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10Вестник Сыктывкарского университета. Сер Л. Вып.И.2010
УДК 519.61
CHOOSABILITY OF P5-FREE GRAPHS1 Petr A. Golovachj Pinar Heggernes
A graph is fc-choosable if it admits a proper coloring of its vertices for every assignment of k (possibly different) allowed colors to choose from for each vertex. It is NP-hard to decide whether a given graph is fc-choosable for k > 3, and this problem is considered strictly harder than the fc-coloring problem. Only few positive results are known on input graphs with a given structure. Here, we prove that the problem is fixed parameter tractable on P5-free graphs when parameterized by k. This graph class contains the well known and widely studied class of cographs. Our result is surprising since the parameterized complexity of fc-coloring is still open on p5-free graphs. To give a complete picture, we show that the problem remains NP-hard on p5-free graphs when k is a part of the input.
1. Introduction
Graph coloring is one of the most well known and intensively studied problems in graph theory. The ^-COLORING problem asks whether the vertices of an input graph G can be colored with k colors such that no pair of adjacent vertices receive the same color (such coloring is also called a proper coloring). This problem is known to be NP-complete even when k > 3 is not a part of the input but a fixed constant.
Vizing [23] and ErdHos et al. [7] introduced a version of graph coloring called list coloring. In list coloring, a set L(v) of allowed colors is given for each vertex v of the input graph, and we want to decide whether a proper coloring of the graph exists such that each vertex v receives a color from L(v). If G has a list coloring for every assignment of lists of cardinality k to its vertices, then G is said to be fc-choosable. Hence the
1 Preliminary extended abstract of this paper appeared in the proceedings of MFCS'09 [10]
© Petr A. Golovach, Pinar Heggernes, 2010.
fc-choosablllty problem asks whether an input graph G is fc-choosable. List coloring has received increasing attention since the beginning of 90's, and there are very good surveys [1,21] and books [14] on the subject. It is proved to be a very difficult problem; Gutner and Tarsi [11] proved that fc-choosablllty is Il^-complete for bipartite graphs for any fixed k > 3, whereas 2-Choosability can be solved in polynomial time [7]. The 3-Choosability and 4-Choosability problems remain Il^-complete for planar graphs, whereas any planar graph is 5-choosable [20]. Due to these hardness results, upto the assumption that NP is not equal to co-NP, Choosability is strictly harder than Coloring on general graphs [1].
Despite being a difficult problem to deal with, Choosability has applications in a large variety of areas, like various kinds of scheduling problems, VLSI design, and frequency assignments [1]. Consequently, any attempt to solve this problem is of interest, and we attack it using structural information on the input and parameterized algorithms. A problem is fixed parameter tractable (FPT) if its input can be partitioned into a main part (typically the input graph) of size n and a parameter (typically an integer) k so that there is an algorithm that solves the problem in time 0(nc • /(&)), where / is a computable function dependent only on £;, and c is a fixed constant independent of input [6]. In this case, we say that the problem is FPT when parameterized by k. The field of parameterized algorithms and fixed parameter complexity/tractability has been flourishing during the last decade, with many new results appearing every year in high level conferences and journals, and it has been enriched by several new books [8,18].
In this paper, we show that ^-choosability is fixed parameter tractable on P5-free graphs. These are graphs containing no induced copy of a simple path on 5 vertices, and this graph class contains the class of cographs that has been subject to extensive theoretical study [3]. An interesting point to mention is that the fixed parameter tractability of /c-Coloring on P5-free graphs is still open [12]. As mentioned above, Choosability is more difficult than Coloring on general graphs. Our result indicates that the opposite might be true for the class of P5-free graphs. Hoang et al. showed that ^-coloring can be solved in polynomial time for any fixed k on P5-free graphs [12], but in their running time k contributes to the degree of the polynomial. Furthermore, ^-coloring is NP-complete on P5-free graphs when A; is a part of input [15]. To give a complete picture, here we show that ^-choosability is NP-hard on p5-free graphs when A; is a part of input. Thus fixed parameter tractability is the best we can expect to achieve for /c-Choosability on this graph class.
To mention other existing results on the coloring problem on graphs
that do not contain long induced paths, 3-coloring has a polynomial-time solution on P6-free graphs [19], 5-Coloring is NP-complete for P8-free graphs [24], 4-Coloring is NP-complete for p9-free graphs [16], and 6-Coloring is NP-complete for p7-free graphs [4]. Also, the List Coloring problem is NP-complete for lists of size at most three even for complete bipartite graphs [13] (see also Lemma 1).
2. Definitions and preliminaries
We consider finite undirected graphs without loops or multiple edges. A graph is denoted by G — (V, E), where V = V(G) is the set of vertices and E = E(G) is the set of edges. For a vertex v e V, the set of vertices that are adjacent to v is called the neighborhood of v and denoted by Nq(v) (we may omit index if the graph under consideration is clear from the context). The degree of a vertex v is deg(v) — |TV(^) |. The average degree of G is d(G) = i^i deg(v). For a vertex subset U c V the subgraph of G
induced by U is denoted by G[U]. A set U c V is a clique if all vertices in U are pairwise adjacent in G. A set of vertices U is a dominating set if for each vertex v e V, either v e U or there is a vertex u e U such that v e N(u). We also say that a subgraph H of G is dominating if V(H) is a dominating set. We denote by G — U the graph G[V\U], and by G — u the graph G[V \ {u}] for u e V.
A vertex coloring of a graph G = (V, E) is an assignment c: V —>> N of a positive integer (color) to each vertex of G. The coloring c is proper if adjacent vertices receive distinct colors. Assume that each vertex v e V is assigned a color list L(v) c N, which is the set of admissible colors for v. A mapping c: V N is a list coloring of G if c is a proper vertex coloring and c(y) e L(v) for every v e V. For a positive integer £;, G is k-choosable if G has a list coloring for every assignment of color lists L(v) with \L(v)\ = k for all v e V. The choice number (also called list chromatic number) of G, denoted ch(G), is the minimum integer k such that G is fc-choosable. The /c-Choosability problem asks for a given graph G and a positive integer h, whether G is fc-choosable. It is known that dense graphs have large choice number [1], as indicated by the following result.
Proposition 1 ( [1]). Let G be a graph and s be an integer. If
then ch(G) > s.
By Pr we denote the graph on vertex set • • • ,vr} and edge set
• • • ,vr-ivr}. A graph is Pr-free if it does not contain Pr as an induced subgraph. Cographs are the class of iVfree graphs, and they are contained in the class of P5-free graphs. These graph classes can be recognized in polynomial time. The following structural property of P5-free graphs was proved by Bacso and Tuza [2].
Proposition 2 ( [2]). Every connected P^-free graph has either a dominating clique or a dominating P3.
It follows from the results of [2] that such a clique or path can be constructed in polynomial time.
Finally, we distinguish between the parameterized and the non-parameterized versions of our problem. In the Choosability problem, G and k are input. We denote by /c-Choosability the version of the problem parameterized by k.
3. Fixed parameter tract ability of choosability problems on P5-free graphs
3.1. fc-Choosability is FPT on P5-free graphs
In this section we prove that fc-choosability is fixed parameter tractable on P5-free graphs.
Theorem 1. The fc-Choosability problem is FPT on P^-free graphs.
Proof. We give a constructive proof of this theorem by describing a recursive algorithm based on Propositions 1 and 2 that checks whether ch(G) < k. We assume that k > 3, since for k < 2, fc-Choosability can be solved in polynomial time for general graphs [7]. If G is disconnected, then ch(G) is equal to the maximum choice number of the connected components of G. Thus we also assume that G is connected.
Our algorithm uses as its main tool a procedure called Color, given in Algorithm 1. This procedure takes as input a connected P5-free graph G and a set W = ..., wr} C V(G) with a sequence of color
lists C = (L(w\),..., L(wr)), each of size k. For the notation in this procedure, we let L = L(w\) U ••• U L(wr), and we denote I = maxjmax L(w1),..., max L(wr)}. Let also L = L(w1) X • • • X L{wr) and X = 2l. We say that vertices ..., wr are colored by c = (ci,..., cr) £ Li if each Wi is colored by c*. Set H = G — W. Procedure Color produces an output which either contains a list of different sets X = (Xi,..., Xs), Xi G X, such that for any assignment of color lists
Procedure Color(G, W, C)
Find a dominating set U — {ui,..., up} of H — G — W, such that U is a clique or U induces a P3; Let X = 0;
if p > k then Return(NO), Halt;
if d(G[W UU]) > d then Return(NO), Halt;
forall Color lists L(u\),..., L(up) C {1,I + 1,..., I + kp}, s.t. \L(ui)\ — k do
if u = V(H) then Let X = 0;
forall List colorings s of H do
|_ Let I:=lU{c£l: c(wi) ^ s(uj) if WiUj £ E(G)}-, [_ if X ^ 0 then Add(AT,X); else Return(NO), Halt;
if U ^ V(H) then
Let Hi,..., Hq be the connected components of H — U, and let Ft = G[W UUU V(Hi)] for i £ {1,..., g}; Let £' — (L{ui),..., L(up)), L'=Lx L{ui) X • • • X L(up); for i = 1 to q do
Color(ii, W U U, C U £'); if the output is NO then |_ Return(NO), Halt;
else
|_ Let Xi be the output;
Let y - Afi; for i = 2 to q do Let Z = 0;
forall X £ Xi and Y £ y do
if X n Y ^ 0 then kdd{Z, X n X')-, |_ else Return(NO), Halt; L Let y =
forall Z e Z do
Let X = {(c(ioi),... , c(«v)): c £ Z, c(wi)
c(uj) if WiUj £ E(G) and c(ui) c(uj) if UiUj £
E(G)}-,
if X / 0 then kdd{X, X); |_ else Return(NO), Halt;
if X = 0 then Return(NO), Halt; else Return(AT).
Algorithm 1: Pseudo code for the procedure Color
of size k to vertices of H, there is a set Xi with the property that any c G Xi can be used for coloring of W with respect to adjacencies between vertices in W and vertices in V(H), or the output contains "NO" if there is a list assignment for vertices of H such that no list coloring exists. Denote d = 4(fefc4) log(2(fefc4)). The subroutine Add(A, a) adds the element a to the set A if a A, and the subroutine Halt stops the algorithm. Our main algorithm calls Procedure Color (G, 0, 0). To simplify the description of the algorithm it is assumed that for W = 0, L contains unique zero coloring (i.e. L is non empty). If the output is "NO" then G is not fc-choosable, and otherwise G is fc-choosable.
To prove the correctness of the algorithm, let us analyze one call of Procedure Color. Since each induced subgraph of a P5-free graph is P5-free, by Proposition 2 it is possible to construct the desired dominating set U in the beginning of the procedure. If \U\ > k > 3 then U is a clique in G and ch{G) > ch(G[U]) > k. If d(G[W U U]) > d then ch(G) > ch(G[W U U]) > k by Proposition 1. Otherwise we proceed and consider color lists for vertices of U. It should be observed here that it is sufficient to consider only color lists with elements from the set L U {/ + 1,..., I + kp}, since we have to take into account only intersections of these lists which each other and with lists for vertices of W. If U = V(H) then the output is created by checking all possible list colorings of H. If U ^ V(H) then we proceed with our decomposition of G. Graphs ..., Fq are constructed and Procedure Color is called recursively for them. It is possible to consider these graphs independently since vertices of different graphs Hi and Hj are not adjacent. Then outputs for Fx,..., Fq are combined and the output for G is created by checking all possible list colorings of U.
Now we analyze the running time of this algorithm. To estimate the depth of the recursion tree we assume that h sets U are created recursively without halting and denote them by Since \Ui\ < fc,
\Ui U • • • U Uh\ < kh. Notice that each set Ui is a dominating set for Ui+1,..., Uh. Hence ^ degF(,y) > h — 1, where F = G[E/iU- • -UUh],
veUi
and XI deg(v) > h(h- 1). This means that d(F) > and if h > vev(F)
kd + 1 = 4k(h*) log(2(fefc4)) + 1 then Procedure Color stops. Therefore
the depth of the recursion tree is at most kd+ 1 = 4log(2(fefc)) + 1. It can be easily noted that the number of leaves in the recursion tree is at most n = |V(G)|, and the number of calls of Color is at most (4log(2(fefc4)) + 1 )n = 0(fc5 • 2fe4 • n). Let us analyze the number of
operations used for each call of this procedure. The set U can be constructed in polynomial time by the results of [2]. If \U\ > k then the algorithm finishes its work. Assume that \U\ < k. Since the depth of the recursion tree is at most kd + 1, color lists for vertices of U are chosen from the set {1,..., (kd+ 1 )fc2}, and the number of all such sets is ((kd+Vk So,
there are at most ((kd+Vk ) (0r 2°^8'2fe )) possibilities to assign color lists to vertices of U. The number of all list colorings of vertices of U is at most kk. Recall that the output of Color is either "NO" or a list of different sets X = (Xi,..., Xs) where Xi £ X. Since the depth of the recursion tree is at most kd+1 and each set U contains at most k elements (if the algorithm does not stop), the size of W is at most k(kd +1). Hence the output
contains at most 2k(kd+1^ (or 2°(k6'2h )) sets. Using these bounds and the observation that q < n, we can conclude that the number of operations for each call of Color is 2°(k8'2h ) -nc for some positive constant c. Taking into account the total number of calls of the procedure we can bound the the running time of our algorithm as 2°(k8'2h ) • ns for some positive constant 8. □
3.2. (j, fc)-choosability is FPT on P5-free graphs
Our result can be generalized for the case when sizes of lists of colors are defined by some function. Let G be a graph and f: V(G) —t N be a mapping which assigns positive integers to vertices of G. It is said (see [7]) that G is f-choosable for a function f if G has a list coloring for every assignment of color lists L(v) with \L(v)\ = f(v) for all v £ V. The (j, fc)-choosability problem asks for a given graph G, positive integers j and k (j < fc), and a function f: V(G) —t {j,..., fc}, whether G is /-choosable for /.
Using exactly same arguments as in Section 3.1, it is possible to prove the following claim.
Theorem 2. The (j, fc)-choosability problem is FPT on P^-free graphs when parameterized by k.
3.3. fc-choosability for cographs
For the special case of cographs, it is possible to improve our algorithm. Here we sketch the ideas which can be used in this case.
Recall that if Gi and G2 are two disjoint graphs, then the disjoint union of G1 and G2 is the graph with the vertex set V(G1) U V(G2) and the edge set E(G 1) U E(G2). The join of G\ and G2 is the graph with the vertex set V(GX) U V(G2) and the edge set -B(Gi) U E(G2) U {uv: u £
V(Gi), v G V(G2)}. It is well known (see e.g. [3]) that any cographs can be constructed from isolated vertices by means of these operations, and such decomposition of any cograph can be constructed in linear time [5].
We also need following properties of the choice number of complete bipartite graphs:
Proposition 3 ( [7]). • ch(Kr^rv) > r; • for r> (2fe-x), ch(Kr,r) > k.
Suppose that G is a connected cograph with at least k vertices. Then G is a join of two cographs Gi and G2. Let ni = |V(Gi)|, n2 = |V(G2)| and assume that ni < n2. If ni > then ch(G) > k by
Proposition 3. Suppose that k < ni < If > kk then
ch(G) > k by Proposition 3. Otherwise the number of vertices of G is at most + kk — 2 and it is possible to check whether ch(G) < k by
the brute force algorithm. It remains to consider the case n 1 < k. Notice that V(G 1) is a dominating set in G and we can apply same arguments as in Section 3.1. Now we consider recursively components of G2. It can be assumed that the depth of our recursion is at most fc, since otherwise G contains Kk+i as a subgraph and hence ch(G) > k.
4. Choosability is NP-hard on P5-free graphs
In this section we show that choosability, with input G and fc, remains NP-hard when the input graph is restricted to P5-free graphs.
Theorem 3. The Choosability problem is NP-hard on P^-free graphs.
Proof. We reduce the not-all-equal 3-Satisfiability (NAE 3-SAT) problem with only positive literals [9] to Choosability. For a given set of Boolean variables X = {,..., xn}, and a set C = {Ci,..., Cm} of three-literal clauses over X in which all literals are positive, this problem asks whether there is a truth assignment for X such that each clause contains at least one true literal and at least one false literal. NAE 3-SAT is NP-complete [9].
Our reduction has two stages. First we reduce NAE 3-SAT to List Coloring by constructing a graph with color lists for its vertices. Then we build on this graph to complete the reduction from NAE 3-SAT to Choosability.
At the first stage of the reduction we construct a complete bipartite graph (Kn92m) H with the vertex set {x1,..., xn} U ..., C^} U
{Ci2\...,C£}, where {x1,...,xn} and ({Cj1},..., C£)}U) ({C[2\ ..., Cff}) is the bipartition of the vertex set. Hence on the one side of bipartition we have a vertex for each variable, and on the other side we have two vertices for each clause. We define color lists for vertices of H as follows: L(xi) = {2i — 1,2i} for i £ {1 , ...,n}, L(cf]) = {2 p -1,2 q- 1, 2 r - 1} and L(cf]) = {2 p, 2 q, 2 r} if the clause Cj contains literals xp, xq, for j £ {1,..., m}. Construction of H is shown in Figure 1.
{2p-\,2p} Xp {2^-1,2q)
• t • ♦ •
% {2r -1,2r\
{2/p-i, 2^-1,2r-1} Cj2) {2^,2^,2r}
Phc. 1: Graph ii.
Lemma 1. The graph H has a list coloring if and only if there is a truth assignment for the variables in X such that each clause contains at least one true literal and at least one false literal.
Proof. Assume that H has a list coloring. Set the value of variable Xi = true if vertex xi is colored by 2i — 1, and set xi = false otherwise. For each clause Cj with literals xp,xq,xri at least one literal has value true since at least one color from the list {2p, 2<7, 2r} is used for coloring vertex Cj , and at least one literal has value false, since at least one color from
the list {2p — 1,2q — 1,2r — 1} is used for coloring vertex
Suppose now that there is a truth assignment for the variables in X such that each clause contains at least one true literal and at least one false literal. For each variable xi, we color vertex xi by the color 2i — 1 if Xi = true, and we color xi by the color 2i otherwise. Then any two vertices C^ and which correspond to the clause Cj with literals
ejCp, tjcqy x<p, can
be properly colored, since at least one color from each of lists {2p —l,2qr — l,2r — 1} and {2p, 2qr, 2r} is not used for coloring of vertices xi9..., xn. □
Now we proceed with our reduction and add to H a clique with fc = n + 4nm — 4m vertices ixi,..., u^. For each vertex xi, we add edges XiUi for t G {1,..., fc}, ^ ^ 2i — 1, 2i. For vertices Cj1^ and Cj2^ which correspond to clause Cj with literals xp,xq,xr, edges C^ui such that
i ^ 2p - 1, 2qr - 1, 2r - 1 and edges such that i ^ 2p, 2q, 2r
are added for i G {1,..., fc}. We denote the obtained graph by G.
We claim that G is fc-choosable if and only if there is a truth assignment for the variables in X such that each clause contains at least one true literal and at least one false literal.
For the first direction of the proof of this claim, suppose that for any truth assignment there is a clause all of whose literals have the same value. Then we consider a list coloring for G with same color list {1,..., fc} for each vertex. Assume without loss of a generality that Ui is colored by color i for i G {1,..., fc}. Then each vertex xi can be colored only by colors 2i — 1, 2i, each vertex Cj1^ can be colored only by colors 2p — 1,2q — 1, 2r — 1 and each vertex Cj can be colored only by colors 2p, 2q, 2r if
Cf correspond to the clause with literals xp, xq, xr. By Lemma 1, it is impossible to extend the coloring of vertices ixi,..., u^ to a list coloring of G.
For the other direction, assume now that there is a truth assignment for the variables in X such that each clause contains at least one true literal and at least one false literal. Assign arbitrarily a color list L(v) of size fc to each vertex v G V(G). We show how to construct a list coloring of G. Denote by U the set of vertices {i62n+i? • • • ? uk}• Notice that U is a clique whose vertices are adjacent to all vertices of G. We start coloring the vertices of U and reducing G according to this coloring, using following rules:
1. If there is a non colored vertex v G U such that L(v) contains a color c which was not used for coloring the vertices of U and there is a vertex «7 G {xu ..., xn}U {cf \ ..., C£>}U{Ci2\ ..., such that c (£ L(w), then color v by c. Otherwise choose a non colored vertex v G U arbitrarily and color it by the first available color.
2. If, after coloring some vertex in [/, there is a vertex xi such that at least 2m — 1 colors that are not included in L(xi) are used for coloring [/, then delete Xi.
3. If, after coloring some vertex in [/, there is a vertex C^ with s G {1,2} such that at least n — 2 colors that are not included in L(C^)
are used for coloring E/, then delete Cj8\
This coloring of U can be constructed due the property that for each v G E7, \L(v)\ = fc and \U\ = fc — 2n < k. Rule 2 is correct since degG(xi) = k + 2m — 2, and therefore if at least 2m — 1 colors that are not included in L(xi) are used for coloring E/, then any extension of the coloring of U to the coloring of G — Xi can be further extended to the coloring of G, since there is at least one color in L(xi) which is not used for the coloring of neighborhood of this vertex. By same arguments, we can show the correctness of Rule 3 using the fact that degG(Cj^) = fc + n —3.
If after coloring the vertices of [/, all vertices of {xl9..., xn} U ..., CU {C[2\ ..., Cffl} are deleted then we color remaining vertices ul9..., u2n greedily, and then we can claim that a list coloring of G exists by the correctness of Rules 2 and 3. Assume that at least one vertex of {xu ..., xn} U {Cix),..., C^} U {Cj2\ ..., was not deleted,
and denote the set of such remaining vertices by W. Let v G U be the last colored vertex of U. Since \U\ = k — 2n = n + 4nm — 4m — 2n = n(2m — 1) + 2m(n — 2), the color list L(v) contains at least 2n colors which are not used for coloring the vertices of U. Furthermore, for each w G W, all these 2n colors are included in L{w), due to the way we colored the vertices of U and since w was not deleted by Rules 2 or 3. We denote these unused colors by 1,..., 2n and let L = {1,..., 2n}. We proceed with coloring of G by coloring the vertices ul9..., u2n by the greedy algorithm using the first available color. Assume without loss of generality that if some vertex Ui is colored by the color from L then it is colored by the color i. Now it remains to color the vertices of W. Notice that G[W] is an induced subgraph of H. For each w G W, denote by L'(w) the colors from L(w) which are not used for coloring vertices from the set {ixi,..., that are adjacent to w. It can be easily seen that for any Xi G W, 2i — 1,2¿6 V{xi), for any cj1^ G W which corresponds to clause with literals xp9 xq, xr, 2p — 1, 2q — 1, 2r — 1 G and
for any Cj G W which corresponds to clause with literals xp,xq,xri 2p,2q,2r G £'(cj2)). Since there
is a truth assignment for variables X such that each clause contains at least one true literal and at least one false literal, by Lemma 1 we can color the vertices of IV.
To conclude the proof of the theorem, it remains to prove that G is P5-free. Suppose that P is an induced path in G. Since H is a complete bipartite graph, P can contain at most 3 vertices of H and if it contains 3 vertices then these vertices have to be consecutive in P (notice that if P
contains vertices only from one set of the bipartition of H, then the number of such vertices is at most 2 since they have to be joined by subpaths of P which go through vertices from the clique {iXi,..., гх&}). Also P can contain at most 2 vertices from the clique {txi,..., гц.}, and if it has 2 vertices then they are consecutive. Hence, P has at most 5 vertices, and if P has 5 vertices then either P = utlut2Cj^ XiC^ or P =
UtxUt2XixCj8^Xi2. Assume that P = utlut2Cj^ XiCj82\ Since P is an induced path, vertices utl, ut2 are not adjacent to Xi. By the construction of G, it means that {il912} = {2г — 1, 2г}. But then Cj^ is adjacent either utl or ut2. Suppose that P = u^u^x^C^ Xi2. Again by the construction
of G, {tut2} = {2г2 - 1,2г2} and cjs) is adjacent to utl or ut2. By these contradictions, P has at most 4 vertices. □
5. Conclusion and open problems
We proved that the fc-Choosability problem is FPT for P5-free graphs when parameterized by k. It can be noted that our algorithm described in the proof of Theorem 1 does not explicitly use the absence of induced paths P5. It is based on the property that any induced subgraph of a fc-choosable P5-free graph has a dominating set of bounded (by some function of k) size. It would be interesting to construct a more efficient algorithm for fc-choosability which actively exploits the fact that the input graph has no induced P5.
Another interesting question is whether it is possible to extend our result for Pr-free graphs for some r > 6? Particularly, it is known [22] that any P6-free graph contains either a dominating biclique or a dominating induced cycle C6. Is it possible to prove that fc-Choosability is FPT for P6-free graphs using this fact?
Also, we proved that fc-Choosability is NP-hard for P5-free graphs. Is this problem П^-complete?
Finally, is it possible to improve our algorithm for cographs which is double exponential in fc?
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Department of Informatics;
University of Bergen, N-5020 Bergen, Norway,
{Peter. Golovach I Pinar. Heggernes}@ii. nib .no Поступила 21.10.2009
