Parameterized Complexity of Connected Even/Odd Subgraph Problems

Paameteized Complexity of Connected Even/Odd Subgaph Poblems Fedo V. Fomin 1 and Pet A. Golovach 2 1 Depatment of Infomatics, Univesity of Begen PB 7803, 5020 Begen, Noway fedo.fomin@ii.uib.no 2 School of Engineeing and Computing Sciences, Duham Univesity Science Laboatoies, South Road, Duham DH1 3LE, UK pet.golovach@duham.ac.uk Abstact Cai and Yang initiated the systematic paameteized complexity study of the following set of poblems aound Euleian gaphs. Fo a given gaph G and intege k, the task is to decide if G contains a (connected) subgaph with k vetices (edges) with all vetices of even (odd) degees. They succeed to establish the paameteized complexity of all cases except two, when we ask about a connected k-edge subgaph with all vetices of odd degees, the poblem known as k-edge Connected Odd Subgaph; and a connected k- vetex induced subgaph with all vetices of even degees, the poblem known as k-vetex Euleian Subgaph. We esolve both open poblems and thus complete the chaacteization of even/odd subgaph poblems fom paameteized complexity pespective. We show that k-edge Connected Odd Subgaph is FPT and that k-vetex Euleian Subgaph is W[1]-had. Ou FPT algoithm is based on a novel combinatoial esult on the teewidth of minimal connected odd gaphs with even amount of edges. 1998 ACM Subject Classification F.2.2 Nonnumeical Algoithms and Poblems, G.2.1 Combinatoics, G.2.2 Gaph Theoy Keywods and phases Paameteized complexity, Eule gaph, even gaph, odd gaph, teewidth Digital Object Identifie 10.4230/LIPIcs.STACS.2012.432 1 Intoduction An even gaph (espectively, odd gaph) is a gaph whee each vetex has an even (odd) degee. Recall that an Euleian gaph is a connected even gaph. Let Π be one of the following fou gaph classes: Euleian gaphs, even gaphs, odd gaphs, and connected odd gaphs. In [4], Cai and Yang initiated the study of paameteized complexity of subgaph poblems motivated by Euleian gaphs. Fo each Π, they defined the following paameteized subgaph and induced subgaph poblems: This wok is suppoted by EPSRC (EP/G043434/1), Royal Society (JP100692), and the Euopean Reseach Council (ERC) via gant Rigoous Theoy of Pepocessing, efeence 267959. Fedo V. Fomin and Pet A. Golovach; licensed unde Ceative Commons License NC-ND 29th Symposium on Theoetical Aspects of Compute Science (STACS 12). Editos: Chistoph Dü, Thomas Wilke; pp. 432 440 Leibniz Intenational Poceedings in Infomatics Schloss Dagstuhl Leibniz-Zentum fü Infomatik, Dagstuhl Publishing, Gemany

F.V. Fomin and P.A. Golovach 433 k-edge Π Subgaph (esp. k-vetex Π Subgaph) Instance: A gaph G and non-negative intege k. Paamete: k. Question: Does G contain a subgaph with k edges fom Π (esp. an induced subgaph on k vetices fom Π)? Cai and Yang established the paameteized complexity of all vaiants of the poblem except k-edge Connected Odd Subgaph and k-vetex Euleian Subgaph, see Table 1. It was conjectued that k-edge Connected Odd Subgaph is FPT and k- Vetex Euleian Subgaph is W[1]-had. We esolve these open poblems and confim both conjectues. Euleian Even Odd Connected Odd k-edge FPT [4] FPT [4] FPT [4] FPT Thm. 3 k-vetex W[1]-had Thm. 4 FPT [4] FPT [4] FPT [4] Table 1 Paameteized complexity of k-edge Π Subgaph and k-vetex Π Subgaph. The emaining pat of the pape is oganized as follows. In Section 2, we povide definitions and give peliminay esults. In Section 3, we show that k-edge Connected Odd Subgaph is FPT. Ou algoithmic esult is based on an uppe bound fo the teewidth of a minimal connected odd gaphs with an even numbe of edges. We show that the teewidth of such gaphs is always at most 3. The poof of this combinatoial esult, which we find inteesting in its own, is non-tivial and is given in Section 4. The bound on the teewidth is tight complete gaph on fou vetices K 4 is a minimal connected odd gaph with an even numbe of edges and its teewidth is 3. In Section 5, we pove that k-vetex Euleian Subgaph is W[1]-had and obseve that the poblem emains W[1]-had if we ask about (not necessay induced) Euleian subgaph on k vetices. We conclude the pape in Section 6 with some open poblems. 2 Definitions and Peliminay Results Gaphs. We conside finite undiected gaphs without loops o multiple edges. The vetex set of a gaph G is denoted by V (G) and its edge set by E(G). A set S V (G) of paiwise adjacent vetices is called a clique. Fo a vetex v, we denote by N G (v) its (open) neighbohood, that is, the set of vetices which ae adjacent to v. Distance between two vetices u, v V (G) (i.e., the length of the shotest (u, v)-path in the gaph) is denoted by dist G (u, v). Fo a vetex v and a positive intege k, N (k) G [v] = {u V (G) dist G(u, v) k}. The degee of a vetex v is denoted by d G (v), and (G) is the maximum degee of G. Fo a set of vetices S V (G), G[S] denotes the subgaph of G induced by S, and by G S we denote the gaph obtained fom G by the emoval of all the vetices of S, i.e. the subgaph of G induced by V (G) \ S. Paameteized Complexity. Paameteized complexity is a two dimensional famewok fo studying the computational complexity of a poblem. One dimension is the input size n and anothe one is a paamete k. It is said that a poblem is fixed paamete tactable (o FPT), if it can be solved in time f(k) n O(1) fo some function f. One of basic assumptions of the Paameteized Complexity theoy is the conjectue that the complexity class W[1] FPT, S TAC S 1 2

434 Paameteized Complexity of Connected Even/Odd Subgaph Poblem and it is unlikely that a W[1]-had poblem could be solved in FPT-time. We efe to the books of Downey and Fellows [6], Flum and Gohe [7], and Niedemeie [8] fo detailed intoductions to paameteized complexity. Teewidth. A tee decomposition of a gaph G is a pai (X, T ) whee T is a tee and X = {X i i V (T )} is a collection of subsets (called bags) of V (G) such that: 1. i V (T ) X i = V (G), 2. fo each edge {x, y} E(G), x, y X i fo some i V (T ), and 3. fo each x V (G) the set {i x X i } induces a connected subtee of T. The width of a tee decomposition ({X i i V (T )}, T ) is max i V (T ) { X i 1}. The teewidth of a gaph G (denoted as tw(g)) is the minimum width ove all tee decompositions of G. Minimal odd gaphs with even numbe of edges. We say that a gaph G is odd if all vetices of G ae of odd degee. Let be a vetex of G. We assume that G is ooted in. Let G be a connected odd gaph with an even numbe of edges. We say that G is a minimal if G has no pope connected odd subgaphs with an even numbe of edges containing. The impotance of minimal odd subgaphs with even numbes of edges is cucial fo ou algoithm because of the following combinatoial esult. Theoem 1. Let G be a minimal connected odd gaph with an even numbe of edges with a oot. Then tw(g) 3. Fo non-ooted gaphs, we also have the following coollay. Coollay 2. Fo any minimal connected odd gaph G with an even numbe of edges, tw(g) 3. Let us emak that the bound in Theoem 1 is tight complete gaph K 4 with a oot vetex is a minimal odd gaph with even numbe of edges and of teewidth 3. The poof of Theoem 1 is given in Section 4. This poof is non-tivial and technical, and we find the combinatoial esult of Theoem 1 to be inteesting in its own. Fom algoithmic pespective, Theoem 1 is a conestone of ou algoithm; combined with colo coding technique of Alon, Yuste and Zwick in [1] it implies that k-edge Connected Odd Subgaph is FPT. We give this algoithm in the next section. 3 Algoithm fo k-edge Connected Odd Subgaph To give an algoithm fo k-edge Connected Odd Subgaph, in addition to Theoem 1, we also need the following esult of Alon, Yuste and Zwick fom [1] obtained by a poweful colo-coding technique. Poposition 1 ([1]). Let H be a gaph on k vetices with teewidth t. Let G be a n-vetex gaph. A subgaph of G isomophic to H, if one exists, can be found in O(2 O(k) n t+1 ) expected time and in O(2 O(k) n t+1 log n) wost-case time. We ae eady to pove the main algoithmic esult of this pape. Theoem 3. k-edge Connected Odd Subgaph can be solved in time O(2 O(k log k) n 4 log n) fo n-vetex gaphs. Poof. Let (G, k) be an instance of the poblem. We apply the following algoithm. Step 1. If k is odd and (G) k, then etun Yes. Else if k is odd but (G) < k, then go to Step 3.

F.V. Fomin and P.A. Golovach 435 Step 2. If k is even and (G) k, then we enumeate all odd connected gaphs H with k edges of teewidth at most 3. Fo each odd gaph H of teewidth at most 3 and with k edges, we use Poposition 1 to check whethe G has a subgaph isomophic to H. The algoithm etuns Yes if such a gaph H exists. Othewise, we constuct a new gaph G by emoving fom the old gaph G all vetices of degee at least k. Step 3. Fo each vetex v, check whethe thee is a connected odd subgaph H with k edges that contains v. To do it, we enumeate all connected subgaphs with p = 0,..., k edges that include v using the following obsevation. Fo evey connected subgaph H of G with p 1 edges such that v V (H), thee is a connected subgaph H with p 1 edges such that v V (H ) and H is a subgaph of H. Hence, given all connected subgaphs with p 1 edges, we can enumeate all subgaphs with p edges by a bute-foce algoithm. The algoithm etuns Yes if a connected odd subgaph H with k edges exists fo some vetex v, and it etuns No othewise. 1 In what follows we discuss the coectness of the algoithm and evaluate its unning time. If k is odd and (G) k, then the sta K 1,k is a subgaph of G. Hence, G has a connected odd subgaph with k edges. Let k be even and let V (G) be a vetex with d G () k. If G has a connected odd subgaph with k edges containing, then G has a minimal connected odd subgaph H with even numbe of edges ooted in. Let l = E(H). Gaph H contains at most l vetices in N G (). It follows that thee ae k l vetices v 1,..., v k l N G () \ V (H). Denote by H the subgaph of G with the vetex set V (H) {v 1,..., v k l } and the edge set E(H) {v 1,..., v k l }. Since k and l ae even, we have that H is an odd gaph. By Theoem 1, tw(h) 3. Gaph H is obtained fom H by adding some vetices of degee 1, and, theefoe, tw(h ) 3. This means that when G has a connected odd subgaph H with k edges containing, then thee is a connected odd subgaph H with k edges containing and of teewidth at most thee. But then in Step 2, we find such a gaph H with k edges. If no connected odd subgaph with k edges was found in Step 2, then if such a gaph exist, it contains no vetex of degee (in G) at least k. Theefoe all such vetices can be emoved fom G without changing the solution. Finally, in Step 3, tying all possible connected subgaphs with k edges in the obtained gaph of maximum degee at most k 1, we can deduce if G contains an odd subgaph with k edges. Concening the unning time of the algoithm. Thee ae at most ( ) k(k 1)/2 k nonisomophic gaphs with k edges, and we can find all connected odd gaphs with k edges in time 2 O(k log k) and to check in time O(k) if the teewidth of each of the gaphs is at most thee by making use of Bodlaende s algoithm [3]. The unning time of this pat can be educed to 2 O(k), see e.g. [2]. Then fo each gaph H of this type, to check whethe H is a subgaph of G, takes time O(2 O(k) n 4 log n) by Poposition 1. When we aive at Step 3, we have that (G) k 1. We show by induction that fo any p 1, thee ae at most p!k p connected subgaphs with p edges that contain a given vetex v. Clealy, the claim holds fo p = 1. Let p > 1. Any connected subgaph of G with p 1 edges has at most p vetices. Since thee ae at most pk possibilities to add an edge to this subgaph to obtain a connected subgaph with p edges, the claim follows. Theefoe, fo each vetex v, we can enumeate all connected subgaphs H with k edges that include v in 1 The idea of Step 3 is due to anonymous STACS efeee. This allows us to impove the unning time O(2 O(k2 log k) n 4 log n) of the algoithm fom the oiginal vesion. S TAC S 1 2

436 Paameteized Complexity of Connected Even/Odd Subgaph Poblem time O(k!k k ). Hence, Step 3 can be done in time O(2 O(k log k) n). We conclude that the total unning time of the algoithm is O(2 O(k log k) n 4 log n). 4 Minimal connected odd gaphs with even numbe of edges In this section we give a high level desciption of the poof of Theoem 1, the main combinatoial esult of this pape. The poof is inductive, and fo the inductive step we identify specific stuctues in a minimal connected odd gaph with an even numbe of edges. To poceed with the inductive step, we need a stonge vesion of Theoem 1. Let G be a gaph and let x V (G). We say that a gaph G is obtained fom G by splitting x into x 1, x 2, if G is constucted as follows: fo a patition X 1, X 2 of N G (x), we eplace x by two vetices x 1, x 2, and join x 1, x 2 with the vetices of X 1, X 2 espectively. The following claim implies Theoem 1. Claim 1. Let G be a minimal connected odd gaph with an even numbe of edges with a oot. Then tw(g) 3. Moeove, if d G () = 1 and z is the unique neighbo of, then at least one of the following holds: i) thee is a tee decomposition (X, T ) of G of width at most thee such that fo any bag X i X with z X i, X i 3; o ii) fo any gaph G obtained fom G by splitting z into z 1, z 2, tw(g ) 3 and thee is a tee decomposition (X, T ) of G of width at most thee such that thee is a bag X i X containing both z 1 and z 2. To descibe the stuctues in the gaph, we need a notion of a subgaph with teminals. Roughly speaking, a subgaph with teminals is connected to the emaining pat of the gaph only via teminals. Moe fomally, let H be a subgaph of gaph G, and let,..., s V (H). We say that H is a subgaph of G with teminals,..., s if thee is a subgaph F of G such that G = F H; V (F ) V (H) = {,..., s }; and E(F ) E(H) =. Thus evey edge of G having at least one endpoint in a non-teminal vetex of H, should be an edge of H. In paticula, teminal vetices of H sepaate non-teminal vetices of H fom othe vetices of G. We also say that a subgaph H with a given set of teminals is sepaating if the gaph obtained fom G by the emoval of all non-teminal vetices of H and all the edges of H (denoted G H) is not connected. The specific stuctues we ae looking fo in the inductive step ae the subgaphs isomophic to gaphs with teminals fom the set H = {H 1, H 2, H 3, H 4, H 5, H 6 } shown in Fig. 1. We often say that H i H is contained in gaph G (o G has H i ) if G has a subgaph isomophic to H i with the teminals shown in Fig. 1. Notice that H 6 is a subgaph of H 4 and H 5, and we ae looking fo H 6 only if we cannot find H 4 o H 5. The poof of Claim 1 is by induction on the numbe of edges. The basis case is a gaph with 6 edges. Evey connected odd gaph with an even numbe of edges has at least 6 edges, and thee ae only two gaphs with 6 edges that have these popeties, these gaphs ae shown in Fig. 2. Tivially, Claim 1 holds fo these gaphs fo any choice of the oot. Then we assume that a minimal connected odd gaph G with an even numbe of edges has at least 8 edges.

F.V. Fomin and P.A. Golovach 437 s 3 s 4 s 5 H 2 s 3 s 3 s 4 s 3 s 3 s 6 H 1 H 3 H 4 H 5 H 6 Figue 1 The set H. Figue 2 The base of the induction: Minimal gaphs with six edges. If G contains a subgaph R with teminals, shown in Fig. 3 such that / V (R) \ {, } and / E(G), then we eplace R by edge. It is possible to show that the esulting gaph G is a minimal connected odd gaph with an even numbe of edges. Since G has less edges than G, we can use the inductive assumption. Futhemoe we assume that G has no R. R G G Figue 3 Replacement of R. Next step is to pove that if G has no subgaph fom H, then G is one of the gaphs G 1, G 2, G 3 shown in Fig. 4. Fo each of these gaphs the theoem tivially holds. Actually, we will need a stonge esult, saying that if G has no subgaph fom H 2,..., H 6 and evey subgaph of G isomophic to H 1 is of specific fom, namely, this subgaph is not sepaating and is not a non-teminal vetex of H 1, then even in this case, G is one of the gaphs G 1, G 2, G 3 shown in Fig. 4. The poof of this claim is not staightfowad. With this claim we can poceed futhe with an assumption that G contains at least one gaph fom H. Fo the case when is a non-teminal vetex of a subgaph H H, we pove that H = H 1. We emove non-teminal vetices of H, identify teminals,, and add a new oot vetex adjacent to the vetex obtained fom,. Then we pove that this gaph is a minimal connected odd gaph with an even numbe of edges, and then we can apply the induction assumption on this gaph, and deive ou claim fo G. The difficulty hee is to ensue that the teewidth of the gaph G does not incease when we make the inductive step. This equies the assumptions i) and ii) in Claim 1 on the stuctue of tee decompositions. Fom this point, it can be assumed that is not a non-teminal vetex of a subgaph fom H with the coesponding set of teminals. All gaphs H 2,..., H 6 have even numbe of edges and evey teminal vetex of such a gaph is of even degee. This means, that G cannot contain a non-sepaating gaph H fom {H 2,..., H 6 }, because emoving edges and non-teminal vetices of H, would esult in a S TAC S 1 2

438 Paameteized Complexity of Connected Even/Odd Subgaph Poblem G 1 G 2 G 3 Figue 4 Gaphs G 1, G 2, G 3. connected odd subgaph of G with even numbe of edges, which is a contadiction to the minimality of G. Hence, if G contains subgaphs fom H but they ae non-sepaating, G can contain only H 1. Then as we aleady have shown, G is one of the gaphs G 1, G 2, G 3 shown in Fig. 4. = s 3 s 3 x 2 y 2 x 2 y 2 x 1 y 1 x 1 y 1 x 1 s 4 F (1) 1 s 4 F (2) 1 x 2 y 1 s 4 F (3) 1 s 3 y 2 x 1 s 3 x 2 y 1 s 4 F (4) 1 y 2 Figue 5 The case H = H 4, the tees F (1) 1,..., F (4) 1 ae fomed by bold" edges. Thus we can assume that G contains a sepaating subgaph H fom H. Among all such sepaating subgaphs, we select H such that the numbe of edges of the component F 1 of the gaph G = G H containing is minimum. We pove that G has exactly two components F 1,, whee F 1 is a tee. We consequently conside the cases H = H 1,..., H 6 and ague as follows. If H = H 1, then F 1 = K 2 and we apply induction fo ooted in one of the teminals of H. If H = H 2, then we pove that F 1 = K 2. If = K 2, then the poof follows diectly. Othewise, we identify teminals, s 3, and add a new oot adjacent to the vetex obtained fom, s 3. It is possible to show that the constucted gaph is a minimal connected odd gaph with an even numbe of edges, and we can use the induction assumption fo this gaph. The aguments fo the case H = H 3 ae simila. If H = H 4, then we pove that F 1 is one of the tees F (1) 1,..., F (4) 1 shown in Fig. 5. Fo, we pove that tw( ) 2, and use this fact to constuct a tee decomposition of G of width thee. The case H = H 5 is simila. Finally, fo H = H 6, we pove that it can be assumed that,, s 4, s 5 V (F 1 ), s 3, s 6 V ( ), and F 1 is the tee shown in Fig. 6. Then we apply fo the same aguments as in the case H = H 4. In each of the cases, we succeed to educe G to a smalle minimal connected odd gaph G with even numbe of edges and show that tw(g) tw(g ), which completes the induction step. s 4 s 5 s 3 s 6 Figue 6 The case H = H 6, the tee F 1 is induced by bold" edges.

F.V. Fomin and P.A. Golovach 439 5 Complexity of k-vetex Euleian Subgaph In this section we pove that k-vetex Euleian Subgaph is W[1]-had. Theoem 4. The k-vetex Euleian Subgaph is W[1]-had. Poof. We educe fom the well-known W[1]-complete k-clique poblem (see e.g. [6]): k-clique Instance: A gaph G and non-negative intege k. Paamete: k. Question: Does G contain a clique with k vetices? Notice that the poblem emains W[1]-complete when the paamete k is esticted to be odd. It follows immediately fom the obsevation that the existence of a clique with k vetices in a gaph G is equivalent to the existence of a clique with k + 1 vetices in the gaph obtained fom G by the addition of a univesal vetex adjacent to all the vetices of G. Fom now it is assumed that k > 1 is an odd intege. Let G be a gaph. We constuct the gaph G by subdividing edges of G by k 2 vetices, i.e. each edge xy is eplaced by an (x, y)-path of length k 2 + 1. We say that u V (G ) is a banch vetex if u V (G), and u is a subdivision vetex othewise. We also say that u is a subdivision vetex fo an edge xy E(G) if u is a subdivision vetex of the path obtained fom xy. We claim that G has a clique of size k if and only if G has an induced Euleian subgaph on k = 1 2 (k 1)k3 + k vetices. Suppose that G has a clique K with k vetices. Let H be the subgaph of G induced by K and the subdivision vetices fo all edges xy with x, y K. It is easy to see that H is a connected Euleian gaph on k = 1 2 (k 1)k3 + k vetices. Let now H be an induced Euleian subgaph of G on k = 1 2 (k 1)k3 + k vetices. Denote by U the set of banch vetices of H, and let p = U. Let A = {xy E(G) x, y U, and H has a subdivision vetex fo xy} and let F = (U, A). Let also q = A. Since H is connected, the gaph F is connected as well. Obseve that if u V (H) is a subdivision vetex fo an edge xy E(G), then all subdivision vetices fo xy ae vetices of H and x, y V (H). It follows that H has p + q k 2 = k vetices, and we have p k = ( 1 2 (k 1)k q)k2. Since k 2 is a diviso of p k, p k. Suppose that p > k. Then since k 2 is a diviso of p k, p k 2 + k. Any connected gaph with p vetices has at least p 1 edges, and it means that q k 2 + k 1 > 1 2 (k 1)k. We get that 0 < p k = ( 1 2 (k 1)k q)k2 < 0; a contadiction. We conclude that p = k. Then q = 1 2 (k 1)k and U is a clique with k vetices. Recall that k-vetex Euleian Subgaph asks about an induced Euleian subgaph on k vetices. Fo the gaph G in the poof of Theoem 4, any Euleian subgaph is induced. It gives us the following coollay. Coollay 5. The following poblem: Instance: A gaph G and non-negative intege k. Paamete: k. Question: Does G contain an Euleian subgaph with k vetices? is W[1]-had. S TAC S 1 2

440 Paameteized Complexity of Connected Even/Odd Subgaph Poblem 6 Conclusion We poved that k-edge Connected Odd Subgaph is FPT and k-vetex Euleian Subgaph is W[1]-had. This completes the chaacteization of even/odd subgaph poblems with exactly k edges o vetices fom paameteized complexity pespective. While it is tivial to decide whethe a gaph G has a (connected) even o odd subgaph with at most k edges o vetices, the question about a subgaph with at least k edges o vetices seems to be much moe complicated. Fo At Least k-edge Odd Subgaph and At Least k-vetex Odd Subgaph, following the lines of the poofs fom [4] fo k-edge Odd Subgaph and k-vetex Odd Subgaph, it is possible to show that these poblems ae in FPT. Fo othe cases, the appoaches used in [4] and in ou pape, do not seem to wok. Cai and Yang in [4] also consideed dual poblems whee the aim is to find an even o odd subgaph of a gaph G with V (G) k vetices o E(G) k edges espectively. Recently, these esults wee complemented by Cygan et al. [5]. Howeve, the complexity of the dual poblem to k-edge Connected Odd Subgaph, namely, obtaining connected odd subgaph with E(G) k edges, emains open. Acknowledgments. The authos ae gateful to the anonymous efeees fo thei constuctive suggestions and emaks. Refeences 1 Noga Alon, Raphael Yuste, and Ui Zwick. Colo-coding. J. ACM, 42(4):844 856, 1995. 2 Omid Amini, Fedo V. Fomin, and Saket Sauabh. Counting subgaphs via homomophisms. In Poceedings of the 36th Intenational Colloquium on Automata, Languages and Pogamming (ICALP 2009), volume 5555 of Lectue Notes Comp. Sci., pages 71 82. Spinge, 2009. 3 Hans L. Bodlaende. A linea-time algoithm fo finding tee-decompositions of small teewidth. SIAM J. Comput., 25(6):1305 1317, 1996. 4 Leizhen Cai and Boting Yang. Paameteized complexity of even/odd subgaph poblems. J. Discete Algoithms, 9(3):231 240, 2011. 5 Maek Cygan, Dániel Max, Macin Pilipczuk, MichałPilipczuk, and Ildikó Schlotte. Paameteized complexity of euleian deletion poblems. In Poceedings of the 37th Intenational Wokshop on Gaph-Theoetic Concepts in Compute Science (WG 2011), volume 6986 of Lectue Notes Comp. Sci., page31 142. Spinge, 2011. 6 Rodney G. Downey and Michael R. Fellows. Paameteized complexity. Monogaphs in Compute Science. Spinge-Velag, New Yok, 1999. 7 Jög Flum and Matin Gohe. Paameteized complexity theoy. Texts in Theoetical Compute Science. An EATCS Seies. Spinge-Velag, Belin, 2006. 8 Rolf Niedemeie. Invitation to fixed-paamete algoithms, volume 31 of Oxfod Lectue Seies in Mathematics and its Applications. Oxfod Univesity Pess, Oxfod, 2006.