Anti-Monotonic Overlap-Graph Support Measures

Transcription

1 Anti-Monotoni Overlp-Grph Support Mesures Toon Clers Einhoven University of Tehnology Jn Rmon Ktholieke Universiteit Leuven Dries Vn Dyk Hsselt University, Trnsntionl University of Limurg Astrt In grph mining, frequeny mesure is nti-monotoni if the frequeny of pttern never exees the frequeny of supttern. The effiieny n orretness of most grph pttern miners relies ritilly on this property. We stuy the se where the tset is single grph. Vnetik, Gues n Shimony lrey gve suffiient n neessry onitions for nti-monotoniity of mesures epening only on the ege-overlps etween the intnes of the pttern in lele grph. We exten these results to homomorphisms, isomorphisms n homeomorphisms on oth lele n unlele, irete n unirete grphs, for vertex n ege overlp. We show set of reutions etween the ifferent morphisms tht preserve overlp. We lso prove tht the populr mximum inepenent set mesure ssigns the miniml possile meningful frequeny, introue new mesure se on the minimum lique prtition tht ssigns the mximum possile meningful frequeny n introue new mesure snwihe etween the former two se on the poly-time omputle Lovász θ-funtion. Introution Reently, grph mining hs emerge s new fiel within ontemporry t mining tht got lot of ttention over the lst severl yers. The entrl tsk is to fin sugrphs, lle ptterns tht our frequently in either olletion of grphs, or in one lrge grph. Espeilly in the single-grph setting, the notion of frequeny, however, is not t ll strightforwr. For exmple, the nïve solution of tking the numer of instnes of the pttern s its frequeny hs the unesirle property tht extening pttern (i.e., mking it more restritive), my inrese its frequeny. Hene, s pointe out y Vnetik, Gues n Shimony [], goo frequeny mesure must e suh tht the frequeny of super-pttern is lwys t most s high s tht of supttern. This property is lle nti-monotoniity. Also for resons of effiieny, nti-monotoniity of the frequeny mesure is highly esirle, s it llows for pruning lrge prts of the serh spe. The effiieny n orretness of most grph pttern miners relies ritilly on the nti-monotoni property of the use frequeny mesure. An importnt lss of nti-monotoni support mesures in the single grph setting is se on the notion of n overlp grph grph in whih eh vertex orrespons to mth of the pttern n two verties re onnete y n ege if the orresponing mthes overlp. Vnetik, Gues n Shimony prove neessry n suffiient onitions for nti-monotoniity in the single, lele grph setting, in whih the verties of the overlp grph represent sugrphs of the t set isomorphi to the pttern, n the eges represent ege overlp [] etween the sugrphs. In the ontext of grph mining, however, not only sugrph isomorphism n lele grphs re importnt. On the one hn, the importne of homeomorphi se grph mining inrese rstilly with the stuy of iologil networks [, 6]. On the other hn, in pplitions where verties n ply severl roles (e.g. soil networks) homomorphism is more suitle. Homomorphism in the ontext of t mining hs een thoroughly investigte in the fiel of inutive logi progrmming []. In this pper we exten the results of Vnetik, Gues n Shimony to these settings s well. Our min ontriutions re:. We stuy systemtilly ll 4 omintions of iso-, homo-, or homeomorphism, on lele or unlele, irete or unirete grphs, with ege- or vertexoverlp n exten the nti-monotoniity results.. In our proofs, we use reutions whih re lso of interest in their own right, s they llow to trnsfer results for ifferent types of morphisms n overlp from one setting to nother. For n overview of the ifferent reutions, see Figure.. An interesting onsequene of the reutions is tht ny unlele, unirete grph is potentil vertexn ege-overlp grph in ll onsiere settings.

2 G, Homo e v R G, Homo v v G λ e v R Gλ v v, Homo (Th. ), Homo (Th. ) G, Homeo G, Homeo G, Iso Gλ R, Iso [], Th. R e v R Gλ v v, Homeo G, Iso Gλ R, Iso [], Th. R e v R Gλ, Homeo v v Figure. Overview of the reutions. R, R, n R re proven respetively in Theorems 5, 9, n ; the she n the otte rrows in Prop.. Arrows representing reutions hnging γ-overlp into γ -overlp re lele with γ γ. 4. We show tht (uner resonle ssumptions) the mximum inepenent set mesure (MIS) of Vnetik, Gues n Shimony [] is the smllest ntimonotoni mesure in the lss of overlp-grph se frequeny mesures. We lso introue the new minimum lique prtition mesure (MCP) whih represents the lrgest possile one. 5. In generl, oth the MIS mesure n the MCP mesure re NP-hr to ompute in the size of the overlp grph. The Lovász mesure is omputle in polynomil time n is snwihe etween the former two mesures. We show tht the Lovász mesure inues n nti-monotoni overlp-grph se frequeny mesure. Due to spe limittions, we omit the proofs for our theorems n only stte the results n intuitions in this version. In Setion we review si onepts from grph theory. In Setion we introue overlp grphs, support mesures n reutions etween the ifferent settings. Next, in Setion 4 we present our results on miniml, mximl n polytime omputle meningful overlp mesures. In Setion 5 we present our reutions n use them to exten the results to ll 4 settings. Finlly, we onlue in Setion 6. Preliminries We ssume tht the reer is fmilir with most si grph theoreti notions n omputtionl omplexity. Any textook in these res, suh s [] n [] supply neessry kgroun. Grphs A grph G = (V, E) is pir in whih V is (non-empty) set of verties or noes n E is either set of eges E {{v, w} v, w V, v w} or set of rs E {(v, w) v, w V, v w}. In the ltter se we ll the grph irete. A lele grph is quruple G = (V, E, Σ, λ), with (V, E) grph, Σ non-empty finite, totlly orere set of lels, n λ funtion V Σ ssigning lels to the verties. We use the nottion V (G), E(G) n λ G to refer to the set of verties, the set of rs (eges) n the leling funtion of grph G, respetively. By G, we enote the lss of ll grphs; y G (G ), the restrition to irete (unirete) grphs; n y G λ (G ) the restrition to lele (unlele) grphs. We often omine nottion; e.g., G for irete, unlele grphs. Morphisms The following onepts introue for Gλ re lso vli for unirete n/or unlele grphs y ropping the iretion of the eges n/or the vertex-lels. A homomorphism π from H = (V H, E H, Σ, λ H ) to G = (V, E, Σ, λ) is mpping from V H V, suh tht (v, w) E H : (π(v), π(w)) E. If suh π exists, we sy tht H is homomorphi to G.We ll π ege-surjetive if (v, w ) E (v, w) E H : π(v) = v π(w) = w n ll it surjetive if it is oth vertex- n ege-surjetive. An isomorphism from H to G is ijetive homomorphism π from H to G. In tht se, we sy tht H is isomorphi to G n write H = G. We use H G to enote tht H = g, for some sugrph g of G. A pth of length k in G is sequene of verties (v 0,..., v k ) with (v i, v i ) E. The verties v,..., v k re lle the inner verties n v 0, v k the en verties of the pth. Two pths P n P of G re lle isjoint or inepenent if no inner noe of P is in P n vie vers. The set of ll pths of G is enote P G, n of ll pths with en verties v n w, P G (v, w). A sugrph homeomorphism π from H to G is pir of injetive mppings from V (H) V (G) n from E(H) P G, suh tht (v, w) E(H): π((v, w)) P G (π(v), π(w)) x π((v, w)) : y V (H) \ {v, w} : π(y) x, n (v, w), (x, y) E(H):

3 P 4 5 P 4 () () 5 G 5 () 4 () Figure. Exmples of the ifferent morphisms. An isomorphi imge of P (), P (), homomorphi imge of P () n homeomorphi imge of P (). The eges of the sugrph to whih pttern is mppe re in ol. The imge of vertex of the pttern is lele with its ientifier. (v, w) (x, y) π((v, w)) n π((x, y)) isjoint [0]. We ll π surjetive if v V (G) n e E(G): [( v V (H) : v = π(v)) ( e E(H) : v π(e))] [ e E(H) : e π(e)]. By Homo, Iso n Homeo, we enote the lss of grph homomorphisms, isomorphisms n homeomorphisms, respetively. If for π : H G {Homo, Iso, Homeo} it hols tht λ H (v) = λ G (π(v)), we ll π lel-preserving. We will lwys impliitly ssume tht π is lel-preserving when H, G G λ. Exmple. Fig. illustrtes the introue morphisms for unlele, unirete grphs. Note tht in () the verties,4 n,5 re mppe to the sme vertex n in () the eges {, } n {, } re mppe to pths of length. Support mesures n overlp grphs Definition. A support mesure on Gβ α is funtion f : Gβ α Gα β N tht mps (P, G) to f(p, G) where P is lle the pttern, G is lle the tset grph n f(p, G) is lle the support of P in G. For effiieny resons, most grph mining lgorithms use level-wise or epth-first pproh to generte frequent ptterns, expning smller ptterns to lrger ones, whih requires n nti-monotoni support mesure: Definition. A support mesure f on Gβ α is nti-monotoni iff p, P, G Gβ α : p P f(p, G) f(p, G). Most support mesures re se on the mthes of pttern in grph: Definition 4. Let K {Homo, Iso, Homeo} n P, G Gβ α, α {, }, β {λ, }. A K-mth of P in G is sugrph g G for whih there exists surjetive mpping π K from P to g. An iniviul mpping π from P to g is lle n emeing of P in G. We ll n Iso-mth of P in G n instne of P in G. However, just ounting the numer of K-mthes of pttern in G oes not result in n nti-monotoni support mesure, s lrger ptterns my hve more mthes (e.g, in Figure, there re more instnes of P in G thn tringles).. Overlp grph Most nti-monotoni mesures in single grph setting re se on the notion of n overlp grph G γ P [, 9] : Definition 5. Let P, G Gβ α, α {, }, β {λ, }. Two sugrphs g n g of G hve vertex-overlp if V (g ) V (g ) n ege-overlp if E(g ) E(g ). Let γ {vertex, ege} n K {Homo, Iso, Homeo}. The K-γ-overlp grph G γ P of pttern P in the tset G is n unirete, unlele grph in whih eh vertex orrespons to K-mth of the pttern P n two verties re onnete if the orresponing K-mthes hve γ-overlp. Note tht G γ P is lwys unirete n tht the eges epen on the use notion of overlp. For exmple, G γ P will e enser for vertex-overlp thn for ege-overlp euse the ltter implies the former. Let p, P, G Gβ α, γ {vertex, ege}, α {, }, β {λ, }, n K {Homo, Iso, Homeo}. Throughout the rtile, P enotes the (super)pttern, p P the supttern n G the tset, single grph. G γ P (Gγ p ) is the K-γ-overlp grph of P (p) in G. Vnetik, Gues n Shimony [] onsier three opertions on the overlp grph G γ P : lique ontrtion, ege removl n vertex ition, s efine elow. Definition 6. Let G = (V, E) G. Let K G e lique in G. The lique ontrtion CC(G, K) yiels new grph G = (V, E ) in whih the sugrph K G is reple y new vertex k / V jent to {w v V (K) : {v, w} E}: V = V \ V (K) {k} E = E \ {{v, w} {v, w} V (K) } {{k, w} v V (K) : {v, w} E}. [] uses the term instne grph inste of overlp grph. The term instne suggests the use of isomorphisms, n we onsier support mesures se on ny kin of morphism, we follow the terminology of [9] to voi onfusion.

4 The ege removl ER(G, e) of the ege e = {v, w} yiels new grph G = (V, E \ {{v, w}}). The vertex ition VA(G, v) of the vertex v / V yiels new grph G = (V {v}, E {{v, w} w V }). The rtionle ehin these opertions is tht the K-γoverlp grph of P n e trnsforme into the K-γ-overlp grph of p y mens of these opertions. This n e seen se on the following two oservtions: Oservtion Any K-mth of P ontins K-mth of p. Oservtion Let g, g e two K-mthes of P n g g n g g e two K-mthes of p. If g n g hve γ-overlp, so o g n g. These onitions hol for ll settings onsiere in this rtile. We quikly sketh the min ies of the trnsformtion proess n refer to [] for the full etils. For resons of simpliity we ssume tht p ontins t lest one ege. Let g G e mth of p, n let super(g ) e ll mthes of P in G ontining g. Beuse of Oservtion, every mth g of P in G must e in t lest one super(g ). Beuse of Oservtion, super(g ) forms lique in G γ P, s they ll overlp on g. Furthermore, if there is n ege {g, g } in G γ p, there is n ege etween ny two g super(g ) n g super(g ) in Gγ P. As suh, n inue sugrph of G γ p n e forme y susequently ontrting the liques super(g ) until for ll g G γ p, either super(g ) is empty, or singleton. It is esy to see tht one n go from n inue sugrph of G γ p to G γ p: first ll verties not in the inue sugrph with noe itions, n then remove spurious eges with ege removls.. Overlp support mesure Definition 7. A grph mesure is funtion ˆf : G N. Let o e grph opertion tht trnsforms grph G into grph o(g). A grph mesure ˆf is inresing uner o if n only if G G : ˆf(G) ˆf(o(G)). Definition 8. Let α {, }, β {λ, }, γ {vertex, ege} n K {Homo, Iso, Homeo}. A support mesure f on Gβ α is K-γ-overlp support mesure on Gβ α, if there exists grph mesure ˆf suh tht P, G G α β : f(p, G) = ˆf(G γ P ). Informlly, n overlp support mesure is support mesure tht only epens on the overlp grph. Note tht the ssoite grph mesure ˆf is lwys unique. An exmple of n nti-monotoni overlp support mesure is the mesure tht ssigns to every pttern P the size of the mximum inepenent set (MIS) [] of G γ P ; tht is, the support is the mximl numer of mthes tht fit in G without overlp. The min result of this rtile is the generliztion of the following theorem of Vnetik, Gues n Shimony []: Theorem 9 (Vnetik, Gues, Shimony). Let α {, }. Any Iso-ege-overlp support mesure f on Gλ α is nti-monotoni if n only if the ssoite grph mesure ˆf is non-eresing uner lique ontrtion, ege removl n vertex ition. We exten it to the omplete spe efine y the prmeters α, β, K n γ. More formlly: Theorem 0. Let α {, }, β {λ, }, K {Iso, Homo, Homeo}, n γ {vertex, ege}. Any K-γ-overlp support mesure f on Gβ α is nti-monotoni if n only if the ssoite grph mesure ˆf is inresing uner lique ontrtion, ege removl n vertex ition. The proof of suffiieny, i.e., tht ny K-γ-overlp support mesure f is nti-monotoni if the ssoite grph mesure is inresing uner CC, VA n ER follows immeitely from the ft tht G γ P n e trnsforme into Gγ p y these opertions. To prove neessity, Vnetik, Gues n Shimony onstrut for every unlele grph H n every opertion o, triple (P, p, G) (where P is super-pttern, p supttern n G tset) suh tht G γ P = H n G γ p = o(h). So if f woul not e inresing uner some o {CC, ER, VA}, there woul e H suh tht f(h) > f(o(h)) n one oul onstrut G, P n p suh tht f(g, P ) > f(g, p), whih woul men tht f is not nti-monotoni. We follow the sme pproh.. Reutions The neessity proofs for most settings re se on reutions from K-mthes for G α β to K -mthes for G α β. Definition. Let K, K {Iso, Homo, Homeo}, α, α {, }, β, β {, λ}, n γ, γ {ege, vertex}. A K, γ-overlp on Gβ α to K, γ -overlp on Gβ α reution is funtion R : (Gβ α) (Gβ α ) tht mps triplet (p, P, G) to triplet (p, P, G ) suh tht: () p P iff p P n () G γ p = G γ p Gγ P = G γ P. Note tht this efinition oes not utomtilly imply tht the numer of K-emeings of P in G equls the numer of emeings of P in G, s P might hve more/less utomorphisms thn P. The following property gives reutions from unlele to lele grphs, n from unirete to irete grphs. Property. For ll α {, }, γ {vertex, ege}, β {λ, }, K {Iso, Homo, Homeo}, there exist reutions:

5 from K, γ-overlp on G α to K, γ-overlp on Gα λ ; n from K, γ-overlp on G β to K, γ-overlp on G β. An overview of the reutions whih follow from our results in Setion 5 re shown in Figure. G P G e P Miniml, Mximl n PTIME overlp support mesures Let G = (V (G), {{v, w} v, w V } \ E(G)), enote the omplement grph of G G. E.g., for the omplete grph on k verties, K k = ({v,..., v k }, {{v i, v j } i j k}), K k is the grph with k isolte verties. We ll n overlp support mesure f meningful if it is nti-monotoni n ssigns the frequeny k to k nonoverlpping mthes, i.e., ˆf(K k ) = k. An inepenent set of G is suset I of V (G) suh tht v, w I : {v, w} / E(G). A mximum inepenent set (MIS) of G is n inepenent set of mximum rinlity n its size is notte s mis(g). Up to now, ll meningful overlp support mesures f we re wre of re MISmesures, i.e., the support of f(p, G) = mis(g γ P ). MIS ws introue n proven to e nti-monotoni in []. A more ompt proof n e foun in [4]. 4. MCP-mesure We introue new nti-monotoni overlp support mesure, inspire y the CC-opertion: Definition. A lique prtition of G G is prtitioning of V (G) into {V,..., V k } suh tht eh V i inues lique in G. A minimum lique prtition (MCP) is lique prtition of minimum size. Its size is enote mp(g). The MCP-mesure is efine y MCP(P, G) : (P, G) mp(g γ P ). Theorem 4. Let K {Iso, Homo, Homeo}, γ {vertex, ege}, n α {, }, β {λ, }. The MCPmesure is n nti-monotoni K-γ-overlp mesure on G α β. It is interesting to ompre MCP with MIS. Let χ(g) e the hromti numer of G, i.e., the miniml numer of olors to olor the verties of G suh tht no two verties with the sme olor re jent, n let ω(g) e the lique numer; the size of the lrgest lique in G. First, it is known tht mp(g) = χ(g) n mis(g) = ω(g) (see, e.g., [5], setion 5.5.). Consequently, mp(g) mis(g), G G, sine the size of mximum lique is lower oun for the hromti numer. Informlly, it is esy to see why this is so: let {V,..., V k } e n MCP n I MIS for G. We know tht I ontins t most one vertex v i of eh V i, i k. Figure. Left: A pttern P n grph G. The 5 Iso-mthes of P in G re inite y the imge in G of the eges outsie the tringle. Right: The Iso-ege-overlp grph G e P with MCP (she ellipses) n MIS (white verties). In other wors, to eie whether we n inlue mth of V i, MIS fores us to hoose either no mth or extly one mth v i, whih must e inepenent of ll hosen v j V j. MCP, however, llows us to ount mth in V i s soon there is mth in V i whih oes not overlp with mth in V j. Tht is, we n mke nother hoie for eh (V i, V j ) pir. Exmple 5. Let us look t n exmple: onsier pttern P n the grph G s shown in Figure. The 5 Iso-mthes of P re inite y n ientifier on the imge in G of the eges outsie the tringle of P. The Iso-ege-overlp grph G e P of P in G is shown on the right in Figure n is isomorphi to pentgon. The white verties mrk the MIS {, } n the she ellipses mrk the MCP {{}, {, }, {4, 5}} of G e P. Hene, if we ount mth with MIS, we n only tke mth or mth 4 s seon inepenent mth, euse n 4 overlp, leing to MIS-support of. This is it unnturl, euse eh of the mthes of the tringle n e extene to mth of P in wy tht they o not overlp with eh other, whih woul le to support of of P. This more nturl notion of ounting inepenent mthes is extly wht MCP-support llows us to o: we o not ount iniviul mthes, ut groups of mthes of P shring mth of supttern p ( tringle) n llow to swith mthes to eie whether group is inepenent of n other. In this exmple, the group {} is inepenent of the groups {, } n {4, 5}, euse it oes not overlp with mth respetively mth 4 n the group {, } is inepenent of the group {4, 5} euse, for instne, mth n mth 5 o not overlp. 4. Bouning theorem n PTIME overlp support mesure Interestingly, MIS n MCP turn out to e the miniml n the mximl possile meningful overlp mesures:

6 Theorem 6. Let K {Iso, Homo, Homeo}, γ {vertex, ege}, α {, }, n β {λ, }. For every meningful K-γ-overlp mesure f on Gβ α, n every P, G Gβ α, it hols tht: MIS(P, G) f(p, G) MCP(P, G). Proof. We use Theorem 0 to show oth the minimlity of MIS n the mximlity of MCP. Let H = G P, let mis(h) = k, n let I = {v,..., v k } e MIS for H. Strting from the grph ({v,..., v k }, ) we n the verties V (H) \ I using VA n remove eges not in E(H) y ER. Sine f is meningful, it is ntimonotoni n therefore ˆf nnot erese fter eh step, strting from ˆf(({v,..., v k }, )) = k. As suh, ˆf(H) is lrger thn or equl to k = mis(h). On the other hn, let mp(h) = k, n let {V,..., V k } e n MCP for H n let H = CC(... CC(CC(H, V ), V )..., V k ). H oes not hve eges: if it woul, then joining the two liques tht were ontrte to two onnete verties of H woul give us smller lique prtition. Beuse f is nti-monotoni, ˆf is inresing uner CC n thus ˆf(H) ˆf(CC(H, V )) ˆf(H ) = ˆf(K k ) = k. Unfortuntely, oth mis n mp re known to e NPhr to ompute in the size of the overlp grph. This les us to the following question: oes there exist meningful overlp mesure whih is effiiently omputle? A well-known mesure tht is snwihe etween mis n mp n tht n e ompute in polynomil time, is the thet funtion, lso known s the Lovász funtion [8]. There re severl equivlent hrteriztions of this funtion. The most onise efinition is proly: θ(g) = min A λ mx (A), where λ mx (A) enotes the lrgest eigenvlue of mtrix A n the minimum is tken over ll fesile mtries A suh tht A = A, A ii = n A ij = if (i, j) E(G). Theorem 7. θ is meningful overlp mesure. 5 Neessity for other morphisms n grph settings In the previous setions we onsiere mesures on overlp grphs tht re nti-monotoni w.r.t. the opertions VA, ER n CC. Let us now return to the onnetion etween mesures with this property n nti-monotoni grph support mesures se on overlp grphs. [] showe tht when onsiering unlele, unirete grphs uner sugrph isomorphism, grph support mesure se on the ege-overlp grph is nti-monotoni if n only if the orresponing grph mesure on the ege-overlp grph is ntimonotoni w.r.t. VA, ER n CC. In this setion, we generlize this result to ll 4 settings. We o this y proving se se n then pplying reutions from ll the other settings to this se se. 5. Neessity for lele homomorphisms n isomorphisms As se se, we prove the neessity of the noneresingness of ˆf uner the three grph opertions for the nti-monotoniity of f for K, γ-overlp on Gλ α, for ll omintions of K {Iso, Homo}, γ {vertex, ege}, n α {, }. The proof will not rely on reutions, ut show the neessity iretly for these ses. We show only the unirete se, s the proof for the irete se is very similr. Notie lso tht the irete se follows from the unirete-to-irete reutions shown in Property. We will essentilly use invrints uner sugrph homomorphism to fore n injetive homomorphism; i.e., to ensure isomorphism. Let G G. The o girth g o (G) of G is the size of smllest yle of o length in G. The istne G (v, w) is equl to the length of shortest pth from v to w in G. If no suh yle respetively shortest pth exist, we efine g o (G) respetively G (v, w) equl to. We will use the following well known invrints [7] to fore eh sugrph homomorphism into n isomorphism: Proposition 8. If H, G Gβ α, α {, }, β {λ, } for whih there exists homomorphism π : H G, then. g o (H) g o (G),. v, w V (H) : H (v, w) G (π(v), π(w)). Remrk tht the first invrint implies tht yle of length k, with k o, n only e mppe y homomorphism to n o yle of length t most k. We will use the following grphs s ptterns: Definition 9. Let k + e n o integer. B k enotes the grph in Gλ efine y: V (B k ) = V V V V, V = {,..., k,,..., k }, V = {,..., k+ }, V = {,..., k+ }, V = {,..., k,,..., k+ }, E(B k ) = E E E E k i= {{ i, i }, { i, i }} { k+, k+ }, E = k i= { i, i }, E = { k+, } k i= { i, i+ }, E = { k+, } k i= { i, i+ }, E = k i= { i, i }, λ Bk (u) = x, u V x, x =,,,. We ll the eges { i, i } rms, the eges { i, i } legs, i k, the yle inue y V the torso n the yle inue y V the hip.

7 torso rm B k L k hip Figure 4. The grphs B k n L k. leg L k Gλ enotes the sugrph of B k inue y V V n is lle the lower oy of B k. An illustrtion of oth grphs is shown in Figure 4. Let P [], P [] n p[], p[] e two instnes of P = B k respetively p = L k in lrger grph G n let super(g) e equl to ll mthes of P in G ontining g G. We will use four types of overlp (see Figure 5): lower oy overlp: P [] n P [] shre the omplete lower oy, whih is single instne of p, resulting in two jent verties in G γ P n single vertex in Gγ p, leg overlp: P [] n P [] shre leg, resulting in two jent verties in G γ P n two jent verties in Gγ p, rm overlp: P [] n P [] shre n rm, resulting in two jent verties in G γ P n two inepenent verties in Gγ p, prtil leg overlp: p[] P [] shres leg with p[], with super(p[]) =, resulting in single vertex in G γ P n two jent verties in G γ p. Note tht in eh type of overlp, there is lwys vertex overlp if n only if there is ege overlp. When three or more instnes of P or p overlp, we will lwys mke sure tht n rm or leg is shre y t most two instnes of P or p, whih is lwys possile y tking k suffiiently lrge. When two instnes overlp, they lwys overlp one, i.e., if P [] n P [] hve n overlp of type x, they o not hve n itionl overlp of type y x. We will ll these restritions the overlp onition n ssume impliitly tht they re oeye t ll times when onstruting grphs y overlpping instnes of P n p. Lemm 0. Let G Gλ e grph onstrute from n overlpping instnes P [],..., P [n] of P = B k. Then, the P [],..., P [n] re the only Homo-mthes of P in G. () () Theorem. Let α {, }, γ {vertex, ege} n K {Homo, Iso}. Any unirete grph H is K-γ () () Figure 5. The four overlp types: () lower oy overlp () leg overlp () rm overlp () prtil leg overlp overlp grph, i.e., there lwys exist P, G Gλ α suh tht H = G γ P. Theorem. Let α {, }, γ {vertex, ege} n K {Homo, Iso}. Any K-γ-overlp support mesure f on Gλ α is nti-monotoni only if the ssoite grph mesure ˆf is inresing uner lique ontrtion, ege removl n vertex ition. 5. Lele isomorphisms to homeomorphisms We will now show tht we n reue isomorphi mppings to homeomorphi mppings while preserving egen vertex-overlp. First we show how to reue isomorphi mppings to homeomorphi mppings for the lele se, n then we show how to reue the lele homeomorphisms to unlele homeomorphisms. We will only give the proofs for the unirete ses, s the irete ses n either e proven in very similr wy (reple ll eges y rs), or y omposition of the reution for the unirete se with the reution from unirete to irete of Prop.. We first prove some invrints uner sugrph homeomorphism whih will e importnt in the proof of orretness of oth reutions in this setion. The egree of vertex v in grph G G α is efine s G (v) := #{w {v, w} E(G)}. The mximum egree of G is then (G) := mx v V (G) G (v).

8 Definition. The onnetion strength etween v n w is efine s s G (v, w) := mx{ P : P P G (v, w) pths in P pirwise isjoint} n the mximl onnetion strength etween two noes of G s: CS G = mx v,w VG s G (v, w). For G G, s G (v, w) is often lle the lol onnetivity of v n w. Notie tht onnetion strength etween noes v n w is well known to e equivlent with vertex-onnetivity (Menger s theorem []); i.e., the miniml numer of verties tht nee to e remove to mke v n w isonnete. Lemm 4. Let π e sugrph homeomorphism from H to G. Then,. V (H) V (G) n E(H) E(G) ;. v V (H) : H (v) G (π(v));. v, w V (H) : s H (v, w) s G (π(v), π(w)). 5.. Lele isomorphisms to lele homeomorphisms We first present the lele se, i.e., Iso, γ-overlp on Gλ to Homeo, γ-overlp on G λ reution. The reution R reples eh ege e y n inue sugrph (V e, E e ) ontining the originl en verties n some new verties. We mke sure tht no new vertex n e the imge of n originl vertex y leling them with new lel. Formlly, let G Gλ with lel lphet Σ. Let R : G R (G) Gλ n e = {u, w} E(G). We efine (V e, E e ) s follows: V e = {u, w} {v i e i 5}, V e V (G) = {u, w}, E e = {{u, v e }, {u, v e }, {v e, v e }, {v e, v e }, {v e, v4 e }, {v e, v5 e }, {v4 e, w}, {v5 e, w}} Let e = {u, w } E(G). For e e, we mke sure tht V e V e = {u, w} {u, w } n E e E e =. We re now rey to efine R (G): V (R (G)) = V (G) e E(G) V e, E(R (G)) = e E(G) E e, λ R(G)(u) = λ G (u), u V (G), λ R(G)(v e ) = l / Σ, v e V (R (G)) \ V (G). An illustrtion of the sugrph repling n ege from vertex lele to vertex lele is shown in Figure 6. We n prove the following using onnetion strength: Theorem 5. (P, p, G) (R (P ), R (p), R (G)) is Iso, γ-overlp on G λ to Homeo, γ-overlp on G λ reution, γ {vertex, ege}. l l l Figure 6. The sugrph (V e, E e ) use in the reution to reple n ege e = {u, w}, with λ(u) = n λ(w) =. L L L Figure 7. Lel grphs for n = 5.. Lele to unlele homeomorphisms We now show the reution from the lele se to the unlele one; i.e., from vertex-overlp of Gλ to γ-overlp of G for ll γ {vertex, ege}. We will use the following speil lel-grphs L n i to reple the lels Σ = {l,..., l n }. l l Definition 6. Let s < k e integers. Ck s enotes the grph in G with noes { 0,..., k } n eges E := {{ i, j } j = (i + ) mo k... (i + s) mo k}. Let i n e integers. L n i enotes the grph Cn i+ (n+i). Hene, Ck s is yle of length k, with itionl eges: every noe is onnete to its s suessors (n hene lso its s preeessors). The grphs L n i will e use in the proof to reple lels. In Figure 7, the lel grphs for n lphet of size hve een given. Intuitively, we will reple the lels of the verties y tthing n pproprite L n i to the noe. For grph over the lphet Σ = {l,..., l n }, L n i will e use to reple l i. A first essentil piee of the proof is tht for given n, no lel-grph L n i n e mppe to nother lel-grph L n j, j i uner homeomorphism. Lemm 7. Let i, j n e integers. There exists homeomorphism from L n i to Ln j if n only if i = j. In ll wht follows, n will enote the size of the lphet Σ = {l,..., l n }. We ssume funtion ι tht mps the lel l i to its inex i. We will slightly use the nottion ι, n use ι(v) to enote ι(λ(v)). Let G e grph in Gλ over the lphet Σ, with verties V = {v,..., v k } n eges E = {e,..., e m }. In the proofs we will nee mny opies of the sme lel grph L n i. Therefore, we will nee to renme the verties in these grphs to voi onfusion. We will use L[v i ] to enote the following isomorphi opy of L n ι(λ(v : i))

9 Theorem 9. Let p, P, G Gλ, n let = mx{ (G), (P ), (p), n} +. The funtion R tht mps (p, P, G) to (R (p), R (P ), R (G)) is n Homeo, vertex-overlp on Gλ α to Homeo, γ-overlp on Gα reution, for ll α {, } n γ {vertex, ege}. 5. From lele to unlele homomorphisms n isomorphisms Figure 8. Reution for removing lels in the se of homeomorphisms. The tringle in the mile is the originl grph G. The lels of the top, left, right noes were respetively l, l, n l. V (L[v i ]) = { i j j V (L n ι(v )}, n E(L[v i) i]) = {{ i j, i k } { j, k } E(L n ι(v i) )}. As suh, ny two L[v] n L[w] re isjoint whenever v w, even if λ(v) = λ(w). We now efine the reution, prmeterize y. Definition 8. For every v i V (G), let the following sets of verties e given. B(v i ) = { i j j =... } C(v i) = V (L[v i ]) D(v i ) = { i j j =... } We ssume ll these sets re isjoint. Furthermore, 0 (v i ) enotes the noe i 0; i.e., the first noe in L[v i ]. R (G) is the following grph in G : V (R(G)) := V (G) ( ) C(v) D(v) B(v) v V E(R(G)) := E(G) E(L[v]) v V {{ 0 (v), } D(v)} v V {{v, }, {, 0 (v)} B(v)} v V An exmple of the reution hs een given in Figure 8. Intuitively, the rtionle ehin the reution is s follows: the su-grphs L[v] reple the lel of v. The noes D(v) re e in orer to inrese the egree of ll noes 0 (v) to t lest +. All other noes hve egree t most. This llows us to use egree-rguments to show tht ll 0 - noes re mppe to 0 -noes. The noes B(v) re e to onnet ny lel grph L[v] to the right noe v ( of etween). Their numer will e hosen so we n use onnetion strength rguments to show tht in mth lwys lel-grphs will e ssoite with the right noe. Using the rguments ove, we n prove the following: Finlly, we exten the results for homomorphism n isomorphism to unlele grphs. First, we will show tht our onstrutions for lele grphs n e extene to unlele grphs y using speil sugrphs in the unlele se to enoe the lels from the lele se. We will fous on the most iffiult se, homomorphism. For isomorphism, muh simpler onstrutions re possile. Also, we will isuss only the unirete se here. The irete se is nlogous. The key ie for emulting lels with unlele sugrphs uner homomorphism follows from the ft tht liques re lwys mppe on liques of the sme size. Lemm 0. Let G G. Let π e homomorphism from K k to G (where K k is the omplete grph with k verties). Then, π is sugrph isomorphism mpping, i.e. π(k k ) is k-lique of G. Aprt from the nottions introue in Definition, we will lso use V w j = {v w,j, v w,j+ mo σ(w),... v w,j+k mo σ(w) }. The sugrphs tthe to the originl verties to represent the lels re isomorphi to the grphs C σ(w) K s in Definition 6 n re illustrte in Figure 7. We now formlize the enoing of lels with unirete sugrphs: Definition. Let G Gλ. Let k e (n upper oun on) the lique numer of G. A Shem for Leling with Unlele Sugrphs (SLUS) for G is pir (K, σ) suh tht K mx(k +, Σ ); σ : Σ N is n injetive funtion mpping every element from the lphet Σ of lels on istint o integer suh tht l Σ : 4(K + ) < σ(l) < 5(K + ). When it is ler tht w is vertex, we will use σ(w) s shorthn for σ(λ G (w)). We now efine trnsformtion from lele to unlele grphs:

10 Definition. Let G Gλ Let (K, σ) e SLUS for G. Then, we efine the trnsforme (unlele) grph R K,σ (G) y the verties of R K,σ (G) re V (R K,σ (G)) = w V (G) V w where V w = {w j 0 j < σ(w)} where for ll w, w 0 = w n w j, j =... σ(w) re new verties. the eges of R K,σ (G) re E(R K,σ (G)) = E(G) E l K,σ(G) with E l K,σ (G) = w V (G)E w where E w = {{v w,j, v w,j+i mo σ(w) } We n prove the following 0 j < σ(w) i K} Theorem. Let p, P, G Gλ suh tht G hs no two jent verties with the sme lel, n let (K, σ) e SLUS for p, P n G. Then, the funtion R K,σ tht mps (p, P, G) to (R K,σ (p), R K,σ (P ), R K,σ (G)) is Homo, vertex-overlp on Gλ α to Homeo, γ-overlp on Gα reution, for ll α {, } n γ {vertex, ege}. To generlize this result so tht G is not require to hve two jent verties of the sme lel, we n first perform n itionl trnsformtion, splitting ll eges in two new eges n leling the new verties in the mile of the originl eges with new mi-ege lel. 6 Disussion n onlusion We extene the results of [] to rnge of ifferent settings. We prove the results for lele homomorphism s se se n provie reutions whih re more generlly pplile to prove the results for the other settings. We showe tht MIS n MCP re miniml n mximl nti-monotoni overlp support mesures. We lso me first step towrs mking the overlp support mesures slle y proving the nti-monotoniity of the Lovász θ-funtion, polynomil-time omputle grph mesure snwihe etween MIS n MCP. Severl extensions of our work re possile, some of those leing to smller overlp grphs. An interesting one onerns lterntive efinitions for mthes. We onsiere mthes to e ll verties (eges) of the emeing of pttern in the tset. Alterntively, we n onsier ptterns where only few istinguishe verties re tken into ount for overlp. Mking the set of verties relevnt for overlp smller reues the size of the overlp grph. The extension is strightforwr in most of the ses onsiere in this pper. As speil se, suppose only one vertex of pttern is onsiere relevnt. The overlp grph is then reue to set of isolte verties of size t most V (G). [] propose mesure f(p, G) = min v P {w V (G) : π Iso : (π(p ) G) (w V (π(p )))}. One n see this s the minimum over severl mesures, eh onsiering one of the verties of P relevnt. The minimum of nti-monotoni funtions is nti-monotoni itself. There exist lso ifferent notions of overlp. E.g. [4] efines hrmful overlp, whih is se on emeings. Two emeings π n π of pttern P overlp iff v V (P ) : π (v), π (v) π (V (P )) π (V (P )). This notion then results in hrmful overlp grphs. We expet our reutions n e esily pte to generlize lso the hrmful overlp notion to the onsiere omintions of ireteness, leleness n morphism hoie. Referenes [] S. Bnyophyy, R. Shrn, n T. Ieker. Systemti ientifition of funtionl orthologs se on protein network omprison. Genome Res., 6():48 45, Mrh 006. [] B. Bringmnn n S. Nijssen. Wht is frequent in single grph? In Proeeings of Mining n Lerning with Grphs 007, Florene, Itly, 007. [] R. Diestel. Grph Theory, Thir eition. Springer-Verlg, 005. [4] M. Fieler n C. Borgelt. Support omputtion for mining frequent sugrphs in single grph. In Proeeings of Mining n Lerning with Grphs 007, Florene, Itly, 007. [5] J. L. Gross n J. Yellen. Hnook of Grph Theory. CRC Press, 004. [6] Grunewl, Stefn, Forslun, Kristoffer, Dress, Anres, Moulton, n Vinent. Qnet: An gglomertive metho for the onstrution of phylogeneti networks from weighte qurtets. Moleulr Biology n Evolution, 4():5 58, Ferury 007. [7] P. Hell n J. Nešetřil. Grphs n homomorphisms. Oxfor University Press, 004. [8] D. E. Knuth. The snwih theorem. Eletron. J. Comin., :48 pp., 994. [9] M. Kurmohi n G. Krypis. Fining frequent ptterns in lrge sprse grph. Dt Min. Knowl. Disov., ():4 7, 005. [0] A. S. LPugh n R. L. Rivest. The sugrph homeomorphism prolem. In STOC 78: Proeeings of the tenth nnul ACM symposium on Theory of omputing, pges 40 50, New York, NY, USA, 978. ACM Press. [] S. Muggleton n L. De Ret. Inutive logi progrmming : Theory n methos. Journl of Logi Progrmming, 9,0:69 679, 994. [] C. H. Ppimitriou. Computtionl Complexity. Aison- Wesley, 994.

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