Theoretcal Computer Scence 481 (2013) 74 84 Contents lsts avalable at ScVerse ScenceDrect Theoretcal Computer Scence journal homepage: www.elsever.com/locate/tcs Increasng the mnmum degree of a graph by contractons Petr A. Golovach a, Marcn Kamńsk b, Danël Paulusma a,, Dmtros M. Thlkos c a School of Engneerng and Computng Scences, Durham Unversty, Scence Laboratores, South Road, Durham DH1 3LE, Unted Kngdom b Département d Informatque, Unversté Lbre de Bruxelles, Belgum c Department of Mathematcs, Natonal and Kapodstran Unversty of Athens, Panepstmoupols, GR15784 Athens, Greece a r t c l e n f o a b s t r a c t Artcle hstory: Receved 5 November 2011 Receved n revsed form 5 February 2013 Accepted 20 February 2013 Communcated by V. Th. Paschos The Degree Contractblty problem s to test whether a gven graph G can be modfed to a graph of mnmum degree at least d by usng at most k contractons. We prove the followng three results. Frst, Degree Contractblty s NP-complete even when d = 14. Second, t s fxed-parameter tractable when parameterzed by k and d. Thrd, t s W[1]-hard when parameterzed by k. We also study ts varant where the nput graph s weghted,.e., has some edge weghtng and the contractons preserve these weghts. The Weghted Degree Contractblty problem s to test f a weghted graph G can be contracted to a weghted graph of mnmum weghted degree at least d by usng at most k weghted contractons. We show that ths problem s NP-complete and that t s fxed-parameter tractable when parameterzed by k. In addton, we pnpont a relatonshp wth the problem of fndng a mnmal edge-cut of maxmum sze n a graph and study the parameterzed complexty of ths problem and ts varants. 2013 Elsever B.V. All rghts reserved. 1. Introducton Throughout the paper we consder undrected fnte graphs that have no loops. Unless we explctly ndcate ths, they do not have multple edges ether. We denote the vertex set and the edge set of a graph G by V G and E G, respectvely. If no confuson s possble, we may omt subscrpts. We refer the reader to the text book of Destel [10] for undefned graph termnology and to the monographs of Downey and Fellows [12] and Nedermeer [27] for more on parameterzed complexty. A graph modfcaton problem has as nput a graph G and an nteger k. The queston s whether G can be modfed to belong to some specfed graph class that satsfes further propertes by usng at most k operatons of a certan specfed type such as deletng a vertex or deletng an edge. In our paper the permtted operaton s the contracton of an edge, whch removes both end-vertces of the edge and replaces them by a new vertex adjacent to precsely those vertces to whch the two end-vertces were adjacent. We contnue a very recent study [17 19] of the followng graph modfcaton problem called Π-Contractblty, where Π s some prespecfed graph class. Π-Contractblty Instance: a graph G and an nteger k. Queston: can G be modfed to a graph n Π by at most k contractons? Ths paper has been supported by EPSRC (EP/G043434/1), and an extended abstract of t appeared n the Proceedngs of IPEC 2011. Correspondng author. Tel.: +44 0 191 33 41723; fax: +44 0 191 33 41701. E-mal addresses: petr.golovach@durham.ac.uk (P.A. Golovach), marcn.kamnsk@ulb.ac.be (M. Kamńsk), danel.paulusma@durham.ac.uk (D. Paulusma), sedthlk@math.uoa.gr (D.M. Thlkos). 0304-3975/$ see front matter 2013 Elsever B.V. All rghts reserved. http://dx.do.org/10.1016/j.tcs.2013.02.030
P.A. Golovach et al. / Theoretcal Computer Scence 481 (2013) 74 84 75 1.1. Prevous results Research on the Π-Contractblty problem dates back to the early eghtes, when Watanabe, Ae and Nakamura [29,30] showed that Π-Contractblty s NP-complete f Π s fntely characterzable by 3-connected graphs. Ther result was generalzed by Asano and Hrata [2] who showed that Π-Contractblty s NP-complete whenever Π s a graph class that fulflls the followng condtons. Frst, Π must be closed under contractons. Second, Π must be descrbed by a property that s satsfed by nfntely many connected graphs and volated by nfntely many other connected graphs. Thrd, a graph belongs to Π f and only f each of ts bconnected components belong to Π. Examples [2] of such graph classes Π nclude planar graphs, outerplanar graphs, seres parallel graphs, and also forests, chordal graphs, or more generally, graphs wth no cycles of length at least l for some fxed nteger l 3. The problem Π-Contractblty s closely related to the problem H-Contractblty, whch s to test whether a gven graph G can be contracted to a fxed graph H (.e., whch s not part of the nput). Brouwer and Veldman [8] showed that the H-Contractblty problem s NP-complete whenever H s a trangle-free graph that contans no vertex adjacent to all the other vertces. Ther work has been extended by a seres of other papers [20,23,24] showng both polynomal-tme solvable and NP-complete cases. Determnng a full complexty classfcaton for H-Contractblty s open, although such results restrctng the nput graph G to be n a specal graph class have been obtaned [3,4,22]. If Π s the class of paths or cycles, then Π-Contractblty s polynomally equvalent to the problems of determnng the length of a longest path and a longest cycle, respectvely, to whch a gven graph can be contracted. The frst problem has been shown to be NP-complete by van t Hof, Paulusma and Woegnger [21] even for graphs wth no nduced path on 6 vertces. The second problem has been shown to be NP-complete by Hammack [16]. Eppsten [13] showed that t s NP-complete to decde f a graph contans a complete graph K p as a mnor for some gven nteger p. Ths problem s equvalent to decdng f a graph s contractble to K p. As a drect consequence, Π-Contractblty s NP-complete f Π s the class of complete graphs. Due to a close relatonshp wth the problem that s to test whether a gven graph contans a so-called dsconnected cut set, Martn and Paulusma [25] have shown that Π-Contractblty s NP-complete f Π s the class of bclques K p,q wth p, q 2. Recently, more papers appeared that study the Π-Contractblty problem, and n partcular, ts parameterzed complexty where the parameter s the number k of edges that may be contracted. Heggernes et al. [19] gave an 4 k+o(log2 k) + n O(1) tme algorthm for Π-Contractblty f Π s the class of paths. Moreover, they showed that n ths case the problem has a lnear kernel. When Π s the class of trees, they showed that the problem can be solved n 4.88 k n O(1) tme and that a polynomal kernel does not exst unless conp NP \ poly. When the nput graph s a chordal graph wth n vertces and m edges, Heggernes et al. [17] could show that Π-Contractblty can be solved n O(n + m) tme when Π s the class of trees and n O(nm) tme when Π s the class of paths. When Π s the class of bpartte graphs, Heggernes et al. [18] observed that Π-Contractblty s NP-complete and showed that Π-Contractblty s fxed-parameter tractable when parameterzed by k. Later on, Marx, O Sullvan, and Razgon [26] obtaned ths result for bpartte graphs as a corollary from ther result on generalzed bpartzaton. Bodlaender, Koster and Wolle [6] ntroduced the related noton of contracton degeneracy as a useful tool to mprove lower bound heurstcs for treewdth. The contracton degeneracy of a graph G s the largest mnmum degree of any mnor of G. When G s connected, the contracton degeneracy of G s equal to the largest mnmum degree of any graph to whch G can be contracted [6]. The Contracton Degeneracy problem s to test whether the contracton degeneracy of a gven graph s at least d for some gven nteger d (see also [28] for extensons of ths problem). Bodlaender, Koster and Wolle [6] proved that ths problem s NP-complete, even for bpartte graphs, and that t s fxed-parameter tractable when parameterzed by d. They also evaluated a number of heurstcs for computng the contracton degeneracy. 1.2. Our results In order to defne the class Π of graphs that we consder, we need the followng termnology. A vertex u n a graph G = (V, E) s a neghbor of a vertex u f uv E G. We let N(u) = {uv v V} denote the neghborhood of u. The degree of a vertex u s denoted d(u) = N(u). We let δ = mn{d(v) v V} denote the mnmum degree of G. We study the Π-Contractblty problem where Π s the class of graphs of mnmum (vertex) degree at least d for some nteger d. Note that ths class of graphs does not satsfy the frst and thrd property of Asano and Hrata [2]. Moreover, for ths class of graphs, we allow the nteger d to be part of the nput as well. Ths leads to the followng problem. Degree Contractblty Instance: a graph G and two ntegers d and k. Queston: can G be modfed to a graph of mnmum degree at least d by at most k contractons? We observe that the problem becomes equvalent to the Contracton Degeneracy problem when the nput s restrcted to connected graphs G = (V, E) and k E. In Secton 2 we show that Degree Contractblty s fxed-parameter tractable when parameterzed by k and d. However, when ether k or d s part of the nput, Degree Contractblty becomes hard n the followng sense. Frst, f k s part of the
76 P.A. Golovach et al. / Theoretcal Computer Scence 481 (2013) 74 84 Table 1 An overvew of our results for the problems DC and WDC. Input Parameter DC WDC d, k NP-complete NP-complete d k W[1]-hard FPT k d para-np-complete para-np-complete d, k FPT FPT nput, then Degree Contractblty s NP-complete for any fxed d 14. Second, f d s part of the nput, then Degree Contractblty s W[1]-hard when parameterzed by k. These results complement the result of Amn, Sau and Saurabh [1] who showed that detectng a subgraph that has at most k vertces and mnmum degree at least d s W[1]-hard for any fxed d 4 when parameterzed by k. In Secton 3 we study a weghted verson of Degree Contractblty. In order to defne ths varant, let G = (V, E) be a weghted graph,.e., wth some edge weghtng w: E R >0 where R >0 denotes the set of postve real numbers. The weghted degree d w (u) of a vertex u s the sum of the weghts of the edges ncdent wth u n G,.e., d w (u) = v N(u) w(uv). We let δ w = mn{d w (v) v V} denote the mnmum weghted degree of G. The weghted contracton of an edge e = uv s a contracton of e where the weghts on the edges ncdent wth the new vertex x uv are defned as follows: w(x uv y) = w(uy) f y s adjacent to u and not adjacent to v; w(x uv y) = w(vy) f y s adjacent to v and not adjacent to u; w(x uv y) = w(uy) + w(vy) f y s adjacent to u and v. We can now state the new varant. Weghted Degree Contractblty Instance: a weghted graph G and two ntegers d and k. Queston: can G be modfed to a weghted graph of mnmum weghted degree at least d by at most k weghted contractons? Because the weght of an edge x uv y wth y adjacent to both u and v s the accumulated weght of the two orgnal edges uy and vy, Degree Contractblty s not a specal (unweghted) case of Weghted Degree Contractblty. However, we can make the followng observaton. A smple contracton s the operaton on loopless multgraphs that dentfes both end-vertces of the edges, keeps multple edges, but removes the loop that was created. The Weghted Degree Contractblty problem for nteger edge weghts on a graph G s equvalent to the varant of Degree Contractblty, where smple contractons are used on the loopless multgraph G obtaned from G by replacng each edge uv by w(uv) parallel edges. Contrary to the aforementoned W[1]-hardness result for Degree Contractblty when parameterzed by k, accumulatng the weghts after contractng an edge results n the problem not beng hard anymore,.e., we prove that Weghted Degree Contractblty s fxed-parameter tractable when parameterzed by k even when d s part of the nput. If both d and k are parts of the nput, then Weghted Degree Contractblty s NP-complete n the strong sense for nteger edge weghts, even n the case when k E. The latter case s equvalent to the case, n whch there s no upper bound mposed on the number of weghted contractons. We denote ths specal case as the problem Weghted Contracton Degeneracy Instance: a weghted graph G and an nteger d. Queston: can G be modfed to a weghted graph of mnmum weghted degree at least d by weghted contractons? We prove that ths problem s fxed-parameter tractable when parameterzed by d. Both ths result and the aforementoned NP-completeness result are based on an equvalence, whch we pnpont, between Weghted Contracton Degeneracy and the problem that s to test whether a connected graph has a mnmal edge-cut of some gven sze. Table 1 summarzes our results for the Degree Contractblty (DC) problem and the Weghted Degree Contractblty (WDC) problem. In ths table, the para-np-completeness of the problem Weghted Degree Contractblty parameterzed by d mmedately follows from our result that Weghted Degree Contractblty s NP-complete n the strong sense for nteger edge weghts, whch allows us to fx d = 1 after dvdng all edge weghts by d. However, when we restrct the edge weghtng to be nteger, the correspondng problem s stll open. In Secton 4 we dentfy two related problem settngs. In the frst settng weghted plane graphs are consdered, where the edge weghtngs defne weghted degrees of the faces. In the second settng, party constrants are consdered nstead of degree constrants. We show how these problem settngs are related wth our prevous problems, and we study them usng results of the prevous sectons. 1.3. Prelmnares Let G = (V, E) be a (weghted) graph. A subset U V s a clque f there s an edge n G between any two vertces of U, and U s an ndependent set f there s no edge n G between any two vertces of U. We wrte G[U] to denote the subgraph of G
P.A. Golovach et al. / Theoretcal Computer Scence 481 (2013) 74 84 77 nduced by U V,.e., the graph on vertex set U and an edge between any two vertces f and only f there s an edge between them n G. We let G/e denote the (weghted) graph obtaned from G by the (weghted) contracton of e. If a (weghted) graph H s obtaned from G by a sequence of (weghted) contractons, then H s a (weghted) contracton of G. For a weghted graph G wth an edge weghtng w and X E, we wrte w(x) = e X w(e). Let G and H be two graphs. An H-wtness structure W s a vertex partton of G nto V H (nonempty) sets W(x) called H-wtness bags, such that () each W(x) nduces a connected subgraph of G; () for all x, y V H wth x y, bags W(x) and W(y) are adjacent n G f and only f x and y are adjacent n H; By contractng all bags to sngletons we observe that H s a contracton of G f and only f G has an H-wtness structure such that condtons () () hold. Note that a graph may have more than one H-wtness structure. 2. Contractons Frst, we observe that Degree Contractblty s FPT when parameterzed by k and d. Proposton 1. Degree Contractblty can be solved n tme O(d k (n + m)) for graphs wth n vertces and m edges. Proof. Let G be a graph wth n vertces and m edges. We gve the followng branchng algorthm. Let d G (u) < d for some vertex u V G. We consder all edges e ncdent wth u, and call our algorthm recursvely for G/e and parameter k = k 1. The algorthm returns Yes, f for at least one of the new nstances the answer s Yes, and t returns No otherwse. Snce for each recursve call of our algorthm, we create at most d 1 nstances of the problem, and the depth of the recurson s at most k, the algorthms runs n tme O(d k (n + m)). A graph G s r-degenerate for some nteger r f δ(h) r for every subgraph H of G. A graph class G s contracton-closed f G/e G for every graph G G and every e E G. Proposton 1 has the followng consequence. Corollary 1. Let G be a contracton-closed graph class so that all graphs n G are r-degenerate for some nteger r 0. Then Degree Contractblty can be solved n tme O(r k (n + m)) for every G G wth n vertces and m edges. Proof. If d > r, then we cannot modfy a graph G G to a graph of mnmum degree at least d by edge contractons. Otherwse, we apply Proposton 1. Corollary 1 holds, for example, for the class of planar graphs whch are 5-degenerate, or more general, for bounded-genus graph classes and excluded-mnor graph classes. For general graphs, we observe that Degree Contractblty s n XP when parameterzed by k,.e., t can be solved n n f (k) tme for n-vertex graphs by checkng all sequences of at most k contractons (here f s some functon dependng only on k). However, we show that t s unlkely to be solvable n FPT-tme. Theorem 1. Degree Contractblty parameterzed by k s W[1]-hard. Proof. The problem Multcolored Clque s to test whether a graph wth a proper k-colorng contans a clque of sze k wth exactly one vertex from each color class. Fellows et al. [11] proved that ths problem s W[1]-hard when parameterzed by k. Consequently, ts complementary problem, the problem Multcolored Independent Set, whch s to test whether a graph wth a partton X 1,..., X k of the vertex set, where each X s a clque, has an ndependent set of sze k wth exactly one vertex from each X, s W[1]-hard as well when parameterzed by k. Our am s to reduce Degree Contractblty to ths problem. Let (G, k) wth a partton X 1,..., X k of V G be an nstance of Multcolored Independent Set. Let clque X = {x 1,..., x n } for {1,..., k} where we assume wthout loss of generalty that n 2. Let d = n(4k + 3) + 1. From G we construct a graph G n the followng way. Recall that connectng two vertces means addng an edge between them. 1. Construct a clque W wth vertces w 1,..., w d+1. 2. Connect every x wth w 1,..., w t where t = d d G (x ) n 4k 2. 3. Add vertces y 1,..., y k. 4. For every x, construct a clque Q wth vertces r, a (1),..., a (2k+1), b (1) wth x and y. Moreover, connect every r wth w 1,..., w d (4k+3), every a (s) w 4k+5,..., w d+1. 5. Construct a clque C wth vertces c 1,..., c k+2, z 1,..., z k. 6. For h = 1,..., k + 2, connect c h wth w 1,..., w d 2k+1 7. For = 1,..., k, connect z wth every vertex of X.,..., b (2k+1) and connect every vertex of Q wth w 1,..., w d (4k+3), and every b (s) wth Stages 1 4 of the constructon are shown n Fg. 1(a), and Stages 5 7 are shown n Fg. 1(b). We let k = k(2k + 3) and clam that G has an ndependent set wth exactly one vertex from each X f and only f G can be modfed to a graph H wth mnmum degree at least d by at most k contractons.
78 P.A. Golovach et al. / Theoretcal Computer Scence 481 (2013) 74 84 Fg. 1. The constructon of G. Frst suppose that {x 1j1,..., x kjk } s an ndependent set n G. For = 1,..., k, let A = {a (1) b (1),..., a (2k+1) b (2k+1), r y, x z } be a set of 2k + 3 edges n G. We contract every edge n every A. Then the total number of contractons s k(2k + 3) = k. Moreover, the resultng graph has mnmum degree at least d. Now suppose that G can be modfed to a graph H wth mnmum degree at least d by at most k contractons. Let W be an H-wtness structure of G. For each bag W of W, we choose an arbtrary spannng tree of G [W]. Let A E G denote the unon of the sets of edges of these trees. Because we obtan H by contractng the edges of A, we fnd that A k. Clam 1. A = A 1 A k, where each A = {x z, y f, g 1 h 1,..., g 2k+1 h 2k+1 } wth {f, g 1,..., g 2k+1, h 1,..., h 2k+1 } = Q. We prove Clam 1 as follows. Let 1 k. Because d G (y ) < d, at least one edge ncdent wth y must be ncluded n A. Assume that y f A for some f Q. Note that after contractng y f, all 4k + 2 vertces of Q \ {f } have degrees less that d. Hence, at least 2k + 1 edges ncdent wth these vertces must be contracted. We also note that d G (z ) < d. Therefore, at least one edge ncdent wth z must be n A. Suppose that z t A for some t C. Then, after contractng z t, all other 2k vertces of C have degrees less than d. Hence, we must contract at least k edges ncdent wth these vertces. Because the total number of contractons s k = k(2k + 3) and we also need to contract at least 2k + 3 edges for every h, ths s not possble. We conclude that z x A for some 1 j n and that A = {x z, y f, g 1 h 1,..., g 2k+1 h 2k+1 } wth {f, g 1,..., g 2k+1, h 1,..., h 2k+1 } = Q. We now consder x and observe that by contractng the edges g 1 h 1,..., g 2k+1 h 2k+1 we decreased the degree of x by 2k + 1. Hence, j = j and Clam 1 follows. Due to Clam 1, we can defne the set {x 1j1,..., x kjk } wth x z A for = 1,..., k. We prove that ths s an ndependent set n G. In order to obtan a contradcton, assume that there s an edge x x j E G. Recall that d G (x j) = d G (x j) + (n 1) + (4k + 3) + 1 + t = d + 1. Contractng those edges of A that have both end-vertces n Q decreases the degree of x j by 2k + 1. Moreover, after contractng the edges n A and A, the edges z z and x x j have been replaced by one edge. Because z s adjacent to all vertces n C \{z }, ths means that the degree of the vertex of H obtaned by contractng x jz s at most d + 1 (2k + 1) 2 + (2k + 1) = d 1. Ths s not possble. Hence, {x 1j1,..., x kjk } s an ndependent set n G wth a vertex, namely x j, from each X, as desred. Ths completes the proof of Theorem 1. If we parameterze the problem only by d, then Degree Contractblty becomes hard even f d s a fxed nteger. Theorem 2. For any fxed d 14, Degree Contractblty s NP-complete. Proof. The ncluson of the problem n NP s obvous. For smplcty, we prove NP-hardness for d = 14. We reduce from the NP-complete Set Cover problem [14]. Ths problem s defned as follows. Gven a set U = {u 1,..., u m }, a famly of subsets X 1,..., X n U and an nteger r, are there at most r subsets that cover U,.e., ther unon s U? It s known [14] that ths problem remans NP-complete even f () each X has cardnalty 3, and () each u j s ncluded n ether two or three subsets of X 1,..., X n. We consder an nstance (U, X 1,..., X n ) of Set Cover wth restrctons () and (). We construct a graph G n the followng way; also see Fg. 2. We say that we connect a vertex wth some other vertex f we add an edge between them. 1. Construct a clque wth 13 vertces w 1,..., w 13. 2. Add two new vertces s, t and connect each of them wth w 1,..., w 13. 3. For = 1,..., n, add a vertex x and connect t wth s, t. 4. For = 1,..., n, add two adjacent vertces p (1), p (2), connect p (1) wth s, w 1,..., w 11, x, and connect p (2) s, w 3,..., w 13, x. 5. For j = 1,..., m, add a vertex u j and connect t wth t. wth
P.A. Golovach et al. / Theoretcal Computer Scence 481 (2013) 74 84 79 Fg. 2. The constructon of G. 6. Connect x and u j whenever u j X. In that case also add two adjacent vertces q (1), q (2), connect q (1) wth x, u j, w 1,..., w 11 and connect q (2) wth x, u j, w 3,..., w 13. 7. For j = 1,..., m, connect u j wth w 1,..., w 8 f u j occurs n two subsets of X 1,..., X n, and connect u j wth w 1,..., w 6 f u j occurs n three subsets. We set k = n + r and clam that U can be covered by at most r subsets of {X 1,..., X n } f and only f G can be modfed to a graph wth mnmum degree at least d = 14 by at most k contractons. Frst suppose that X 1,..., X r s a set cover of U,.e., U = X 1 X r. For j = 1,..., r, we contract the edges sx and p (1) j p (2) j. We also contract the edge x t for every / { 1,..., r }. The total number of contractons s 2r + (n r) = n + r = k. Moreover, the resultng graph s readly seen to have mnmum degree at least 14, as desred. Now suppose G can be modfed to a graph H wth mnmum degree at least d = 14 by at most k contractons. Let W be an H-wtness structure of G. For each bag W of W, we choose an arbtrary spannng tree of G[W]. Let A E G denote the unon of the sets of edges of these trees. Because H s obtaned by contractng the edges of A, we fnd that A k. For each X, we defne a set of edges E E G as follows. The set E ncludes all edges ncdent wth x, p (1), p (2), and all edges ncdent wth q (1), q (2) for every u j X. Moreover, we choose one vertex u j X and also add all (other) edges ncdent wth u j to E. The sets E 1,..., E n have the followng propertes. 1. E E j = for 1 < j n. 2. E A for = 1,..., n. 3. The number of sets E wth E A 2 s at most r. Property 1 s true by defnton. Property 2 follows from the fact that d G (x ) = 13 < 14 = d; therefore, at least one edge ncdent wth x must be contracted. Property 3 follows from propertes 1 and 2 and the aforementoned observaton that A k = n + r. Let I = { E A 2}. We clam that I X = U. In order to obtan a contradcton, assume that there s a vertex u j U \ I X. Then, for each X wth u j X, we fnd that E contans a unque edge e E A. Because d G (x ) = 13 < d, e s ncdent wth x. If e = sx, then contractng e decreases the degree of p (1) and p (2). Because they both have degree 14, at least one edge ncdent wth them must be contracted as well. Hence, e sx. Smlarly, f e = x p (1) then contractng e decreases the degree of p (2). Hence, e x p (1). We apply the same arguments on the other edges n E and conclude that the only possblty s e = x t. Now we consder two cases. Case 1. u j s ncluded n exactly two sets X 1, X 2. Then edges x 1 t, x 2 t are contracted, whereas all other edges ncdent wth x 1, x 2 and also edges p (1) 1 p (2) 1, p (1) 2 p (2) 2, q (1) 1 j q(2) 1 j, q(1) 2 j q(2) 2 j are not contracted. Moreover, no edges ncdent wth u j are contracted, because these belong to E 1 E 2. However, then u j has degree at most 13 < d n H, a contradcton. Case 2. u j s ncluded n three sets X 1, X 2, X 3. By the same arguments as n Case 1, we fnd that the degree of u j n H s at most 13 < d, a contradcton. We conclude that {X I} s a set cover, whch contans at most r sets due to Property 3. Ths completes the proof of Theorem 2. Whle t can be easly seen that for any fxed d 3, Degree Contractblty can be solved n polynomal tme, determnng the complexty for 4 d 13 s an open queston. 3. Weghted contractons 3.1. Weghted Degree Contractblty parameterzed by k We frst show that Weghted Degree Contractblty s n FPT when parameterzed by k. Recall that x uv denotes the vertex obtaned from u and v after contractng an edge uv n a graph.
80 P.A. Golovach et al. / Theoretcal Computer Scence 481 (2013) 74 84 Theorem 3. Weghted Degree Contractblty can be solved n tme O(2 k k 2k (n + m)) for weghted graphs wth n vertces and m edges. Proof. Let d and k be two ntegers, and let G be a weghted graph wth n vertces and m edges. Below we present our algorthm for decdng whether G can be modfed to a weghted graph of mnmum weghted degree at least d by at most k weghted contractons. Let U = {u V G d w G (u) < d} and let r = U. Trvally, f r = 0, then the answer s Yes. If r 1, then we branch accordng to the followng four cases. Case 1. r > 2k. The algorthm returns No. The reason s that at least one edge ncdent wth each vertex of U must be contracted to get a graph of mnmum weghted degree at least d, and every edge s ncdent wth at most two vertces of U. Case 2. r 2k and there s a vertex u U wth d G (u) k. At least one edge ncdent wth u must be contracted to obtan a graph of mnmum degree at least d. Hence, for each edge e ncdent wth u, we call our algorthm recursvely for G/e and parameter k = k 1. The algorthm returns Yes f for at least one of the new nstances the answer s Yes, and No otherwse. Case 3. k < r 2k and d G (u) k + 1 for all u U. If G can be contracted to a graph of mnmum weghted degree at least d, then at least one edge wth both ts end-vertces n U must be contracted. Note that there at most k(2k 1) such edges. If there are no such edges, then the algorthm returns No. Otherwse, for each e = xy wth x, y U, we call our algorthm recursvely for G/e and parameter k = k 1. The algorthm returns Yes f for at least one of the new nstances the answer s Yes, and No otherwse. Case 4. r k and d G (u) k + 1 for all u U. Let U = {u 1,..., u r }. Each u s adjacent to at least two vertces n V G \ U. For = 1,..., r, we do the followng. Let y, z be two neghbors of u n V G \ U, where we assume that w(u y) w(u z). Let G = G/u y. Then we deduce that d w G (x u y) = d w G (u ) + d w G (y) 2w(u y) w(u y) + w(u z) + d w G (y) 2w(u y) = d w G (y) w(u y) + w(u z) d w G (y) d. Hence, we contract u y and recursvely proceed wth G and U = U \ {u }. Note that the weghted contracton of u y does not change the weghted degrees of the other vertces. Consequently, each vertex n U s adjacent to at least two vertces of weghted degree at least d n G, and U s the set of vertces of weghted degree at most d 1 n G. Then after processng u r, we obtan a graph of mnmum degree at least d by usng r k weghted contractons. Hence, our algorthm always returns Yes n ths case. To estmate the runnng tme, observe that for each recursve call of our algorthm, we create at most k(2k 1) nstances of the problem, and the depth of the recurson s at most k. Hence, the algorthm runs n tme O(2 k k 2k (n + m)). 3.2. Weghted Contracton Degeneracy Recall that we call the specal case of Weghted Degree Contractblty. n whch there s no upper bound on the number of weghted contractons,.e., n whch k = E G, the Weghted Contracton Degeneracy problem. In ths secton we prove that Weghted Degree Contractblty s NP-complete n the strong sense but FPT when parameterzed by d f all edge weghts are ntegers. We frst ntroduce some extra termnology. Let G = (V, E) be a connected graph. For a proper subset U V, the set of edges that have one end-vertex n U and the other one n U = V \ U s called an edge-cut denoted E(U, U). An edge-cut C s mnmal f G has no edge cut C C. The followng lemma s well known (see e.g [10]). Lemma 1. Let G = (V, E) be a connected graph and U V. Then E(U, U) s a mnmal edge-cut of G f and only f G[U] and G[U] are both connected. For the proofs of our results we also need the followng lemma. Lemma 2. Let G be a connected weghted graph wth an edge weghtng w, and let d R >0. Then G has a weghted contracton H wth δ w (H) d f and only f G has a mnmal edge-cut C wth w(c) d. Proof. Frst suppose that H s a weghted contracton of G wth δ w (H) d. Let W be a correspondng H-wtness structure. Let x be a vertex of H that s not a cut-vertex. Then the subgraphs of G nduced by U = W(x) and by U, respectvely, are connected. Lemma 1 tells us that C = E(U, U) s a mnmal edge-cut of G. Because d w H (x) d, we deduce that w(c) d.
P.A. Golovach et al. / Theoretcal Computer Scence 481 (2013) 74 84 81 Now suppose that G has a mnmal edge-cut C = E(U, U) wth w(c) d. Then G[U] and G[U] are connected graphs due to Lemma 1. Contractng the edges of G[U] and G[U] yelds a graph H that has two vertces and one edge wth weght at least d. Hence, δ w (H) d. Lemma 2 mples that the Weghted Contracton Degeneracy problem s equvalent to the Maxmum Mnmal Cut problem that s to test whether a connected graph G wth an edge weghtng w has a mnmal edge-cut C wth w(c) d for some gven nteger d. A problem s sad to be NP-complete n the strong sense, f t remans NP-complete even when all of ts numercal parameters are bounded by a polynomal n the sze of the nput. We prove that Weghted Degree Contractblty s NP-complete n the strong sense. Theorem 4. Weghted Contracton Degeneracy wth nteger edge weghts s NP-complete n the strong sense. Proof. It s clear that the problem s n NP. In order to prove NP-hardness, we reduce from the Max-Cut problem. Ths NP-complete problem [7] s to test whether a connected graph G has an edge-cut C wth at least s edges for some gven nteger s. Gven an nstance (G, s) of Max-Cut, we construct a weghted graph G as follows. We add two new adjacent vertces u and v to G by makng each of them adjacent to all vertces of G. We set w(uv) = n + m, and w(e) = 1 for all other edges n G. We let d = 2n + m + s. In ths way we obtaned an nstance (G, d) of Weghted Contracton Degeneracy. Observe that all numercal parameters of ths nstance,.e., all edge weghts and d are polynomally bounded n n and m. We clam that G has an edge-cut C wth C s f and only f G has a mnmal edge-cut C wth w(c ) d. Frst suppose that G has an edge-cut C = E G (U, U) wth C s. We defne U = U {u} and U = (V G \ U) {v}. Because both u and v are adjacent to all vertces of G, we fnd that both G [U ] and G [U ] are connected. Then Lemma 1 tells us that C = E G (U, U ) s a mnmal edge-cut of G. We also deduce that w(c ) w(uv) + n + s = n + m + n + s = d. Now suppose that G has a mnmal edge-cut C = E G (U, U ) wth w(c ) d. If u, v U or (symmetrcally) u, v U, then C E G +2n = 2n+m < d. Because C only contans edges of weght 1, we obtan w(c ) < d, whch s not possble. Hence, ether u U, v U or v U, u U. We may assume wthout loss of generalty that u U, v U. Then C = E G (U \ {u}, U \ {v}) s an edge-cut n G wth C = w(c ) w(uv) n d (n + m) n = 2n + m + s n m n = s. To conclude the proof, t remans to observe that, by Lemma 2, G has a weghted contracton H wth δ w (H) d f and only f G has a mnmal edge-cut C wth w(c ) d. Due to Theorem 4, we mmedately obtan the followng result by takng k = E. Corollary 2. Weghted Degree Contractblty wth nteger edge weghts s NP-complete n the strong sense. Recall that Contracton Degeneracy s n FPT when parameterzed by d [6]. We now show that Weghted Contracton Degeneracy s also FPT when parameterzed by d. For dong ths we frst gve some extra termnology. A tree decomposton of a graph G s a par (X, T) where T s a tree, the vertces of whch are called nodes, and X = {X V T } s a collecton of subsets (called bags) of V G such that the followng three condtons are satsfed: 1. V T X = V G ; 2. for each edge xy E G, the vertces x, y are n a bag X for some V T ; 3. for each x V G, the set { x X } nduces a connected subtree of T. The wdth of tree decomposton (X, T) s max VT { X 1}. The treewdth of a graph G s the mnmum wdth over all tree decompostons of G. Theorem 5. Weghted Contracton Degeneracy wth nteger edge weghts s n FPT when parameterzed by d. Proof. Let G = (V, E) be a weghted graph and d be a nonnegatve nteger. We may assume wthout loss of generalty that G s connected, as otherwse we can consder each connected component of G separately. We use Bodlaender s algorthm [5] to check n lnear tme whether the treewdth of G s at most 2d 2. Frst suppose that the treewdth of G s at most 2d 2. Because all edge weghts are ntegers, for a gven d, we can express Weghted Contracton Degeneracy n monadc second order logc usng Lemmas 1 and 2. Then we apply the well-known result of Courcelle [9] to solve the problem n lnear tme. Now suppose that the treewdth of G s at least 2d 1. We clam that G contans a mnmal edge-cut C wth w(c) d. In order to obtan a contradcton, we assume that all mnmal edge-cuts of G have weght less than d. We recursvely construct a tree decomposton of G as follows. Durng ts constructon we mantan the followng property. Suppose that at some moment we have constructed a tree decomposton (X, T) of a subgraph of G nduced by a subset U V. Then for each mnmal edge-cut C E G (U, U) of G the set Z = {z U z s ncdent wth an edge of C}
82 P.A. Golovach et al. / Theoretcal Computer Scence 481 (2013) 74 84 must be ncluded n some bag of (X, T). To ensure ths property, ntally, we set U = {u} for an arbtrary vertex u V. Then we recursvely extend U untl we get V as follows. Let C E G (U, U) be a mnmal edge-cut of G. Defne the sets Z = {z U z s ncdent wth an edge of C} Z = {z U z s ncdent wth an edge of C}. and Let X be a bag of the tree decomposton of G[U] that contans Z. We set U = U Z. To obtan a tree decomposton of G[U ], we add a pendant node j adjacent to node and defne the bag X j as Z Z. Then Z Z 2 C 2d 2, where the latter nequalty follows from the fact that C < d, as all edge weghts are postve ntegers and we assume that w(c) < d. For every mnmal edge-cut C E G (U, U ), the set of vertces n U ncdent wth an edge of C s ether contaned n Z or n Z. In the frst case we take the bag X, and n the second case we take the bag X j. Hence, we may proceed wth U and the obtaned tree decomposton of G[U ]. Ths eventually leads to a tree decomposton of G that has wdth at most 2d 2; a contradcton. By ths clam and Lemma 2, we conclude that the answer s always a Yes f the treewdth of G s at least 2d 1. 4. Concludng remarks We leave the problem of determnng the complexty of Weghted Degree Contractblty wth nteger edge weghts, when parameterzed by d, as an open problem. We conclude our paper wth two addtonal results. 4.1. Weghted Face Degree Subgraph We show that our algorthm from Theorem 3 can be appled for weghted faces n plane graphs; for the defntons of a plane graph and the geometrc dual of a plane graph and other related notons we refer the reader to the text-book of Destel [10]. The weghted face degree of a face f of a plane weghted graph G s the sum of all the weghts of the edges of G ncdent wth f. The Weghted Face Degree Subgraph problem s to test whether a plane weghted graph G can be modfed to a plane weghted graph of mnmum weghted face degree at least d by usng at most k edge removals. Theorem 6. Weghted Face Degree Subgraph s n FPT when parameterzed by k. Proof. Gven an nstance of Weghted Face Degree Subgraph wth a plane weghted graph G and an nteger d we do as follows. Let G denote the geometrc dual of G. There s a one-to-one correspondence between the edges of G and the edges of G. Let e be the edge of G that corresponds to the edge e n G. We assgn weghts to the edges of G n the followng way: w G (e ) = w G (e). An embedded contracton of an edge e of a plane graph s a contracton of e that respects the embeddng and keeps multple edges f they appear (that s, f the endponts of e have common neghbors). We observe that the dual of the graph obtaned from a plane graph G by removng an edge e s the graph obtaned from G by an embedded contracton of e. We apply our algorthm from Theorem 3 for Weghted Degree Contractblty for G and degree d. Note that there s a one-to-one correspondence between the faces of G and the vertces of G. Therefore, weghted contractons can smulate the face degree transformatons of a graph wth embedded contractons and multple edges. Due to the equvalence between edge removals n a plane graph and embedded edge contractons n ts dual, the sequence of k edges of G that s a soluton to the Weghted Degree Contractblty problem can be transformed nto a sequence of k edge removals n G. Hence, Weghted Face Degree Subgraph can be solved n FPT tme. 4.2. Euleran Smple-Contractblty As future work, one may consder other varants of constraned contractblty problems, for example by mposng party constrans. To show that ths may be nterestng, we spot an algorthm that solves such a problem. Recall that a smple contracton s the operaton on loopless multgraphs that dentfes both end-vertces of the edge and keeps multple edges but removes the loop that was created. A connected multgraph s Euleran f all ts vertces have even degree. The problem Euleran Smple-Contractblty s to fnd the mnmum number of smple contractons that transforms a connected multgraph nto an Euleran graph. Whle t has never been stated explctly n the lterature, Euleran Smple-Contractblty admts a polynomal-tme algorthm. We show ths below. Let G be a multgraph. A vertex of G s odd (or even) when ts degree s odd (or even). An odd-vertex parng s a subset of edges, whose removal from G yelds a multgraph that contans no odd vertces. Hadlock [15] consdered mnmum oddvertex parngs and proved the followng result. Lemma 3 ([15]). Let P be a subset of edges of a multgraph G. Then P s a mnmum odd-vertex parng f and only f P forms a collecton of edge-dsjont paths wth the odd vertces of G as endponts, usng each as an endpont once, and wth mnmum sum of path lengths.
P.A. Golovach et al. / Theoretcal Computer Scence 481 (2013) 74 84 83 We use Lemma 3 to show the followng lemma. Lemma 4. Let P be a subset of edges of a multgraph G. Then P s a mnmum set of edges whose smple contractons transform G nto an Euleran graph f and only f P s a mnmum odd-vertex parng of G. Proof. Frst suppose that P s a mnmum set of edges whose smple contractons transform G nto an Euleran graph. We wll prove that P s a mnmum odd-vertex parng by nducton on P. Let P = 1. Then the endponts of the only edge e n P must be odd n G, and they must be the only two odd vertces n G, because P s a mnmum set of edges whose smple contractons transform G nto an Euleran graph. Hence, P s an odd-vertex parng. Because P = 1, we deduce that P s mnmum. Let P > 1. Let C be a connected component of P. Because P s mnmum, C s a tree. Consder an edge vw n P such that v s a leaf of C (and w s ts neghbor n C). Let x vw be the new vertex obtaned as the result of a smple contracton of vw. Also, let P be the resultng set of edges obtaned from P, and let G be the resultng multgraph obtaned from G, after performng the smple contracton of vw. Then P s a mnmum set of edges whose smple contractons transform G nto an Euleran graph. By our nducton hypothess, P s a mnmum odd-vertex parng of G. If v s even n G, then the smple contractons of all the edges n P except the edge ncdent wth ths leaf would transform G nto an Euleran graph; a contradcton wth the choce of P. Hence, v s an odd vertex n G. If w s odd n G, then x vw s even n G, because v s odd n G. Ths s not possble, because P s a mnmum odd-vertex parng of G, and by Lemma 3 ths means that P forms a collecton of edge-dsjont paths wth the odd vertces of G as endponts, usng each as an endpont once, Hence, w s even n G. Consequently, P forms a collecton of edge-dsjont paths wth the odd vertces of G as endponts, usng each as an endpont once. Moreover, P s mnmum because we need at least one edge to cover v. By Lemma 3, we then fnd that P s a mnmum odd-vertex parng of G. Now suppose that P s a mnmum odd-vertex parng. Then Lemma 3 tells us that P forms a collecton of edge-dsjont paths wth the odd vertces of G as endponts, usng each as an endpont once, and wth mnmum sum of path lengths. We observe that the vertex obtaned by the smple contracton of an edge xy E s odd f and only f one of {x, y} s even and the other one s odd. Hence, the smple contractons of all edges n P transform G nto an Euleran graph. Suppose that there exsts a smaller set P of edges whose smple contractons transform G nto an Euleran graph. Then we may assume wthout loss of generalty that P s mnmum. However, as shown n the forward mplcaton, P s a mnmum odd-vertex parng as well. Ths s not possble. Hence, P s mnmum. Ths completes the proof of Lemma 4. Hadlock [15] observed that a mnmum odd-vertex parng can be computed n polynomal tme by a reducton to a maxmum matchng problem. 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