Sparsification Lower Bounds for List H-Coloring

We investigate the List H-Coloring problem, the generalization of graph coloring that asks whether an input graph G admits a homomorphism to the undirected graph H (possibly with loops), such that each vertex v ∈ V(G) is mapped to a vertex on its list L(v) ⊆ V(H). An important result by Feder, Hell, and Huang [JGT 2003] states that List H-Coloring is polynomial-time solvable if H is a so-called bi-arc graph, and NP-complete otherwise. We investigate the NP-complete cases of the problem from the perspective of polynomial-time sparsification: can an n-vertex instance be efficiently reduced to an equivalent instance of bitsize \(\mathcal {O} (n^{2-\varepsilon })\) (n2-ɛ) for some ɛ > 0? We prove that if H is not a bi-arc graph, then List H-Coloring does not admit such a sparsification algorithm unless \(\mathsf {NP \subseteq coNP/poly}\) . Our proofs combine techniques from kernelization lower bounds with a study of the structure of graphs H which are not bi-graphs.

be a neighbor (in graph H) of the value assigned to v, and for each vertex v ∈ V (G) there is a constraint that the value of v belongs to L(v).Hence any NP-hard List H-Coloring problem translates into a CSP with a non-Boolean domain in which constraints have arity at most two.Recent work [4,29] has led to a number of nontrivial advances in the study of sparsification for CSPs with a Boolean domain.A natural next step in that line of research is to target non-Boolean CSPs, of which the List H-Coloring problems form a rich subset.
The last motivation for studying sparsification for List H-Coloring is that it forms the logical next step in the study of sparsification for coloring problems.Recent work [25] showed that Graph (List) q-Colorability does not admit nontrivial polynomial-time sparsification for q ≥ 3 unless NP ⊆ coNP/poly, but left the case of List H-Coloring open.

Our results
We prove that for all undirected, possibly non-simple, graphs H for which List H-Coloring is NP-complete, the problem does not admit nontrivial sparsification unless an unlikely complexity-theoretic collapse occurs.Our proofs combine techniques from kernelization lower bounds with a careful analysis of the common structures of hard graphs H.To state our sparsification lower bounds in full generality, we use the notion of generalized kernelization (see Definition 2), where the number of vertices n of the instance plays the role of the complexity parameter k.A generalized kernelization for List H-Coloring of size f (n) is therefore a polynomial-time algorithm that maps any n-vertex input G, to an equivalent instance (of a potentially different but fixed decision problem) of bitsize f (n).Since a polynomial-time sparsification algorithm mapping to instances of bitsize f (n) yields a generalized kernelization of size f (n), lower bounds on the latter also apply to the former.Theorem 1.If H is an undirected graph that is not a bi-arc graph, possibly with loops, then List H-Coloring parameterized by the number of vertices n admits no generalized kernel of size O(n 2−ε ) for any ε > 0, unless NP ⊆ coNP/poly.
The techniques employed in the proof of Theorem 1 are rather different from those in the NP-completeness proof for the hard cases of List H-Coloring.Feder, Hell, and Huang [13] establish the NP-completeness of List H-Coloring when H is not a bi-arc graph, by reducing from 3-Coloring.They build gadgets in List H-Coloring instances to mimic the effect of a normal edge in 3-Coloring, and then replace each edge with such a gadget.Although 3-Coloring is known not to admit any nontrivial sparsification unless NP ⊆ coNP/poly [27], the mentioned NP-completeness reduction does not transfer this lower bound from 3-Coloring to List H-Coloring: as the reduction introduces a gadget (with new vertices) for every edge of the 3-Coloring instance, it blows up the number of variables.
Our sparsification lower bound therefore follows a different route.We introduce a technical annotated version of the List P 4 -Coloring problem.For this annotated problem, we prove a sparsification lower bound via cross-composition [2], a technique from kernelization lower bounds.We give a polynomial-time algorithm that embeds a sequence of t 2 instances of the Clique problem, on n vertices each, into a single instance (G , L ) of Annotated List P 4 -Coloring, on O(t • n O (1) ) vertices, which acts as the logical OR of the Clique inputs: there is a list coloring if and only if at least one Clique instance has a solution.The fact that the information from t 2 distinct inputs is packed into a single instance of O(t • n O (1) ) vertices, means that the embedding is very efficient: the t 2 n-vertex instances of Clique carry t 2 • n 2 bits of information (for each instance, which edges are present?), while G has t 2 • n O (1)  potential edges, and therefore carries t 2 • n O(1) bits of information.Applying this reduction for t a polynomial in n whose degree depends on the constant in n O (1) , this intuitively implies that G cannot be sparsified without losing information.Via the framework of crosscomposition [2] we get the formal result that Annotated List P 4 -Coloring parameterized by the number of vertices n does not admit a generalized kernelization of size O(n 2−ε ) for any ε > 0 unless NP ⊆ coNP/poly.
To transfer the lower bound for Annotated List P 4 -Coloring to List H-Coloring for all graphs H which are not bi-arc, we first use a reduction inspired by Feder, Hell, and Huang [13], to reduce to the case of bipartite graphs H. Then we investigate the common structure of simple bipartite non-bi-arc graphs H, which are known to be the simple bipartite graphs H whose complement is not a circular-arc graph [13].We uncover a common structure of such graphs which can be used to prove the incompressibility of the related List H-Coloring problems: we prove all such graphs H contain five vertices (a, b, c, d, e) such that H[{a, b, c, d}] is an induced P 4 , the open neighborhoods N H (a), N H (c), and N H (e) are incomparable (i.e., none of them is contained in another), and such that also the open neighborhoods N H (b), N H (d) are incomparable.This 5-tuple in a bipartite graph H is sufficient to prove hardness of sparsification, which we consider one of the main contributions of the paper: We prove that the 5-tuple can be used to implement certain gadgets to enforce pairs of vertices to receive different colors in List H-Coloring.By applying these gadgets sparingly -and not for all edges -we reduce Annotated List P 4 -Coloring to List H-Coloring without blowing up the number of vertices, and obtain Theorem 1.

Related work
More background on homomorphisms and H-Coloring can be found in the textbook by Hell and Nešetřil [21], or the survey by Hahn and Tardif [19].The classical complexity of H-Coloring has also been investigated when restricted to planar [30], minor-closed [10], and bounded-degree [16,38] input graphs G.The complexity of List H-Coloring was investigated for bounded-degree graphs [14].There is also an interesting line of research concerning the descriptive and space complexity [5,7,8].Finally, the fine-grained complexity of both variants was also investigated [9,17,33,35,36].
Organization Section 2 contains preliminaries on kernelization and graphs.In Section 3 we present a sparsification lower bound for an annotated version of List P 4 -coloring, which forms the keystone of our hardness results.In Section 4 we analyze the structure of hard graphs H, and use that structure to build certain gadgets.These allow us to reduce the annotated problem to standard List H-Coloring problems and prove Theorem 1.

Preliminaries
To denote the set of numbers 1 to n, we use the following notation: Graphs.All graphs considered in this paper are finite and undirected, and do not have parallel edges.We allow self-loops, unless explicitly stated otherwise.The vertex set and the edge set of G are denoted by V (G) and E(G), respectively.An edge {u, v} ∈ E(G) is denoted shortly by uv, and by vv we denote the loop on the vertex v.
Let G and H be graphs.We say that G is H-colorable if there exists a Such a function is also called a homomorphism from G to H. Note that a graph G has a homomorphism to the complete graph K q if and only if G is (properly) q-colorable.If f is a homomorphism from G to H, then we denote it by f : G → H.We write G → H to indicate that some homomorphism from G to H exists.For a graph G and lists L : In our applications, the complexity parameter k will be the number of vertices n.We will use the framework of cross-composition, introduced by Bodlaender, Jansen, and Kratsch [2], to establish kernelization lower bounds.  (1, where t denotes the number of instances, and Q has a polynomial (generalized) kernelization with size bound O(k d−ε ), then NP ⊆ coNP/poly.
We will refer to an or-cross-composition of cost f (t) = √ t log(t) as a degree-2 crosscomposition.By Theorem 5, a degree-2 cross-composition can be used to rule out generalized kernels of size O(k 2−ε ) and thus provides a way to obtain sparsification lower bounds.Generalized kernelization lower bounds can be transferred using the notion of linear-parameter transformations.
Definition 6 (Linear-parameter transformation).Let P, Q ⊆ Σ * × N be two parameterized problems.A linear-parameter transformation from P to Q is a polynomial-time algorithm that, given an instance (x, k) ∈ Σ * × N of P, outputs an instance (x , k ) ∈ Σ * × N of Q such that the following holds: (i) (x, k) ∈ P if and only if (x , k ) ∈ Q, and (ii) k ∈ O(k).
It is well-known [2] that the existence of a linear-parameter transformation from problem P to Q implies that any generalized kernelization lower bound for P, also holds for Q.

3
Lower bound for Annotated List P 4 -Coloring We prove a sparsification lower bound for the following problem, where we take P 4 to be the graph on vertices {a, b, c, d} with edges ab, bc, cd.

Annotated List P4-Coloring
Input: A tuple (G, L, S, F ), such that G is a simple undirected bipartite graph with bipartition , and such that for all {u, v} ∈ F we have f (u) = f (v)?
Intuitively, the annotations allow one to express two types of additional constraints on the coloring f .Using a set S i , one can enforce that at least one vertex is not colored c.Using a pair {u, v} ∈ F , one can enforce that u and v do not receive the same color.While the latter can easily be expressed by simply inserting an edge between u and v in a K q -Coloring instance, this needs a nontrivial gadget for general graphs H. Proof.We will prove this lower bound by giving a degree-2 cross-composition from Clique to Annotated List P 4 -Coloring.We define a polynomial equivalence relation R on instances of Clique.Let any two instances that ask for a clique that is larger than their respective number of vertices be equivalent; these are always no-instances.Let two instances of Clique be equivalent under R, when the input graphs have same number of vertices and the problems ask for a clique of the same size.It is easy to verify that R is indeed a polynomial equivalence relation.
By duplicating one of the inputs multiple times as needed, we can assume the number of inputs to the cross-composition is a square.Therefore, assume we are given t instances of Clique, such that t := √ t is integer and such that each instance has n vertices and asks for a size-k clique.Enumerate the given input instances as X i,j for i, j ∈ [t ] and let G i,j denote the corresponding graph.Label the vertices in each instance arbitrarily as x 1 , . . ., x n .We show how to create an instance (G, L, S, F ) that is a yes-instance for Annotated List P 4 -Coloring if and only if at least one of the given instances for Clique is a yes-instance.Refer to Figure 1 for a . . . . . .
. . .,3).Edges between P and Q are omitted, except for the edges that result from the fact that x3x4 / ∈ E(G2,3).A fat edge between u and v indicates that {u, v} ∈ F .Vertex sets contained in S are marked in blue.White vertices have lists {b, d} while black vertices have list {a, c}.Note that the constructed graph is bipartite with the white and black vertices as partite sets.
that Q i contains 6 k 2 vertices for each ordered pair of vertices in an n-vertex graph; these pairs model edges and self-loops.Let Q : can receive color a or c.Add the pairs {q i e,f , qi e,f } and {r i e,f , r i e,f } to F .Verify that when both q i e,f and r i e,f get color b, then s i e,f and t i e,f must get color c.Recall that the goal of the construction is to ensure that the Annotated List P 4 -Coloring instance (G, L, S, F ) acts as the logical or of the Clique instances X i,j , so that G has a coloring respecting the lists and annotations if and only if some input graph G i,j has a clique of size k.The part of G constructed so far allows colorings of G to encode the vertex set of a k-clique through its behavior on P .Finding a proper list coloring of G entails highlighting vertices from one set P j that correspond to a clique in instance X i,j for some i ∈ [t ].The highlighting property will be enforced by ensuring at least one vertex in each set The index of the vertex that is colored a encodes the m-th vertex in the clique to which the coloring corresponds.The vertices in Q i are then used to verify that the selected vertices form a clique in G i,j .The next steps add additional vertices and edges, in order to achieve these properties.

4.
For each i, j ∈ [t ], consider instance X i,j .For all f ∈ [k]  2 and e = (e 1 , e 2 ) ∈ [n] 2 , connect vertex p j e1,f1 to q i e,f and connect p j e2,f2 to r i e,f whenever Observe also that each vertex q i e,f , r i e,f has a unique neighbor in P j for each j ∈ [t ].The above step will allow using the coloring of vertices s i e,f and t i e,f to verify that the vertices selected in P j correspond to a clique: when x e1 x e2 is not an edge, they will ensure that we cannot select both. 5. Add vertices y j and ŷj for all j ∈ [t ] and let

Similarly, add vertices z i and ẑi for all i ∈ [t ] and let
Add the sets Ŷ and Ẑ to S. Furthermore, for all i ∈ [t ], add {y i , ŷi } and {z i , ẑi } to F .The steps above ensure that at least one vertex y j ∈ Y receives color c and at least one vertex in z i ∈ Z receives color c.This will indicate that instance X i,j is selected.We will now put further constraints on the coloring of P j and Q i when they correspond to a selected instance.

For all
2 and e ∈ [n] 2 , add the set {s i e,f , t i e,f , z i } to S. This concludes the construction of G, L, S and F .Let us start by counting the number of vertices in G: Observe that hereby |V (G)| is properly bounded for a degree-2 cross composition.We continue by showing that G is a valid instance of Annotated List P 4 -Coloring.Verify that G is bipartite with bipartition Hence, V 1 contains all vertices whose lists are a subset of {a, c} and V 2 contains all remaining vertices, and it can be verified that the lists of these vertices are a subset of {b, d}.Observe that indeed each set in F is a subset of either V 1 or V 2 , and each set in S is a subset of Furthermore, it is straightforward to verify that |F | ≤ |V (G)| as promised for Annotated List P 4 -Coloring (note that we only add elements to F in Steps 3 and 7).We can also verify that Step 8 As such, we have created a valid instance of Annotated List P 4 -Coloring.The next two claims show that the constructed graph G indeed acts as the logical or of the given input instances.
Claim 8.If some input graph G i * ,j * has a clique of size k, then G is annotated P 4 -colorable.
Proof.Let such i * , j * ∈ [t ] be given, we create a an annotated P 4 -coloring h : V (G) → {a, b, c, d} for G. First of all, for all j = j * with j ∈ [t ], let h(y j ) := a and let h(ŷ j ) := c.Let h(y j * ) := c and let h(ŷ j * ) := a.Similarly, for i = i * we let h(z i ) := a and let h(ẑ i ) := c.Furthermore define h(z i * ) := c and h(ẑ i * ) := a. Hereby, not all vertices in Ŷ have color c, and not all vertices in Ẑ have color c, such that we satisfy the sets added to S in Step 7 of the construction.For all p ∈ P j for j = j * , let h(p) := c.Furthermore, for all e and  h(s i e,f ) := h(t i e,f ) = c.It remains to color the vertices in P j * and } contains a vertex that receives color a, as desired.We now extend this coloring to Suppose towards a contradiction that indeed both these vertices have color c.By the choice of our coloring, this implies that h(q i * e,f ) = h(r i * e,f ) = d and thus h( that means that q i * e,f and r i * e,f have their unique neighbor in P j * of color a, implying h(p j * e1,f1 ) = h(p j * e2,f2 ) = a.So these edges were constructed in Step 4, implying x e1 x e2 / ∈ E(G i * ,j * ).Since x e1 ∈ K and x e2 ∈ K, this contradicts that K is a clique.Secondly, verify that for all pairs in {u, v} ∈ F , h(u) = h(v): we only add sets to F in Steps 3 and 7. We always ensure in the construction that if {u, v} ∈ F , the two vertices get different colors.
Thirdly, we verify the coloring of endpoints of edges in G. First of all, consider the edges added in Step 3 and observe that we always color the endpoints properly in the description above: if qi e,f gets color d, we color s i e,f with c which is allowed; if qi e,f has color b, we use color a in s i e,f which is again fine.One may verify that the same holds for edges ri e,f t i e,f .Now consider the edges between a vertex u ∈ P and v ∈ Q.If u / ∈ P i * it follows that h(u) = c.Since by the lists, h(v) ∈ {b, d} this implies that this edge is properly colored.Similarly, if v / ∈ Q j * we obtain h(v) = b and since h(u) ∈ {a, c} we are again done.If u ∈ P j * and v ∈ P i * one may observe that the edge uv is properly colored by definition: v has color d only if it has no neighbors of color a (and h(u) ∈ {a, c} thus implies h(u) = c), and otherwise v has color b such that the edge is again properly colored by h(u) ∈ {a, c}.Claim 9.If G has an annotated P 4 -coloring h, then there exist i * , j * ∈ [t ] such that G i * ,j * has a clique of size k.
follows that for all m ∈ [k], there exists i m ∈ [n] such that h(p j * im,m ) = a.Let x 1 , . . ., x n be the vertices of G i * ,j * , define K := {x i1 , . . ., x i k }.We show that K is a size-k clique in G i * ,j * by showing that x im x i m is an edge for all m = m .Observe that this then also proves that all selected vertices are distinct as the input graphs have no self-loops.
Let m, m ∈ [k].Without loss of generality let m < m .Suppose towards a contradiction that x im x i m / ∈ E(G i * ,j * ).Then, in Step 4, we added the edges p j * im,m q i * (im,i m ),{m,m } and p j * i m ,m r i * (im,i m ),{m,m } .Note that since we choose Since b is the only neighbor of a in the P 4 , we get h( ,{m,m } and ri * (im,i m ){m,m } t i * (im,i m ),{m,m } are edges in G (also added in Step 3), we get that h(s i * (im,i m ),{m,m } ) = h(t i * (im,i m ),{m,m } ) = c.However, note that {r i * (im,i m ),{m,m } , r i * (im,i m ),{m,m } , z i * } ∈ S, by Step 9.These three vertices all have color c, contradicting that h is a valid annotated P 4 -coloring of G.
Using the claims above and the bound on the size of V (G) computed earlier, we conclude that we have given a degree-2 cross-composition to annotated P 4 -coloring, such that the lower bound follows from Theorem 5.

4
Gadgets in hard graphs for List H-Coloring

Bi-arc graphs, associated bipartite graphs, and consistent instances
Recall that the complexity dichotomy for List H-Coloring was proven in three steps: 1. for reflexive H, the polynomial cases appear to be interval graphs [11], 2. for irreflexive H, the polynomial cases appear to be bipartite co-circular-arc graphs [12], 3. for general graphs, the polynomial cases are the so-called bi-arc graphs [13].
The main idea of showing the final step of the dichotomy was a reduction to the bipartite case.For a graph H, by H * we denote the associated bipartite graph, defined as follows.The vertex set of H * is the union of two independent sets: Note that the edges of type x x in H * correspond to loops in H.
As we mentioned in the introduction, bi-arc graphs are defined in terms of certain geometric representation, but for us much more convenient will be to use the following characterization in terms of the associated bipartite graph.
Theorem 10 (Feder, Hell, and Huang [13]).Let H be an undirected graph, possibly with loops.The following are equivalent.1. H is a bi-arc graph.The following Proposition follows from the idea of Feder, Hell, and Huang [13], and provides a reduction from List H * -Coloring to List H-Coloring that preserves the vertex set of G. Its exact statement from [33,34].
Proposition 12 (Okrasa et al. [33,34]).Let H be a graph and let (G, L) be a consistent instance of List H * -Coloring.

Hard bipartite graphs H
The following notion was introduced by Feder, Hell, and Huang [12].Definition 13.Let k ≥ 1 and let H be a bipartite graph with bipartition classes X, Y .Let U = {u 0 , . . ., u 2k } ⊆ X and V = {v 0 , . . ., v 2k } ⊆ Y be ordered sets of vertices such that {u 0 v 0 , u 1 v 1 , . . ., u 2k v 2k } is a set of edges of H.We say that (U, V ) is a special edge asteroid (or, in short, an asteroid) of order 2k + 1, if for every i ∈ {0, . . ., 2k} there exists a u i -u i+1 -path P i,i+1 in H (indices are computed modulo 2k + 1), such that (a) there are no edges between {u i , v i } and {v i+k , v i+k+1 } ∪ V (P i+k,i+k+1 ) and (b) there are no edges between {u 0 , v 0 } and {v 1 , . . ., v 2k } ∪ 2k−1 i=1 V (P i,i+1 ).Feder, Hell, and Huang showed the following characterization of hard bipartite cases of List H-Coloring, i.e., bipartite graphs H, whose complement is not a circular-arc graph.

Theorem 14 (Feder et al. [12]). A bipartite graph H is not the complement of a circular-arc graph if and only if H contains an induced cycle with at least 6 vertices or an asteroid.
While induced cycles of length at least 6 and asteroids suffice to prove NP-completeness of List H-Coloring, to prove sparsification lower bounds via Annotated List P 4 -Coloring we need a more local structure.We therefore introduce the following notion.Intuitively, if H contains an extended P 4 gadget, then the P 4 on (a, b, c, d) allows a List H-Coloring instance to express a homomorphism problem to P 4 , while the presence of vertex e and the incomparability of the neighborhoods allows gadgets to be constructed to enforce the semantics of the set F and the sequence S in the definition of Annotated List P 4 -Coloring, thereby allowing a reduction from that problem to the List H-Coloring.The gadgets needed to simulate the pairwise constraints from F are given by the next lemma.

Lemma 16.
Let H be a bipartite graph which contains an induced cycle of at least 6 vertices or an asteroid.Then there exists an extended P 4 gadget (a, b, c, d, e) in H.Moreover, for every We remark that it is actually sufficient to show that every bipartite graph H which contains an induced cycle of at least 6 vertices or an asteroid, contains also an extended P 4 gadget.In this situation, the existence of (G {a,c,e} , L) and (G {b,d} , L) follows from a result in [33,34].However, for the sake of completeness, we include the whole proof.
Before we proceed to the construction of an extended P 4 gadget, let us introduce some definitions.A walk P is a sequence p 1 , . . ., p of vertices of H, such that p i p i+1 ∈ E(H), for every i ∈ [ − 1].We say that P = p 1 , . . ., p is a p 1 -p -walk and call − 1 the length of P. For walks P = p 1 , . . ., p and Q = q 1 , . . ., q m such that p = q 1 , we define P • Q := p 1 , . . ., p , q 2 , . . ., q m .We say that two walks P = p 1 , . . ., p and Q = q 1 , . . ., q m avoid each other if = m, p 1 = q 1 , and p i q i+1 , q i p i+1 ∈ E(H) for every i ∈ [ − 1].For two sets A, B of vertices of a graph, we say that they are anticomplete, if there is no edge with one endvertex in A and another one in B.
We call the set of three vertices T of a bipartite graph H a special triple if there exists an asteroid ({u 0 , u 1 , . . ., u 2k }, {v 0 , v 1 , . . ., v 2k }) (we use the notation introduced in Definition 13), such that T = {u 0 , u 1 , u k+1 }.Observe that the neighborhoods of vertices of every special triple are pairwise incomparable, as edges u 0 v 0 , u 1 v 1 , and u k+1 v k+1 induce a matching.
To make the proof of Lemma 16 easier, we first prove the following auxiliary lemma.

Lemma 18.
Let H be a bipartite, connected graph which contains an asteroid (U, V ).Then there exists a special triple T , an extended P 4 gadget (a, b, c, d, e) in H and: Define F to be a minimal induced subgraph of H which contains any asteroid, and let (U, V ) = ({u 0 , u 1 , . . ., u 2k }, {v 0 , v 1 , . . ., v 2k }) be an asteroid in F .Notice that if the neighborhoods of some vertices are incomparable in F , then so are the neighborhoods of these vertices in H.
For every i ∈ {0, . . ., 2k} we define P i,i+1 as follows.First, we choose P i,i+1 to be a shortest one from all {u i , v i }-{u i+1 , v i+1 }-paths in F that are anticomplete to {u i+k+1 , v i+k+1 } and, if i ∈ {0, 2k}, also to {u 0 , v 0 }.We know that at least one such path exists by the definition of an asteroid.Clearly, exactly one of the vertices u i , v i and exactly one of the vertices u i+1 , v i+1 belong to P i,i+1 (as endvertices).Now if u i (respectively u i+1 ) does not belong to P i,i+1 , append it as the first (resp.last) vertex.This way we obtain P i,i+1 .Observe that by the choice of P i,i+1 the path P i,i+1 is induced.
The minimality of F implies that every vertex of F belongs to (U ∪ V ) or at least one P i,i+1 .For every i we define Clearly this set induces a path in F .Similarly, we define the set P i,i+1 := V (P i,i+1 ) ∪ {v i , v i+1 } and note that P 0,1 and P 2k,0 also induce paths in F .Indeed, let us consider P 0,1 , the case of P 2k,0 is symmetric.Recall that P 0,1 is an induced path.Furthermore, if v 0 (resp.v 1 ) does not belong to P 0,1 , then it is non-adjacent to every vertex from P * 0,1 (by the minimality of P 0,1 ) and also to u 1 (resp.u 0 ) by the property (b) in Definition 13.If it does not lead to confusion, we will sometimes identify sets P * i,i+1 , P 2k,0 and P 0,1 with the paths induced by these sets.In the proof we will consider several cases.First, suppose that Let us describe the first case, as the other one is symmetric -we just need to consider the reversed asteroid.Note that the first two vertices of P 0,1 are either u 0 , v 0 or v 0 , u 0 .Consider the first case, as the other one is symmetric, with roles of u's and v's switched (recall that by Proposition 17 the set {v 0 , v 1 , v k+1 } is also a special triple).
Since |P * 0,1 | ≥ 2, we know that there are vertices b, c, d, such that P 0,1 starts with u 0 , v 0 , b, c, d, and b, c ∈ P  * 0,1 and d ∈ P * 0,1 ∪ {u 1 }.We define an extended P 4 gadget to be the tuple (v 0 , b, c, d, v k+1 ) (recall that P 0,1 is an induced path).The private neighbors of v 0 , c, v k+1 are, respectively, u 0 , d, u k+1 .The private neighbor of b is v 0 , and the private neighbor of d is its successor on P 0,1 , i.e., the fourth vertex of Let R be the shortest d-v 1 -walk using consecutive vertices of P 0,1 .Note that its length is at least one.Define D := R • v 1 , u 1 and B = b, v 0 , u 0 , . . ., v 0 , u 0 , so that D and B have equal lengths.Similarly we define A := v 0 , u 0 , . . ., v 0 , u 0 , C := c, d • D • v 1 , u 1 , and E := v k+1 , u k+1 , . . ., v k+1 , u k+1 , so that they have equal lengths.It is straightforward to verify that these walks satisfy the conditions in the lemma.
So we can assume that P * 0,1 has at most one vertex, and since {u 0 , v 0 } must be anticomplete to {u 1 , v 1 }, we conclude that P * 0,1 contains exactly one vertex, say x.
Repeating the same argument for the reversed asteroid, we obtain that P * 2k,0 has exactly one vertex, say x (it might happen that x = x ).
Let us assume that P 0,1 starts with u 0 , v 0 , as the other case is symmetric.This means that the consecutive vertices of P 0,1 are u it exists by the definition of an asteroid.Note that Q is induced and anticomplete to {u 0 , v 0 }, and exactly one of the vertices u 1 , v 1 and exactly one of the vertices u k+1 , v k+1 belong to Q.
If x has no neighbors in Q * , we can define our gadget to be the tuple is a special triple, each of these vertices has a private neighbor.The private neighbor of x is v 0 , and the private neighbor of d is its successor on Q.We define walks A := v 0 , u 0 and C := v 1 , u 1 and E := v k+1 , u k+1 .Moreover, we define D := Q d and B := x, v 0 , u 0 , . . ., u 0 , so that they are of equal length.
So we can assume that x has a neighbor in Q * .Denote by y the last neighbor of x in Q * and by q the successor of y on Q; it exists, because Q terminates at one of u k+1 , v k+1 and y = v k+1 .Clearly, q ∈ N F (v 0 ), because it belongs to Q which is anticomplete to {u 0 , v 0 }; also q ∈ N F (v 1 ) by the choice of Q: otherwise we would have chosen a shorter path starting with v 1 and then using the consecutive elements of Q q .Note that q might be equal to u k+1 .We now branch on two cases, depending on the size of Q * .
Case 1: , we define our gadget to be the tuple (v 0 , x, y, q, v 1 ).The private neighbors of v 0 , v 1 , and y are u 0 , u 1 , and q, respectively.The private neighbor of x is v 0 .The private neighbor of q is its successor on We define walks A := v 0 , u 0 , . . ., u 0 and C := Q y and E := v 1 , u 1 , . . ., u 1 , so that they have equal lengths.Similarly, we define B := x, v 0 , u 0 , . . ., v 0 , u 0 and D = Q q • u k+1 , v k+1 , u k+1 , so that they have equal lengths.We appended u k+1 , v k+1 , u k+1 at the end of D, so that we do not need to treat the case that q = u k+1 separately.
So assume that y ∈ N F (u 1 ), so it is the first vertex of Q * .Note that in this case q ∈ Q * , so, in particular, q / ∈ {u k+1 , v k+1 }.Therefore q has a successor q in Q.Then we take the tuple (v 0 , x, y, q, v k+1 ) with corresponding private neighbors u 0 , v 0 , u 1 , q , and u k+1 .We define walks as follows: A := v 0 , u 0 and C := y, u 1 , and E := v k+1 , u k+1 .Furthermore, we define B := x, v 0 , u 0 , . . ., u 0 and D := Q q .Note that in the currently considered case the length of D is at least two.
Case 2: |Q * | = 1.So we are left with the case that Q * consists only of a vertex y, which is adjacent to both x and u k+1 .Recall that Q ⊆ P 1,2 ∪ . . .∪ P k,k+1 ∪ P k+1,k+2 . . .P 2k−1,2k , which means that there is non-empty First suppose that there is some i ∈ I \ {k}.This means that there exists = i + k + 1 = 0 such that {u , v , u 0 , v 0 } is anticomplete to P i,i+1 .Furthermore, since P i,i+1 is connected and contains at least 2 vertices, y ∈ P i,i+1 has a neighbor r in P i,i+1 .We define the extended P 4 gadget as the tuple (v 0 , x, y, u k+1 , v ).The corresponding private neighbors are u 0 , v 0 , r, v k+1 , and u .Recall that x ∈ P 0,1 , so it must be non-adjacent to v k+1 .
We define walks B := x, v 0 , u 0 and D := u k+1 , v k+1 , u k+1 .The definition of the remaining three walks is more intricate.Let R be the shortest y-u i -walk using consecutive vertices of P i−1,i For j ∈ [2k], let P j,j+1 be the shortest u j -u j+1 -walk using vertices of P j,j+1 .By P j+1,j we denote the walk P j,j+1 in the reversed order.being any u k -u 1 -walk contained in K, we define E := K and walks A := u k+1 , v k+1 , . . ., u k+1 and C := x, v 0 , u 0 , . . ., u 0 of same length as E.
This means that we can assume that x = x , and thus F contains the structure depicted in Figure 2 (left).
Figure 2 Left: The structure in the last case in the proof of Lemma 18. Dashed edges might exist, but do not have to.A smaller asteroid exists if which contradicts the minimality of F (see Figure 2 (right)).
So suppose that at least one of these edges, say x v k+1 , does not exist (the other case is symmetric).The minimality of F implies that the edge x v 1 also does not exist: otherwise F − x still contains the asteroid (U, V ), where the path between u 0 and v i is u 0 , x , v 1 .In such a case we take the tuple (u k+1 , y, x , v 0 , u 1 ), where their corresponding private neighbors are v k+1 , u k+1 , v 0 , u 0 , and v 1 .The walks are A := u k+1 , v k+1 , u k+1 , B := y, u k+1 , C := x , v 0 , u 0 , D := v 0 , u 0 , and E := u 1 , v 1 , u 1 .This completes the proof of the lemma.Now we proceed to the proof of Lemma 16.

Proof of Lemma 16.
If H contains an induced cycle with consecutive vertices x 0 , x 1 , . . ., x k−1 , x 0 for k ∈ {6, 8}, we define an extended P 4 gadget to be the tuple (x 0 , x 1 , x 2 , x 3 , x 4 ).Clearly, for every pair of distinct i, j ∈ {0, . . ., k} we have N H (x i ) = N H (x j ), as they belong to an induced cycle of length more than 4. Then the appropriate instances (G {x0,x2,x4} , L) are shown in Figure 3.As the cycles are symmetric, we can obtain the instance (G {x1,x3} , L) for C 6 and C 8 by taking the same graph as G {x0,x2,x4} , removing x 4 from the lists of γ 1 , γ 2 , and replacing x i by x i+1 (modulo k) for every element of every list.
Observe that every induced cycle in H on at least 10 vertices, with consecutive vertices x 0 , x 1 , . . ., x k−1 , x 0 , contains an asteroid ({x 0 , x 4 , x 6 }, {x 1 , x 3 , x 7 }): the paths P 0,1 := x 0 , x 1 , x 2 , x 3 , x 4 , and P 1,2 := x 4 , x 5 , x 6 , and P 2,0 := x 6 , x 7 , . . ., x k−1 , x 0 satisfy Definition 13.So now it is sufficient for consider the case that H contains an asteroid.By Lemma 18 we know that in such a case there exist: Recall that A, C, and E are of equal length, say , i.e., each of them has + 1 vertices.We define the instance C(A, C, E) := (G, L) of List H-Coloring, such that G is a path with consecutive vertices y 1 , y 2 , . . ., y +1 , and the list L(y i ) contains the i-th vertex of A, the i-th vertex of C, and the i-th vertex of E. Note that since walks A, C, E avoid each other, for every i ∈ [ + 1] we have |L(y i )| = 3, and, in particular, L(y 1 ) = {a, c, e} and L(y +1 ) = T .
Furthermore, each list homomorphism h from C(A, C, E) to H coincides either with one of A, C, E.More formally, we have the following: 1. for every x ∈ {a, c, e}, there is a list homomorphism h x : C({A, C, E}) → H, such that h x (y 1 ) = x and h x (y +1 ) = σ(x), and 2. for any list homomorphism h : C({A, C, E}) → H there is x ∈ {a, c, e}, such that h(y 1 ) = x and h(y +1 ) = σ(x).
From the gadgets of Lemma 16, we can also make efficient larger gadgets to enforce that in a large group of vertices, at least one vertex is not colored c.The construction is an adaptation of a gadget due to Jaffke and Jansen [23].
Proof.The construction is a small adaptation of a gadget due to Jaffke and Jansen [23], which we present here for completeness.
Let T be the complete graph (triangle) on vertex set {1, 2, 3}.We first show how to construct an instance (G , L ) of List T -Coloring with k distinguished vertices γ 1 , . . ., γ k , such that a mapping f : {γ 1 , . . ., γ k } → {1, 2, 3} can be extended to a proper List T -Coloring if and only if f (γ i ) = 1 for some i ∈ [k].Then, we will transform (G , L ) into an instance (G, L) of List H-Coloring with the desired properties by replacing edges with the gadgets of Lemma 16, without blowing up the number of vertices.
The List T -Coloring instance (G , L ) is constructed as follows.Create a path on 3k vertices x 1 , y 1 , z 1 , x 2 , y 2 , z 2 , . . ., x k , y k , z k .Add vertices γ 1 , . . ., γ k and insert the edge γ i y i for all i ∈ [k].This defines graph G .The lists L are defined as follows: For this instance (G , L ) of List T -Coloring, we first argue that a partial coloring that assigns color 1 to all of γ 1 , . . ., γ k cannot be extended to a proper list T -coloring.To see that, note that due to the edges between γ i and y i , the color 1 is blocked for all vertices y i .This means extending the coloring is equivalent to finding a list coloring on the path x 1 , y 1 , z 1 , . . ., x k , y k , z k , where all x-vertices have list {1, 2} (except x 1 which must be colored 2), where all y-vertices have list {2, 3}, and all z-vertices have list {3, 1} (except z k which must be colored 3).But the path has no proper list coloring under these conditions: Since the color of x 1 is fixed to 2, y 1 must be colored 3, implying z 1 must be colored 1, which propagates throughout the path to imply that y k must be colored 3, which conflicts with the fact that L (z k ) = {3}.Hence a mapping that colors all γ i with 1 cannot be extended to a proper list T -coloring of (G , L ).
Next, we argue that if f : {γ 1 , . . ., γ k } → {1, 2, 3} such that f (γ i ) = 1 for some i ∈ [k], then f can be extended to a proper list T -coloring of (G , L ).Consider such an f , and define i − := min{i | f (γ i ) = 1} and i + := max{i | f (γ i ) = 1}, which are well-defined.Let P be the path (x 1 , y 1 , z 1 , . . ., x k , y k , z k ) in its natural ordering from x 1 to z k , and extend f as follows: Set f (y i ) = 1 for all i ∈ [k] for which f (γ i ) = 1.For all vertices before y i − on P , color the x-vertices 2, the y-vertices 3, and the z-vertices 1.For all vertices after y i + on P , color the x-vertices 1, the y-vertices 2, and the z-vertices 3.
Consider the vertices we have not assigned a color so far (if any).They form subpaths P of P of the form z j , x j+1 , . . ., x j for j < j with f (γ j ), f (γ j ) = 1, while f (γ i ) = 1 for j < i < j .Set f (z j ) = 3, set f (x j ) = 2, and for the remaining vertices of P color the x-vertices 2, the y-vertices 3, and the z-vertices 1.Note that for all γ i which are not colored 1, the corresponding y i gets color 1, while if f (γ i ) = 1 then f (y i ) ∈ {2, 3}.It is straight-forward to verify that the resulting extension of f forms a proper list T -coloring of G .
To construct the gadget for List H-Coloring promised by the lemma statement, we transform (G , L ) into a List H-Coloring instance (G, L) as follows.Let (a, b, c, d, e) be an extended P 4 gadget for H as guaranteed by Lemma 16,and let (G a,c,e , L a,c,e ) with distinguished vertices γ * 1 , γ * 2 .
Initialize (G, L) as a copy of (G , L ).Replace occurrences of color 1 by c, of color 2 by a, and of color 3 by e.
Since G is built by replacing all edges of G by gadgets, which are consistent instances by Lemma 16, and since graph G we start from is a tree and therefore bipartite, it is easy to see that the instance (G, L) is consistent.This concludes the proof.Using these gadgets in the two-step process described in the beginning of Section 4, we now obtain the following.Theorem 1.If H is an undirected graph that is not a bi-arc graph, possibly with loops, then List H-Coloring parameterized by the number of vertices n admits no generalized kernel of size O(n 2−ε ) for any ε > 0, unless NP ⊆ coNP/poly.Proof.We start by showing that for any connected bipartite graph H that is not a bi-arc graph, List H-Coloring allows no nontrivial sparsification.We use a linear-parameter transformation from Annotated List P 4 -Coloring, such that the lower bound follows from Lemma 7.
Since H is bipartite and not a bi-arc graph, it is not the complement of a circular arc graph [13], and it follows from Theorem 14 that H has an induced cycle of length at least six or an asteroid.It then follows from Lemma 16 that H has an extended P 4 gadget on distinguished vertices (a, b, c, d, e) of H. Furthermore, there exist two relevant gadgets as described by Lemma 16.We call the gadget constructed for Q = {a, c, e} the a, c, e-NOT-gadget, and the one constructed for Q = {b, d} the b, d-NOT-gadget.
Let an instance (G, L, S, F ) of Annotated List It remains to show the result for non-bipartite graphs H and bipartite graphs H that are not connected.We start with the latter.Since H is not a bi-arc graph, there must exist a connected component H of H such that H is not a bi-arc graph (and since H is bipartite, H is bipartite).This follows from the fact that by Theorem 14 the graph H has an induced cycle of length at least six or an asteroid, and this structure must be found in one of its components.It now follows from the above, that List H -Coloring does not have a generalized kernel of size O(n 2−ε ), unless containment.There is a straightforward linear-parameter transformation from List H -Coloring to List H-Coloring, taking the exact same instance and using the lists to ensure that only colors from H can be used to color each vertex.Therefore, the lower bound for List H-Coloring for (possibly not connected) bipartite graphs H follows.
We conclude the proof by showing the result for non-bipartite graphs.Let H be an undirected graph that is not a bi-arc graph, such that H is non-bipartite, let H * be the associated bipartite graph of H. Since H is not a bi-arc graph, it follows that H * is not the complement of a circular arc graph [13,Proposition 3.1].Since H * is bipartite and irreflexive it follows that H * is not a bi-arc graph.
As proven above, it follows that List H * -Coloring does not have a generalized kernel of size O(n 2−ε ), unless NP ⊆ coNP/poly.Proposition 12 gives a straightforward linearparameter transformation from List H * -Coloring to List H-Coloring, showing that the same lower bound holds for List H-Coloring.

Conclusion
A natural open question is whether analogous results can be obtained for the (non-list) H-Coloring problem.Despite the obvious similarity of H-Coloring and List H-Coloring, they appear to behave very differently when it comes to proving lower bounds.All hardness proofs for List H-Coloring [11,12,13,14,33], including the proofs in this paper, are purely combinatorial and focus on the local structure of H.In all of them, we first identify some "hard" substructure H in H, and then prove the lower bound for H .This can be done, as we can ignore vertices in V (H) \ V (H ) by not including them in the lists.On the other hand, all proofs for H-Coloring use some algebraic tools [3,20,35,38] which allow capturing the global structure of H.We therefore expect similar difficulties in the case of proving sparsification lower bounds for H-Coloring.
[n] := {1, . . ., n}.For a set S we use the notation S k := {S ⊆ S | |S | = k} to denote the set of all size-k subsets of S, and we define 2 S := |S| k=0 S k .We use the notation S k := {(s 1 , . . ., s k ) | s 1 , . . ., s k ∈ S} to denote the set of all k-tuples with elements from S. In particular, [n] 2 denotes all 2-tuples of elements from [n].

Lemma 7 .
Annotated List P 4 -Coloring parameterized by the number of vertices n admits no generalized kernel of size O(n 2−ε ) for any ε > 0, unless NP ⊆ coNP/poly.

Figure 1 A
Figure 1 A sketch of the created graph G, for n = 4, and k = 3 where x3x4 / ∈ E(G2,3).Edges between P and Q are omitted, except for the edges that result from the fact that x3x4 / ∈ E(G2,3).A fat edge between u and v indicates that {u, v} ∈ F .Vertex sets contained in S are marked in blue.White vertices have lists {b, d} while black vertices have list {a, c}.Note that the constructed graph is bipartite with the white and black vertices as partite sets.

2 .
H * is the complement of a circular-arc graph.Thus the graphs H for which List H-Coloring is NP-hard, are precisely those for which List H * -Coloring is NP-hard: when H * is the complement of a circular-arc graph.Now let us explain how showing the hardness of List H-Coloring can be reduced to showing the hardness of List H * -Coloring.Here we need the notion of a consistent instance of the problem.Definition 11.Let F be a connected bipartite graph with bipartition classes X and Y .An instance (G, L) of List F -Coloring is consistent, if G is bipartite and has a bipartition into classes A, B ⊆ V (G), such that L(a) ⊆ X for all a ∈ A, and L(b) ⊆ Y for all b ∈ B.

Definition 15 .
An extended P 4 gadget in an undirected simple graph H is a tuple (a, b, c, d, e) of distinct vertices in H, such that all of the following hold: 1. H[{a, b, c, d}] is isomorphic to P 4 , 2. the sets N H (a), N H (c), N H (e) are pairwise incomparable, and 3. the sets N H (b), N H (d) are pairwise incomparable.
(a) a special triple T , (b) an extended P 4 gadget (a, b, c, d, e), (c) an injective function σ : {a, c, e} → T , (d) an injective function π : {b, d} → T , (e) walks A, C, and E, starting, respectively, in a, c,, and e, and terminating, respectively, in σ(a), σ(c), σ(e), such that each two of A, C, and E avoid each other, (f) walks B, D, starting, respectively, in b and d, and terminating, respectively, in π(b), π(d), such that B and D avoid each other.Let us show how to construct (G {a,c,e} , L), the construction of (G {b,d} , L) is analogous, we just need to use walks B and D instead of A, C, E.

Figure 4
Figure4 The construction of (G {a,c,e} , L) as a composition of two copies of C(A, C, E) and a copy of F .We have L(γ1) = L(γ2) = {a, c, e} and L(δ1) = L(δ2) = T = {σ(a), σ(b), σ(c)}.Blue lines denote which mappings of γ1, δ1, δ2, γ2 to the vertices on their lists can be extended to a list homomorphism of particular gadgets.

1 with v 1
For each edge e of G , do the following.Let v 1 , v 2 be the endpoints of e. Remove edge v 1 v 2 from G , insert a new copy of the graph (G a,c,e , L a,c,e ) with lists as given by L a,c,e .Let γ * 1 , γ * 2 denote the distinguished vertices of the inserted copy.Identifying γ * and γ * 2 with v 2 .Since G has O(k) vertices and edges, the transformation to G introduces O(k) gadgets, each of which has constant size.Hence |V (G )| ∈ O(k), as required.It is easy to perform the construction in polynomial time.Since the gadget (G a,c,e , L a,c,e ) for distinguished vertices γ * 1 , γ * 2 for List H-Coloring has the same effect as an edge in List T -Coloring, while color 1 ∈ V (T ) was mapped to color c ∈ V (H), it follows that a mapping f : {γ 1 , . . ., γ k } → {a, c, e} can be extended to a proper list H-coloring of (G, L) if and only if

P 4 -
Coloring be given, we show how to create an instance G of List H-Coloring.Initialize G as G (ignoring the annotations), where every vertex in G receives the same list it had in G, where now a, b, c, d, e refer to the vertices of the extended P 4 gadget present in H.For any {u, v} ∈ F , if L(u) ⊆ {a, c, e} (implying also L(v) ⊆ {a, c, e}), add a new a, c, e-NOT-gadget to G. Otherwise, meaning that L(u) ⊆ {b, d} and L(v) ⊆ {b, d}, we add a new b, d-NOT-gadget to G. Identify vertex γ 1 of the added gadget with u, and vertex γ 2 with v.For every S = {s 1 , . . ., s m } ∈ S, add a new gadget as described by Lemma 19 for k = m to G. Note that such a gadget has O(m) vertices.Identify vertex γ i of the gadget with vertex s i for all i ∈ [m].It is easy to observe from the correctness of the added gadgets, that G is list H-colorable if and only if G had a coloring respecting the annotations.We continue by bounding the number of vertices in G. Using that S∈S |S| ≤ 3|V (G)| and |F | ≤ |V (G)| by definition of Annotated List P 4 -Coloring, we get|V ( G)| = |V (G)| init + |V (G)| • O(1) NOT-gadgets + O(|V (G)|) Lemma 19 = O(|V (G)|),which is properly bounded for a linear parameter transformation.The result for bipartite graphs H thus follows from[2, Theorem 3.8].Observe that the constructed graph G is consistent, such that the lower bound holds even for consistent instances of List H-Coloring.
takes time polynomial in |x| + k and outputs an instance (x , k ) such that: (i) |x | and k are bounded by h(k), and (ii) sketch.
f in P j * has color a.Otherwise, let h(q i * e,f ) := d.We color r i * e,f in the same way, thus h(r i * e,f ) := b if its unique neighbor in P j * has color a, and h(r i * e,f ) := d otherwise.Color qi * e,f with the only color in {b, d} \ {h(q i * e,f )} and similarly color ri * e,f with the only color in {b, d} \ {h(r i It remains to show that h is a valid annotated P 4 -coloring of G.We split this into three parts.First of all, we verify that each S ∈ S contains a vertex that does not get color c.For Ŷ and Ẑ this was verified before.Consider a set {y j } ∪ {p j ,m | ∈ [n]} added in Step 8. Observe that if j = j * then y j has color a and we are done.Otherwise, by definition, we have h(p j * e,f )}.Finally, let h(s i * e,f ) := c if h(q i * e,f ) = d and let h(s i * e,f ) := a otherwise.Similarly, let h(t i * e,f ) := c if h(r i * e,f ) = d and let h(t i * e,f ) := a otherwise.This concludes the definition of h.* im,m ) := a and thus indeed this set has a vertex of color a.Now consider a set {s i e,f , t i e,f , z i } added in Step 9.If i = i * , vertex z i has color a and we are done.Otherwise if i = i * , we claim that it cannot be the case that h(s

1 .
walks A, C, E, starting, respectively, in a, c, and e, and terminating in distinct elements of T , such that each two of these walks avoid each other, 2. walks B, D, starting, respectively, in b and d, and terminating in distinct elements of T , such that B and D avoid each other.Proof.Recall that in the definition of an extended P 4 gadget (a, b, c, d, e) we require that the appropriate pairs of neighborhoods are incomparable.Actually, we will show a stronger property, i.e., that each of a, c, e has a private neighbor, which is non-adjacent to the other two vertices.We extend this notion and call vertices in N H (b) \ N H (d) and in N H (d) \ N H (b) private neighbors of b and d, respectively.
19.Let H be a bipartite graph which contains an induced cycle of at least 6 vertices or an asteroid, and let (a, b, c, d, e) be an extended P 4 gadget in H as guaranteed by Lemma 16.For any k ≥ 2 one can construct a consistent List H-Coloring instance (G, L) in polynomial time containing k distinguished vertices γ 1 , . . ., γ k such that |V (G)| ∈ O(k), and such that a mapping f : {γ 1 , . . ., γ k } → {a, c, e} can be extended to a proper list H-coloring of (G, L) if and only if there exists an