that lin. Proposition 2.5. Lt T b a full dcomposition tr of trwidth k for an n-vrtx raph G. Thn V T = n k. Proof. Each nod of th dcomposition tr T contains k + 1 vrtics of G. Each lin of T corrsponds to k vrtics of G. Sinc V G is th union of th nods of T, it follows that n = V G = V T (k + 1) E T k = V T (k + 1) ( V T 1)k = V T + k V T = n k Runnin Exampl Part 1. Fiur 2.1 shows an 8-vrtx raph of trwidth 2 and a full dcomposition tr of width 2. W obsrv that th numbr of nods of th full dcomposition tr is 8 2 = 6.

6 6 Ars Math. Contmp. x (xxxx) 1 x a b f a f c c c b b b b c d h d h Fiur 2.1: A raph, and a full dcomposition tr of width 2, rprsntd as a subraph of a 2-tr. Th fiur on th lft is th raph itslf. Each ray trianl on th riht rprsnts a 3- cliqu of th 2-tr. Each dashd ray lin rprsnts an adjacncy of two 3-cliqus in th 2-tr. Each nod of th full dcomposition tr is th st of vrtics lyin within on of th ray trianls. Th ds drawn within th ray trianls ar a rmindr of th adjacncis in th raph itslf. In 3, th subraphs shown within th ray trianls ar calld nod-framnts. Thorm 2.6. Lt G b a raph of trwidth k. Thn G has a full dcomposition tr of width k. Proof. Lt T b a dcomposition tr of width k for a raph G of trwidth k. If th vrtx st from G in on of two adjacnt nods of T is containd in th othr, thn contract th lin of T that joins thos two nods, and liminat th smallr nod. W obsrv that this opration dos not chan th width of th rsultin tr. Itrat this opration until th rsultin dcomposition tr for G has th followin proprty: (P 1 ) For vry nod X i V T and for ach of its nihborin nods X j, thr is a vrtx in X i that dos not li in th nod X j. For simplicity, w assum that th initial tr T alrady has proprty P 1. Sinc th maximum siz of a nod of T is k + 1, and sinc no two adjacnt nods ar idntical, it follows that th intrsction of any pair of adjacnt nods contains at most k vrtics. If som nod X i of tr T contains fwr than k + 1 vrtics, thn choos a vrtx from any nihborin nod and insrt a copy of it into nod X i. If th nod X i now contains vry vrtx in that nihbor, thn contract th lin of T that joins thos two nods. Itrat until vry nod of th rsultin tr has k + 1 vrtics of G. By construction, this tr has proprty P 1. W may now assum that th initial tr T also has this proprty: (P 2 ) Evry nod X i V (T ) has k + 1 vrtics of G.

7 Ars Math. Contmp. x (xxxx) 1 x 7 REMARK. It would b possibl to dsin an alorithm for nus distribution that dos not rquir th ivn dcomposition tr to b convrtd into a full dcomposition tr T. Th rason w prform this convrsion hr is to simplify subsqunt discussion, for instanc, of th us of partial nus distributions in 4. Now suppos that X i and X j ar adjacnt nods of T such that X i = {v 1,..., v m, u m+1,..., u k+1 } X j = {v 1,..., v m, w m+1,..., w k+1 } and X i X j = {v 1,..., v m } whr m < k. Thn rplac th lin (X i, X j ) in T by th nod path X i = {v 1,..., v m, u m+1,..., u k+1 } X (1) i = {v 1,..., v m, w m+1, u m+2,..., u k+1 } X (2) i = {v 1,..., v m, w m+1, w m+2, u m+3,..., u k+1 } X (k m) i = {v 1,..., v m, w m+1,..., w k, u k+1 } X j = {v 1,..., v m, w m+1,..., w k+1 } Th rsultin tr is a full dcomposition tr for G of width k. Corollary 2.7. For any fixd trwidth k, thr is an alorithm to construct a full dcomposition tr for a ivn raph G of trwidth k in linar tim in V G. Proof. A linar-tim alorithm to construct a dcomposition tr T of width k for th raph G is ivn by [2]. Sinc th numbr of ds of such a tr is linar in V G, it follows that a full dcomposition tr for G of width k can b constructd from T within linar tim. Corollary 2.8. Lt G b a raph of trwidth k, and lt T b a full dcomposition tr for G. Thn thr is a k-tr T k with th followin proprty: Each nod of T is a (k + 1)-cliqu of th k-tr T k. Proof. This is implid by th dfinition of a full dcomposition tr. Th nus distribution of a raph G is to b drivd from th partial nus distributions of th inducd raphs (in G) on th vrtics in th rspctiv nods of th tr T, by itrativ amalamation. Various ky idas for drivin th nus distribution of a raph G by itrativ amalamation of a st of subraphs of G ar dvlopd in [6], [7], [8], [9], [10], [15], [18], [19], and [20].

8 8 Ars Math. Contmp. x (xxxx) 1 x 3 Framnts and Amalamations Th alorithm to calculat th nus distribution of a raph G with boundd trwidth and boundd dr rassmbls th raph G by itrativly amalamatin inducd subraphs on th nods of a full dcomposition tr for G. DEF. A nod-framnt of a raph G with rspct to a dcomposition tr T is th inducd subraph in G on th st of vrtics of G that li within a sinl nod of T. For a laf-nod of T, th corrspondin nod-framnt may b calld a laf-framnt. DEF. Lt (G, H, u) and (G, H, u ) b an ordrd pair of disjoint subraph-rootd raphs, and lt η : H u H u b a raph isomorphism. Graph amalamation is th opration that forms a nw raph G η G from G G by mrin th subraphs H u and H u as prscribd by η. In this papr, th subraph H bcoms th nw root subraph, and a nw pivot is chosn accordin to th post-ordr of th dcomposition tr. This is illustratd in Part (2) of th Runnin Exampl. NOTATIONAL CONVENTION. W dnot th vrtics and ds of th subraph H η H of th amalamatd raph G η G by th sam nams as in th subraph H of raph G, th first amalamand. Th vrtics and ds that ar contributd by only on amalamand rtain th nams usd in that amalamand. DEF. A framnt of a raph G with rspct to a dcomposition tr T for G is ithr a nod-framnt or a compound framnt, by which w man th rsult of amalamatin any two framnts across th inducd subraphs of th vrtics that li in both of two adjacnt nods of th tr T. Evry amalamation in th rassmbly of G from its framnts corrsponds to a lin of th dcomposition tr T. Givn a raph G and a full dcomposition tr T of width k, with T nvisiond as drawn in th plan, it is clar that G can b rassmbld by itrativly amalamatin framnts on pairs of vrtics. W fix an arbitrary laf-nod of th tr T as a root of T, and w dtrmin a postordr travrsal of T basd at that root-nod. W obsrv that durin a post-ordr travrsal, vry lin of T is travrsd twic. W amalamat across a lin whnvr th scond travrsal of that lin occurs. Each framnt of th raph G is a rootd subraph of G. In any framnt F, th root subraph is th inducd raph in G on th nod of T in which th framnt F mts th framnt to which it will b amalamatd. In a non-laf-framnt, th root-subraph is to b chosn accordin to th post-ordr of T. LABELING FRAGMENTS AND SELECTING ROOTS OF FRAGMENTS In ordr to discuss th procss of itrativ amalamation, it is hlpful to hav som ruls for assinin labls to framnts.

9 Ars Math. Contmp. x (xxxx) 1 x 9 DEF. Th labl of a framnt F for a raph G with a full dcomposition tr T of width k is of th form G[x 1,..., x q ; r 1,..., r k ; r k+1 ], whr th vrtx st of th framnt F is {x 1,..., x q, r 1,..., r k, r k+1 } Th vrtics r 1,..., r k, r k+1 ar from whatvr nod of th framnt F will b pastd to anothr framnt in th nxt amalamation involvin F that occurs in th post-ordr. Th root-subraph of framnt F is th inducd subraph on vrtics r 1,..., r k, r k+1. Vrtx r k+1 is th pivot. Th vrtics x 1,..., x q ar th rmainin vrtics of framnt F. DEF. In th cours of rassmbly of a raph from its nod-framnts by itrativ amalamation, for ach framnt F, whthr a nod-framnt or a compound framnt, its partnr framnt F is th framnt to which F is amalamatd durin th rassmbly. Similarly, ach nod-framnt M has a partnr nod-framnt M. Runnin Exampl Part 2. Th root subraph of a nod-framnt is th nod-framnt itslf. In ach nod-framnt of Fiur 3.1, th two vrtics with bold labls ar th first two to b pastd to anothr nod. Th third vrtx in th nod is th initial pivot. Ths ar dtrmind by th post-ordr travrsal. W hav takn th nod G[; c, ; d] at th lowr lft of th tr as th root-nod of th dcomposition tr and usd a countr-clockwis travrsal. a b f a f c c c b b b b c d h d h Fiur 3.1: Roots of a dcomposition 2-tr for a raph G. Hr is th rassmbly squnc for th raph of Fiur Th first nod-framnt is F 0 = G[;, ; h]. 2. Th nod-framnt (at th cntr-riht) adjacnt to F 0 is labld G[;, ; b]. Whn nod-framnts F 0 = G[;, ; h] and its partnr F 0 = G[;, ; b] ar amalamatd, th rsultin framnt is F 1 = G[h; b, ; ]

10 10 Ars Math. Contmp. x (xxxx) 1 x 3. Whn framnt F 1 = G[h; b, ; ] is amalamatd with nod-framnt F 1 = G[; b, ; f], th rsultin framnt is F 2 = G[f, h; b, ; ] 4. Whn framnt F 2 = G[f, h; b, ; ] is amalamatd with nod-framnt F 2 = G[; b, ; c], th rsultin framnt is F 3 = G[f,, h; b, c; ] 5. Whn framnt F 3 = G[f,, h; b, c; ] is amalamatd with nod-framnt F 3 = G[; b, c; a], th rsultin framnt is F 4 = G[a, f,, h; c, ; b] 6. Whn framnt F 4 = G[a, f,, h;, c; b] is amalamatd with nod-framnt F 4 = G[;, c; d], th rsultin framnt is F 5 = G[a, b, f,, h; c, ; d] = G W obsrv th followin proprtis of ach of th amalamation in th squnc: Th pivot of ach of th amalamand framnts F i and F i is not a vrtx of th othr amalamand framnt. Th root-subraph of th amalamatd framnt F i+1 is th root-subraph of on of th amalamand framnts F i and F i. Mor prcisly, it is whichvr of th two root-subraphs will b usd whnvr F i+1 is amalamatd to F i+1. Th pivot of th amalamatd framnt F i+1 is th uniqu vrtx in th rootsubraph that will not b mrd with anothr vrtx in th nxt amalamation stp involvin framnt F i+1. REMARK. Althouh th partnr framnts F i wr all nod-framnts in this xampl, this would not b th cas with a mor complicatd dcomposition tr. 4 Partitionin th Gnus Distribution Partitionin th mbddins of a rootd raph has provn to b a hihly usful tchniqu in calculatin nus distributions. A surfac-by-surfac invntory of th clls of th partition is calld a partitiond nus distribution. Th critria for partitionin has bn th incidnc of fb-walks on th roots. In th simplst cas, w hav a raph (G, v) rootd at a sinl 2-valnt vrtx v. W can thn partition th nus distribution squnc { i (G) : i = 0, 1,...} into two partial nus distribution squncs whr d i (G, v) : i = 0, 1,... and s i (G, v) : i = 0, 1,...

11 Ars Math. Contmp. x (xxxx) 1 x 11 d i (G, v) is th numbr of mbddins of G in which two distinct fb-walks ar incidnt on th root-vrtx v; and s i (G, v) is th numbr of mbddins of G in which a sinl fb-walk is twic incidnt on th root-vrtx v. Th prototypical approach to calculatin ths distributions is to construct simultanous rcursions for d i and s i and to obtain i by addin thir solutions. In th nxt smallst cas, that of two 2-valnt roots u and v, thr ar tn diffrnt partial nus distributions. For instanc, in [10] and [7], w hav dfind dd i (G, u, v) as th numbr of mbddins of G in S i in which thr ar two diffrnt fb-walks incidnt on u, on of which is also incidnt on v, and anothr fb-walk incidnt on v that is not incidnt on u. sd i (G, u, v) as th numbr of mbddins of G in S i in which on fb-walk is twic incidnt on u and also incidnt on v, and anothr fb-walk is incidnt on v and not on u. Whn thr ar multipl roots and/or a root-subraph, th numbr of partial nus distributions squncs rows rapidly with th total numbr of root-vrtics or of vrtics in th root-subraph. W can partition th nus distribution of a raph G into as many partial nus distributions as ndd, th sum of which is th nus distribution squnc { i (G) : i = 0, 1,...}. In th nral cas now undr considration, th nus distribution of a subraphrootd raph (G, H, u) is partitiond hr accordin to th cyclic squnc of incidncs of fb-walks on th vrtics of th root-subraph H and on th pivot u. This sction provids th dtails, an xampl, and an uppr bound on th numbr of rstrictd squncs in th partition corrspondin to a raph of trwidth k and maximum dr. This uppr bound dpnds only on th trwidth and maximum dr of a raph, and not on its numbr of vrtics. DEF. A appd word on th root-vrtics r 1,..., r k, r k+1 of a rootd raph (G, H, r k+1 ) is a word on th alphabt {r 1,..., r k, r k+1, } that contains no two conscutiv occurrncs of th bullt, which is calld a ap. Each appd word rprsnts th cyclic ordr in which th various root-vrtics occur in an fb-walk of an mbddin. Whn two root-vrtics r i and r j ar sparatd by a ap symbol, it mans that th subwalk of th corrspondin fbwalk btwn r i and r j contains on or mor non-root-vrtics. Whn two root-vrtics r i and r j ar adjacnt, it mans that thr is an d r i r j travrsd by th fb-walk. Two appd words ar quivalnt if on is a cyclic prmutation of th othr. Th principal rprsntativ of ach quivalnc class of appd words is th on that is lxicoraphically first. W rard th bullt as lxicoraphically last, i.., aftr all th root-vrtics. DEF. A multi-st of principal appd words (writtn as a tupl) is calld a root-phras if ach root-vrtx occurs in th union of th appd words as many tims as its valnc in G;

12 12 Ars Math. Contmp. x (xxxx) 1 x th principal appd words ar in th ordr of non-incrasin siz, with lxicoraphic ordr usd for ti-brakin. Each root-phras rprsnts a cll of th partition of th mbddins of (G, H, u). Morovr, ach root-phras may b subscriptd by an intr that rprsnts th nus of a surfac. Ths two xampls illustrat how a root-phras is usd in our nralization of partial nus distributions. (With larr root-subraphs, som componnts of a root-phras may hav intr cofficints.) Exampl 4.1. For non-adjacnt roots u and v, dd i (G, u, v) is rprsntd by th rootphras (u v, u, v ) i. Exampl 4.2. For non-adjacnt roots u and v, sd i (G, u, v) is rprsntd by th rootphras (u u v, v ) i. DEF. Th partitiond nus distribution of (G, H, u) is a linar combination of th subscriptd root-phrass, in which th cofficint of ach subscriptd root-phras is th numbr of orintd mbddins of G corrspondin to that root-phras, in th surfac whos nus quals th particular subscript. DEF. W us th abbrviation pd for partitiond nus distribution. DEF. W obsrv that th rstriction of th pd of (G, H, u) to a sinl root-phras, takn ovr all subscripts in th nus ran, is th invntory of mbddins within a sinl partition cll, i.., th cll corrspondin to th ivn root-phras. This invntory is calld a partial nus distribution of (G, H, u). Th word partial, whn usd hr as a noun, is a synonym for root-phras. In this sns, w may say that th pd is a linar combination of th subscriptd partials. Runnin Exampl Part 3. Th nod-framnt G[;, ; h] has th followin pd: (h, h) 0 (4.1) Th only mbddin of that framnt has two (orintd) fb-walks, h and h. Sinc all th vrtics in th only two fb-walks ar root-vrtics, and sinc th root-vrtics ar mutually adjacnt, thr ar no aps in th appd words of th only root-phras. Similarly th nod-framnt G[;, ; b] has th followin pd: Th compound framnt G[h; b, ; ] has th pd (b, b) 0 (4.2) (b, b, ) 0 + (b, b, ) 0 + (b b ) 1 + (b b ) 1 (4.3) To s this most asily, on draws th four mbddins and tracs th fb-walks in ach mbddin.

13 Ars Math. Contmp. x (xxxx) 1 x 13 NOTATION. W us p(n) to dnot th numbr of partitions of a positiv intr n. W rcall th asymptotic formula of Hardy and Ramanujan: p(n) 1 4n 3 π 2n 3 as n W us th numbr of partitions toward an uppr bound on th numbr of partials of a rootd raph. NOTATION. W dnot th dr of a vrtx v of a raph Y by δ Y (v). If Y is a subraph of a raph X, this convntion distinuishs th dr of vrtx v in th subraph Y from its dr in X. Thorm 4.1. Lt F b a family of raphs of trwidth k and maximum dr, ach of which is rootd on an isomorphic copy of a fixd raph H, in which V H = {r 1,..., r k }. Thn th numbr of partials associatd with amalamatin two arbitrary mmbrs of F across root-subraph H is at most [k ]! p(k ) 2k (!) k That is, it has an uppr bound that is indpndnt of th numbr of vrtics of th raphs bin amalamatd. Proof. Hr w rard a root-phras as a raw charactr strin in th alphabt V H = {r 1,..., r k }, into which aps and commas ar vntually insrtd. Th lnth of a raw charactr strin for a raph G is k i=1 δ G(r i ). (W nd to us δ G rathr than δ H bcaus w ar concrnd with th dr of ach vrtx of H within th raph G, not just within H.) Thus, th cardinality of th st of raw charactr strins is [ k i=1 δ G(r i )]! k i=1 δ G(r i )! [k ]! (!) k W may insrt th commas into ach raw charactr strin so that th numbr of charactrs btwn two succssiv commas is non-incrasin, as on rads th strin from binnin to nd. Th cardinality of th st of comma-nrichd raw charactr strins so obtaind is at most p(k ). Th numbr of ways to insrt from 0 to k aps into such a comma-nrichd charactr strin so that no two aps ar adjacnt, no ap occurs immdiatly aftr a comma, and no ap occurs at th binnin of th first word is at most 2 k. W hav pr-normalizd th root-phrass so that no ap occurs at th binnin of a word and so that th appd words ar in th ordr of non-incrasin lnth. Thus, vry normalizd appd word is a mmbr of th st of such ap-nrichd, commanrichd charactr strins. This stablishs th uppr bound of th thorm.

14 14 Ars Math. Contmp. x (xxxx) 1 x Whn w ar rassmblin a raph G of trwidth k and maximum dr from th nod-framnts of a full dcomposition tr, th k-vrtx root-subraph across which two framnts ar amalamatd varis by on vrtx at a tim, as w travrs th dcomposition tr. Th isomorphism typ of th root-subraph across which w mr framnts may vary from on amalamation to th nxt. Thus, rathr than only with th partials associatd with a fixd root-subraph H, w nd to b concrnd with th st of all partials for root-subraphs with a ivn numbr of vrtics. Thorm 4.2. Lt G b a raph of trwidth k and maximum dr. Thn th numbr of partials rquird whn rassmblin G from th nod-framnts of a full dcomposition tr of width k is at most [(k + 1) ]! (!) k+1 p((k + 1) ) 2 (k+1) That is, th numbr of partials has an uppr bound that is indpndnt of th numbr of vrtics of th raph G. Proof. As illustratd in Part 3 of th Runnin Exampl, w nd to considr partials on k + 1 symbols. Th isomorphism typ of th k-vrtx subraph of amalamation may chan from amalamation to amalamation, and w provid for this by prmittin th locations of th aps to vary in th root-phrass. Sinc at most on ap occurs btwn two lttrs of th alphabt, th numbr of ways to insrt th aps is at most 2 (k+1). NOTATION. For a subraph-rootd raph (G, H), w dnot th corrspondin st of partials by P H. 5 Embddins of th Amalamatd Graph DEF. Lt ρ and σ b rotation systms for raphs X and Y, rspctivly, such that Y is a subraph of X. W say that thy ar consistnt at a vrtx v of Y if th rotation σ at v is th rstriction of th rotation ρ at v. If at vry vrtx of V Y, th rotation σ is th rstriction of th rotation ρ, thn ρ and σ ar consistnt rotation systms. Morovr, th mbddins corrspondin to rotations systms ρ and σ ar thn calld consistnt mbddins. Givn rotation systms ρ and σ for th rootd raphs (G, H) and (G, H ) and an isomorphism η : H H, such that th rstriction of σ to η(h) ars with th rstriction of σ to H, w sk to count or iv an uppr bound for th numbr of mbddins of th amalamatd raph (G η G, H η H ) that ar consistnt with ρ and σ. Th balanc of this sction is dirctd toward that objctiv. W considr th confiuration at ach vrtx of th subraph H. A list of citations of arly studis of rstrictd rotation systms is ivn by [24]. ISOLATED VERTICES IN THE SUBGRAPH H Th followin proposition is to b usd whn thr is a vrtx of th root-subraph H that has no nihbors in H.

15 Ars Math. Contmp. x (xxxx) 1 x 15 Proposition 5.1. Lt (G, H) and (G, H ) b subraph-rootd raphs and η : H H an isomorphism. Lt ρ and σ b rotation systms for G and G such that σ = η ρ η 1. Lt v b an isolatd vrtx of th root-subraph H, that is, with δ H (v) = 0. Thn th numbr of rotations at v of (G η G, H η H ) that ar consistnt with ρ and σ is ( ) δg (v) + δ G (v) 1 δ G (v) δ G (v) (5.1) Proof. A rotation at vrtx v in (G η G, H η H ) is a cyclic ordrin of th ds of (G η G, H η H ) that ar incidnt at v. W rard th ds of E G incidnt on vrtx v as partitionin a cycl into δ G (v) compartmnts into which ds of E G incidnt at v ar to b insrtd. Onc on of ths δ G (v) compartmnts is slctd as th location of som arbitrarily slctd first d of E G, thr ar δ G (v) + 1 compartmnts into which th rmainin δ G (v) 1 ds can b insrtd. Sinc th ordr of ths rmainin ds is fixd, w may rard ach of thm as a zro, and partition thm with δ G (v) ons. Thrfor, th numbr of ways to insrt ths δ G (v) 1 ds quals th numbr of binary strins of lnth δ G (v) + δ G (v) 1 with δ G (v) ons. Exampl 5.1. In [10], w amalamat two vrtx-rootd raphs (G, v) and (G, v ) at 2- valnt roots. Thus, δ G (v) = δ G (v) = 2, and δ H (v) = 0. Th numbr of rotations in th amalamatd raph that ar consistnt with a pair of rotations, on from G and on from G, is ( ) ( ) ( ) δg (v) + δ G (v) δ G (v) = 2 = 2 = 6 δ G (v) 2 2 VERTICES OF POSITIVE DEGREE IN THE SUBGRAPH H Th cas in which a vrtx v of th root-subraph H has positiv dr rquirs sufficintly complicatd notation, that it is usful to prcd th nral analysis by a dfinition, a lmma, and an xampl. DEF. Lt α and β b linar ordrins of disjoint sts S = {x 1,..., x p } and T = {y 1,..., y q }, rspctivly. W say that a linar ordrin γ of th st S T is an intrlavin of th ordrins α and β if th rstrictions of γ to S and T ar idntical to α and β, rspctivly. Lmma 5.2. Th numbr of ways to intrlav two squncs of rspctiv lnths p and q is ( ) p + q q Proof. Thr is an obvious bijction from th st of intrlavins to th st of binary strins of lnth p + q with q ons.

16 16 Ars Math. Contmp. x (xxxx) 1 x Exampl 5.2. W dpict in Fiur 5.1 an amalamation G G in which th vrtx v of th root-subraph H has dr 2 in H, dr 7 in G, and dr 5 in G. Th rotations at vrtx v ar ρ σ in H : 1, 2 ρ in G : c 1 1, c 1 2, c 1 3, 1, c 2 1, c 2 2, 2 σ in G : d 1 1, d 1 2, 1, d 2 1, 2 G 2 G' 2 c 2 1 c 1 1 c 3 1 v c 2 2 c 1 2 d 1 1 d 2 1 v d Fiur 5.1: A vrtx v of dr 2 in th (darknd) subraph H, dr 7 in G, and dr 5 in G. Th vrtx v has dr 10 in th amalamatd raph G G. A rotation at v in G G is consistnt with th rotations ρ and σ if and only if th d-sts {c 1 1, c 1 2, c 1 3} and {d 1 1, d 1 2} ar intrlavd btwn 2 and 1 and th d-sts {c 2 1, c 2 2} and {d 2 1} ar intrlavd btwn 1 and 2. By Lmma 5.2, th numbr of rotations at v in G G that ar consistnt with th rotations ρ and σ is ( )( ) Proposition 5.3. Lt (G, H) and (G, H ) b subraph-rootd raphs and η : H H an isomorphism. Lt ρ and σ b rotation systms for G and G such that σ = η ρ η 1. Lt v b a vrtx of H with dr δ H (v) > 0 and th followin rotations: (ρ σ) v in H : 1, 2,..., δh (v) ρ v in G : c 1 1, c 1 2,..., c 1 δ 1, 1, c 2 1, c 2 2,..., c 2 δ 2, 2,..., δh (v) σ v in G : d 1 1, d 1 2,..., d 1 δ 1, 1, d 2 1, d 2 2,..., d 2 δ 2, 2,..., δh (v) Thn th numbr of rotations at v in (G η G, H η H ) that ar consistnt with ρ and σ is δ H (v) i=1 ( δi + δ i ) δ i (5.2)

17 Ars Math. Contmp. x (xxxx) 1 x 17 Proof. A rotation τ v at v in (G η G, H η H ) is consistnt with th rotations ρ v and σ v if and only if it has th followin proprty: In th rotation τ v, btwn two conscutiv ds i 1 and i E H (and btwn δh (v) and 1 ) in th rotation ρ σ, th d squncs c i 1, c i 2,..., c i δ i and d i 1, d i 2,..., d i δ ar intrlavd. i By Lmma 5.2, th conclusion follows. COUNTING CONSISTENT ROTATION SYSTEMS W hav now arrivd at th oal of this sction. Thorm 5.4. Lt (G, H) and (G, H ) b subraph-rootd raphs and η : H H an isomorphism. Lt ρ and σ b rotation systms for G and G such that σ = η ρ η 1. For ach vrtx v V H, w hav th rotation (ρ σ) v in H : v 1, v 2,..., v δ H (v) accordin to th followin paramtr valus: δ H (v) is th dr of v in H. If δ H (v) 0, thn for j = 1,..., δ H (v), lt δ j and δ j b th numbrs of ds of E G and E G, rspctivly, that li btwn th ds j 1 and j in th rotation ρ v and in th rotation σ v, rspctivly (whr j 1 is takn to b n if j = 1). Thn th numbr of rotation systms for (G η G, H η H ) that ar consistnt with ρ and σ is ( ) δ δg (v) + δ G (v) 1 H (v) ( δi (v) + δ i δ G (v) (v) ) (5.3) δ G (v) δ i (v) v V H δ H (v)=0 v V H δ H (v)>0 i=1 Proof. This follows from Proposition 5.1 and Proposition 5.3. Thorm 5.5. Lt F b a family of raphs of trwidth k and maximum dr, ach of which is rootd on an isomorphic copy of a spannin subraph of K k+1. Lt (G, H), (G, H ) F, and lt η : H H b an isomorphism. Lt ρ and σ b rotation systms for G and G such that σ = η ρ η 1. Thn th numbr of rotation systms for (G η G, H η H ) that ar consistnt with both ρ and [ σ is( at most )] k (5.4) That is, it has an uppr bound that is indpndnt of th numbr of vrtics of th raph G. Proof. This thorm is a corollary of Thorm 5.4.

18 18 Ars Math. Contmp. x (xxxx) 1 x SPECIAL CASE: AMALGAMATING ACROSS AN EDGE Th simplst cas of amalamatin raphs (G, H) and (G, H ) across isomorphic root-subraphs with ds is th cas in which th root-subraphs ar isomorphic to K 2. Th subcas in which both ndpoints of both root-ds ar 2-valnt was dvlopd in [18]. Corollary 5.6. Lt ρ and σ b rotation systms for th rootd raphs (G, vw) and (G, v w ), whr th pastin matchs vrtics v and w with vrtics v and w, rspctivly. Thn th numbr of mbddins of (G η G, vw) that ar consistnt with ρ and σ is ( dg v + d G v )( 2 dg w + d G w ) 2 d G v 1 d G w 1 (5.5) Proof. Evry mbddin of (G η G, vw) that is consistnt with ρ and σ has its rotations compltly dtrmind by ρ and σ, xcpt at th ima of th vrtics v and v and at th ima of th vrtics w and w. Fiur 5.2 illustrats th situation. Th nams v and w for vrtics in G η G adhr to th notational convntion introducd in 3. d 1 d 2 v G d 1 ' v' d 2 ' G' d 1 ' d 2 ' d 1 d 2 G G' v w w' 1 ' w ' 2 ' ' 3 Fiur 5.2: Amalatin two raphs on an d. Formula (5.5) follows from application of Thorm 5.4. REMARK. In Corollary 5.6, th notation vw for th root-d of th raph G η G is consistnt with th notational convntion ivn in 3 for namin th root-vrtics and rootds of G η G. 6 Production Ruls for Graph Amalamations DEF. A production for a raph opration is a rul that prscribs th ffct on th nus distribution of applyin that opration to its oprands. That is, basd on th partials of th oprands and th nra of thir rspctiv mbddin surfacs, it says how many mbddins th rsultin raph has of ach nus and of ach typ of partial. Th oprands and th opration appar at th tail of a riht-arrow and ar calld th antcdnt of th production. Th consquntial information appars at th had of th riht-arrow and is calld th consqunt of th production.

19 Ars Math. Contmp. x (xxxx) 1 x 19 Runnin Exampl Part 4. A sinl production suffics to calculat th partial nus distribution (4.3) of th compound framnt G[h; b, ; ] from partial nus distributions (4.1) and (4.2), for th nod-framnts G[;, ; h] and G[;, ; b], rspctivly: (h, h) i (b, b) j [; h,h; b,b] (b, b, ) i+j + (b, b, ) i+j +(b b ) i+j+1 + (b b ) i+j+1 REMARK. W obsrv that this production is asymmtric. That is, it convrts th pivot vrtx h of th arlir nod-framnt in th post-ordr into a bullt. This particular form of production is dsind for rassmblin a raph from a full dcomposition tr by itrativ amalamation of framnts. In ach cas, all of th vrtics of ach of th two amalamand-framnts li in a sinl nod within that framnt. For trwidth k, th two nods intrsct in k vrtics. W obsrv that th subscript [; h, h; b, b] of th arrow oprator contains thr lists of ds. 1. Th first list is th st of ds that li in th root-subraphs of both amalamand framnts. 2. Th scond list contains th othr ds that join vrtics within th prsntin nod of th first amalamand framnt. 3. Th third list contains th othr ds that join vrtics within th prsntin nod of th scond amalamand framnt. Th antcdnt of ach production is a root-phras with an intr variabl as subscript, followd by an astrisk dnotin amalamation, and thn anothr root-phras with anothr intr variabl as subscript. Th consqunt of ach production is a sum of rootphrass, ach of which has as its subscript a formula ivin th nus of th surfac of th amalamatd raph that corrsponds to th two oprand mbddins. Th cofficint of ach such subscriptd partial indicats in how many ways an fb-walk corrspondin to that partial can b cratd by amalamatin two oprand mbddins that corrspond to th two partials in th antcdnt. Th sum of th cofficints is subjct to th uppr bound of Formula 5.3. If th two root-phrass in th antcdnt ar inconsistnt on th root-subraph (which can occur for trwidth four or mor), thn th consqunt is mpty. W us th two partials in th antcdnt and th first list of ds in th subscript of th arrow oprator to construct rotation systms for th st of mbddins of th amalamatd raph that ar consistnt with thos two partials. Fiur 6.1 illustrats th situation for Runnin Exampl Part 4. For ach of ths rotation systms, w us th Hfftr-Edmonds fac-tracin alorithm to calculat th numbr of facs in th consqunt mbddins rlativ to th sum of th numbrs in th antcdnt mbddins, aftr which w us th Eulr polyhdral quat to calculat th corrspondin incrmnt or dcrmnt in nus. Th tim rquird dpnds only on th root-subraphs, not on th numbrs of vrtics in th amalamands.

20 20 Ars Math. Contmp. x (xxxx) 1 x h h h h h b b b Fiur 6.1: Drivin a production. b b Thorm 6.1. Lt G b a family of subraph-rootd raphs of trwidth at most k and maximum dr at most, whr k and ar fixd positiv intrs. Lt (G, H), (G, H ) G, and lt η : H H b an isomorphism. Thn th numbr of productions rquird to calculat th pd of (G η G, H) from th pd s of (G, H) and (G, H ) is at most ( [(k + 1) ]! (!) k+1 ) 2 p((k + 1) ) 2 (k+1) That is, th numbr of productions has an uppr bound that is indpndnt of th numbrs of vrtics of th raphs G and G. Proof. On nds a production for ach ordrd pair of partials usd, and no mor. Thus, th numbr of productions ndd is at most th squar of th numbr of partials dtrmind by Thorm 4.2. RECURRENCES AND CLOSED FORMS Productions wr usd in [10], and thn in [7], [18], and [15] to driv systms of simultanous rcurrnc rlations for pds. Usin nratin functions on th simultanous rcurrncs, on can somtims driv closd forms, as in [5]. 7 Gnus Distribution Alorithm A skltal vrsion of th alorithm for calculatin th nus distribution of th raphs in a family G of raphs of trwidth k and maximum dr at most G is straihtforward and intuitiv. Alorithm 7.1 (Gnus Distribution Alorithm). Input: an n-vrtx raph G of trwidth k and maximum dr Output: a pd and th nus distribution for G. Commnt: INITIALIZE 1. Calculat a full dcomposition tr T of width k for G.

21 Ars Math. Contmp. x (xxxx) 1 x Dtrmin a post-ordr for dcomposition-tr T, with th lins l 1, l 2,..., l n k 1 in th post-ordr, so that th j th amalamation is across lin l j. 3. For j = 1,..., n k 1, (a) Construct th subraph-rootd nod-framnt (M j, M j, u j ) as th inducd raph on th j th nod of T in th post-ordr. (b) Idntify th partnr (M j, M j, u j ), and construct th isomorphism η j : M j {u i } M j {u i } Commnt: Partnr is dfind in 3. (c) Calculat th pd of th nod-framnt (M j, M j, u j ). 4. Lt (F 1, H 1, u 1 ) = (M 1, M 1, u 1 ), and lt (F 1, H 1, u 1) b its partnr, i.., lt (F 1, H 1, u 1) = (M 1, M 1, u 1) Commnt: MAIN LOOP REASSEMBLES G 5. For ach lin l j of th dcomposition tr (a) Amalamat th two framnts (F j, H j, u j ) and (F j, H j, u j ) via th isomorphism η j to produc th framnt F j+1. (b) Lt H j+1 b whichvr of M j or M j is th lattr nod-framnt in th postordr. Lt th pivot u j+1 b whichvr vrtx of th root-subraph H j+1 will not b pastd in th nxt amalamation that involvs th framnt F j+1. (c) Us productions to calculat th pd of (F j+1, H j+1, u j+1 ) from th pd s of (F j, H j, u j ) and (F j, H j, u j ). Commnt: CALCULATE i s BY SUMMING. N.B. W rcall from 4 that for a subraph-rootd raph (G, H), w dnot th corrspondin st of partials by P H. 6. For i = 0,..., β(g)/2, calculat i (G) by th formula i (G) = π i (G) π P Hn k BREADTH OF THE GENUS RANGE Analysis of Alorithm 7.1 uss th concpt of bradth of th nus ran. DEF. Th bradth of th nus ran of a raph G is ivn by th quation γ B (G) = γ max (G) γ min (G) + 1 Thus, γ B (G) is qual to th numbr of valus of th indx i such that i (G) 1.

22 22 Ars Math. Contmp. x (xxxx) 1 x Proposition 7.1. Th bradth of th nus ran of any raph G is at most (2 E G + 3)/6. Proof. γ B (G) = γ max (G) γ min (G) + 1 β(g) 2 γ min (G) + 1 E G V G = 2 E G E G 3 V G Corollary 7.2. Lt G b a family of simpl raphs of maximum dr at most G. Thn th bradth of th nus ran of any n-vrtx raph G in G is at most (n G + 3)/6. Proof. Sinc 2 E G n G, by Eulr s thorm on dr sum, this follows immdiatly from Proposition 7.1. ANALYSIS OF THE ALGORITHM As stablishd by [26], assinin an uppr bound to th dr of a raph dos not rduc th computational complxity of th minimum nus problm. Thus, boundin th trwidth is ssntial to th computational complxity of our alorithm. Proposition 7.3. Lt G b a family of subraph-rootd raphs, ach of trwidth at most k and of maximum dr at most, whr k and ar fixd positiv intrs. Lt (G, H, u), (G, H, u ) G and lt η : H {u} H {u } b an isomorphism. Thn th tim rquird to calculat th pd of (G η G, H, u) from th pd s of (G, H, u) and (G, H, u ) is in O( V G V G ). Proof. Th tim dpnds prdominantly on th numbr of applications of productions, which dpnds, in turn, only on th product of th numbrs of partials in th rspctiv pd s of G and G with non-zro subscriptd cofficints. Th numbr of non-zro subscriptd cofficints of th partials dpnds on th bradths of thir nus rans. This thorm now follows from Corollary 7.2 and Thorm 4.2. Thorm 7.4. Lt G b a family of raphs of trwidth at most k and maximum dr at most, whr k and ar fixd positiv intrs. Thn th tim ndd to calculat th nus distribution of an n-vrtx raph G F, startin from a full dcomposition tr T for G, is in O(n 2 ). Proof. W procd stp by stp.

23 Ars Math. Contmp. x (xxxx) 1 x 23 Stp 1. A full dcomposition tr T for th raph G can b calculatd in tim proportional to V T, which is in O(n), by Corollary 2.7. Stp 2. Th post-ordr of th dcomposition tr T can b calculatd in tim proportional to V T, which is in O(n). Substp 3a. Th tim ndd to construct ach of th nod-framnts M 1,..., M n k from th ivn raph G is in O(k 2 ), sinc ach nod of th dcomposition tr T has k + 1 vrtics. Th tim for this substp is in O(1). Substp 3b. Each partnr can b locatd by a post-ordr travrsal. Each isomorphism η j is an artifact of th tr dcomposition. Th tim for this substp is in O(1). Substp 3c. Thr is a fixd uppr bound of (k!) k+1 for th numbr of mbddins of M j, sinc M j is a subraph of th complt raph K k+1. Th tim ndd to dtrmin th appropriat subscriptd partial for ach mbddin of a nod-framnt, by usin th Hfftr-Edmonds alorithm, is in O(k Mj ), that is, linar in th numbr of ds of th nod-framnt, and constant, accordinly, with rspct to V G. Stp 3 total. Sinc thr ar n k nod-framnts, by Proposition 2.5, it follows that th total tim ndd for Stp 3 is in O(n). Stp 4. This initializin stp taks O(1)-tim. Substp 5a. Th tim ndd to calculat a rprsntation of th amalamatd raph F j+1 from rprsntations of th amalamands is proportional to E Fj + E F j, which is linar in V Fj + V F j, bcaus th dr of th raph G is boundd. This substp taks O( V Fj + V F j )-tim. Substp 5b. This substp is achivd by rfrrin to th post-ordr travrsal of tr T. Substp 5c. Th numbr of productions is constant, by Thorm 6.1. Th numbrs of subscriptd partials with non-zro cofficint in th pd s of th amalamand framnts ar proportional to γ B (F j ) and γ B (F j ), rspctivly. By Corollary 7.2, γ B(F j ) and γ B (F j ) ar proportional to V Fj and V F j, rspctivly. By Proposition 7.3, th tim for this substp is O( V Fj V F j ), subjct to th assumption that multiplication of intrs taks constant tim. Stp 5 total. Lt S dnot th squnc of pairs of framnts that occurs in th rassmbly of G from th nod-framnts corrspondin to th dcomposition tr T. Th tim for ach itration of th body of th loop in Stp 4 is dominatd by th tim ndd for Substp 5c. Thus, th total tim ndd to calculat th pd of G is at most (F,F ) S c V F V F = c (F,F ) S V F V F W furthr suppos that th numbr of vrtics in th nod-framnt F i is n i, for i = 1,..., n k, and w obsrv that in th total rassmbly no two nod-framnts occur

24 24 Ars Math. Contmp. x (xxxx) 1 x twic as subraphs of framnts that ar amalamatd. It follows that n k n k V F V F = n i n j n k n 2 i (F,F ) S i=1 j=1 i=1 < (n 1 + n n n k ) 2 = n 2 Stp 6. Of cours, ach trm i (G) in th nus distribution for G is obtaind from th pd by summin all th cofficints of subscriptd partials with subscript i. Th tim ndd for such a calculation of ach i (G) is proportional to th numbr of diffrnt partials, which is boundd by a constant. Thus, th tim ndd to calculat th nus distribution from th pd is proportional to th siz of th nus ran, and thus, in O(n). 8 Conclusions Stahl [22, 23] has calld a family of raphs H-linar if its mmbrs can b drivd by itrativ amalamation of copis of a raph H, and h has introducd a form of production matrics whos lmnts ar univariat polynomials, in which th dr of a trm corrsponds to an incrmnt of nus as an additional copy of th raph H is amalamatd to a rowin linar chain. Th trwidth of th raphs in an H-linar family is th tr width of th raph H. Althouh th siz of such matrics can b vry lar, corrspondin to th numbr of partial nus distributions associatd with a ivn maximum dr, it is of fixd siz. Accordinly, th tim-complxity ndd to tak a powr of such a production matrix dpnds only on th tim ndd to multiply two polynomials of linarly incrasin dr. Practical alorithms for th nus distributions and partitiond nus distributions of th raphs in various intrstin linar familis of raphs, implicit in [23, 10, 15, 18, 11, 12], fall within th quadratic tim-complxity uppr limit ivn by Thorm 7.4. Morovr, quadratic-tim calculation of nus distributions is implicit in [7, 19] for raph familis, includin circular laddrs and Möbius laddrs, that ar not H-linar, but which could b charactrizd as rin-lik. Byond that, thr is a practical quadratic-tim alorithm for th cubic Halin raphs [9], which ar a non-linar family. W obsrv that in a nus distribution calculation by our alorithm, th partials and productions can b nratd dynamically as ndd, rathr than in advanc. This susts th fasibility of implmntin such an alorithm for raphs of rasonably small trwidth and maximum dr, sinc th numbr of productions ndd miht b far smallr than th total numbr possibl for that trwidth and dr. Th xposition hr illustrats onc aain how boundin th trwidth can b usd to rduc othrwis NP-hard calculations rardin mbddins to polynomial tim. Hr w hav also boundd th dr. This immdiatly susts this nral problm: Rsarch Problm 1. Dtrmin whthr th nus distribution of th raphs of boundd trwidth can b calculatd in polynomial-tim, if th dr is not boundd.

25 Ars Math. Contmp. x (xxxx) 1 x 25 Of cours, rathr than bin contnt with an alorithm with such a vast prolifration of partials and productions, on hops for closd formulas or tractabl rcursions for intrstin classs of raphs. A rlatd lin of invstiation would involv boundin th trwidth and prscribin th minimum nus (which ffctivly bounds th avra dr). Th followin two problms may b approachabl. Rsarch Problm 2. Alorithms for calculatin th nus distributions of 3-rular and 4-rular outrplanar raphs ar ivn by [8] and [19], rspctivly. Calculat th nus distributions of arbitrary outrplanar raphs. Rsarch Problm 3. A Halin raph is obtaind from a plan tr with at last four vrtics and no vrtics of dr two, by drawin a cycl throuh th lavs. An alorithm for th nus distribution of any 3-rular Halin raph is ivn by [9]. Calculat th nus distributions of arbitrary Halin raphs. Rsarch Problm 4. A rmainin problm of slf-vidnt thortical intrst is dtrmination of a lowr bound on th tim ndd to calculat th nus distribution of a raph of fixd trwidth and boundd maximum dr. ACKNOWLEDGEMENT Th author wishs to thank Imran Khan and Mhvish Poshni for thir many valuabl commnts on th arly drafts of this papr. Th author also wishs to thank an anonymous rfr for various hlpful sustions. Rfrncs [1] L. W. Bink, R. J. Wilson, J. L. Gross, and T. W. Tuckr, Editors, Topics in Topoloical Graph Thory, Cambrid Univ. Prss, [2] H. L. Bodlandr, A linar-tim alorithm for findin tr-dcompositions of small trwidth, SIAM J. Comput. 25 (1996), [3] H. L. Bodlandr, A partial k-arbortum of raphs with boundd trwidth, Thortical Comp. Sci. 209 (1998), [4] R. B. Bori, R. G. Parkr, and C. A. Tovy, Rcursivly constructd raphs, 2.4 (pp ) of Handbook of Graph Thory, scond dition, ds. J. L. Gross, J. Ylln, and P. Zhan, CRC Prss (imprint of Taylor and Francis), [5] M. L. Furst, J. L. Gross and R. Statman, Gnus distribution for two classs of raphs, J. Combin. Thory (B) 46 (1989),

26 26 Ars Math. Contmp. x (xxxx) 1 x [6] J. L. Gross, Gnus distribution of raphs undr surry: addin ds and splittin vrtics, Nw York J. Math. 16 (2010), [7] J. L. Gross, Gnus distribution of raph amalamations: Slf-pastin at rootvrtics, Australasian J. Combin. 49 (2011), [8] J. L. Gross, Gnus distributions of cubic outrplanar raphs, J. of Graph Alorithms and Applications 15 (2011), [9] J. L. Gross, Embddins of cubic Halin raphs: a surfac-by-surfac invntory, Ars Math. Contmporana 6 (2013), [10] J. L. Gross, I. F. Khan, and M. I. Poshni, Gnus distribution of raph amalamations: Pastin at root-vrtics, Ars Combin. 94 (2010), [11] J. L. Gross, I. F. Khan, and M. I. Poshni, Gnus distribution for itratd claws, prprint 2013, 20pp. [12] J.L. Gross, T. Mansour, T.W. Tuckr, and D.G.L. Wan, Lo-concavity of combinations of squncs and applications to nus distributions, draft manuscript, [13] J. L. Gross and T. W. Tuckr, Topoloical Graph Thory, Dovr, 2001; (oriinal dn. Wily, 1987). [14] K. Kawarabayashi, B. Mohar, and B. Rd, A simplr linar tim alorithm for mbddin raphs into an arbitrary surfac and th nus of raphs of boundd tr-width, Proc. 49th Ann. Symp. on Foundations of Computr Scinc (FOCS 08) IEEE (2008), [15] I. F. Khan, M. I. Poshni, and J. L. Gross, Gnus distribution of raph amalamations at roots of hihr dr, Ars Math. Contmporana 3 (2010), [16] I. F. Khan, M. I. Poshni, and J. L. Gross, Gnus distribution of P 3 P n, Discrt Math. 312 (2012), [17] B. Mohar and C. Thomassn, Graphs on Surfacs, Johns Hopkins Univrsity Prss, [18] M. I. Poshni, I. F. Khan, and J. L. Gross, Gnus distribution of damalamations, Ars Math. Contmporana 3 (2010), [19] M. I. Poshni, I. F. Khan, and J. L. Gross, Gnus distribution of 4-rular outrplanar raphs, Elctronic J. Combin. 18 (2011) #P212, 25pp. [20] M. I. Poshni, I. F. Khan, and J. L. Gross, Gnus distribution of raphs undr slf-d-amalamations, Ars Math. Contmporana 5 (2012), [21] N. Robrtson and P. D. Symour, Graph minors. II, Alorithmic aspcts of tr-width, J. Alorithms 7 (1986),