Packing Bipartite Graphs with Covers of Complete Bipartite Graphs (cid:63)

. For a set S of graphs, a perfect S -packing ( S -factor) of a graph G is a set of mutually vertex-disjoint subgraphs of G that each are isomorphic to a member of S and that together contain all vertices of G . If G allows a covering (locally bijective homomorphism) to a graph H , then G is an H -cover. For some ﬁxed H let S ( H ) consist of all connected H -covers. Let K k,(cid:96) be the complete bipartite graph with partition classes of size k and (cid:96) , respectively. For all ﬁxed k, (cid:96) ≥ 1, we determine the computational complexity of the problem that tests whether a given bipartite graph has a perfect S ( K k,(cid:96) )-packing. Our technique is partially based on exploring a close relationship to pseudo-coverings. A pseudo-covering from a graph G to a graph H is a homomorphism from G to H that becomes a covering to H when restricted to a spanning subgraph of G . We settle the computational complexity of the problem that asks whether a graph allows a pseudo-covering to K k,(cid:96) for all ﬁxed k, (cid:96) ≥ 1.


Introduction
Throughout the paper we consider undirected graphs with no loops and no multiple edges. Let G = (V, E) be a graph and let S be some fixed set of mutually vertex-disjoint graphs. A set of (not necessarily vertex-induced) mutually vertex-disjoint subgraphs of G, each isomorphic to a member of S, is called an S-packing. Packings naturally generalize matchings (the case in which S only contains edges). They arise in many applications, both practical ones such as exam scheduling [12], and theoretical ones such as the study of degree constraint graphs (cf. the survey of Hell [11]). If S consists of a single subgraph S, we write S-packing instead of S-packing. The problem of finding an S-packing of a graph G that packs the maximum number of vertices of G is NP-hard for all fixed connected graphs S on at least three vertices, as shown by Hell and Kirkpatrick [13].
A packing of a graph is perfect if every vertex of the graph belongs to one of the subgraphs of the packing. Perfect packings are also called factors and from now on we call a perfect S-packing an S-factor. We call the corresponding decision problem the S-Factor problem. For a survey on graph factors we refer to the monograph of Plummer [19].
Our Focus. We study a relaxation of K k, -factors, where K k, denotes the biclique (complete connected bipartite graph) with partition classes of size k and , respectively. In order to explain this relaxation we first need to introduce some new terminology.
A homomorphism from a graph G to a graph H is a vertex mapping f : V G → V H satisfying the property that f (u)f (v) belongs to E H whenever the edge uv belongs to E G . If for every u ∈ V G the restriction of f to the neighborhood of u, i.e., the mapping f u : N G (u) → N H (f (u)), is bijective then we say that f is a locally bijective homomorphism or a covering [2,16]. The graph G is then called an H-cover and we write G B − → H. Locally bijective homomorphisms have applications in distributed computing [1] and in constructing highly transitive regular graphs [3]. For a specified graph H, we let S(H) consist of all connected H-covers. In this paper we study S(K k, )-factors of bipartite graphs. (c) a bipartite K2,3-pseudocover that is no K2,3-cover and that has no K2,3-factor. (d) a bipartite graph with a K2,3-factor that is not a K2,3-pseudo-cover. (e) a bipartite graph with an S(K2,3)-factor but with no K2,3-factor and that is not a K2,3-pseudo-cover.
Our Motivation. Since a K 1,1 -factor is a perfect matching, K 1,1 -Factor is polynomial-time solvable. The K k, -Factor problem is known to be NPcomplete for all other k, ≥ 1, due to the aforementioned result of Hell and Kirkpatrick [13]. These results have some consequences for our relaxation. In order to explain this, we make the following observation, which holds because only a tree has a unique cover (namely the tree itself) and the graph K k, is a tree if k = 1 or = 1. Because S(K 1, ) = {K 1, } by Observation 1, the above results immediately imply that S(K 1, )-Factor is only polynomial-time solvable if = 1; it is NPcomplete otherwise. What about our relaxation for k, ≥ 2? Note that, for these values of k, , the size of the set S(K k, ) is unbounded. The only result known so far is for k = = 2; Hell, Kirkpatrick, Kratochvíl and Kříž [14] showed that S(K 2,2 )-Factor is NP-complete for general graphs, as part of their computational complexity classification of finding restricted 2-factors; we explain the reason why an S(K 2,2 )-factor is a restricted 2-factor later.
For bipartite graphs, the following is known. Firstly, Monnot and Toulouse [18] researched path factors in bipartite graphs and showed that the K 2,1 -Factor problem stays NP-complete when restricted to the class of bipartite graphs. Secondly, we observed that as a matter of fact the proof of the NP-completeness result for S(K 2,2 )-Factor in [14] is even a proof for bipartite graphs.
Our interest in bipartite graphs stems from a close relationship of S(K k, )factors of bipartite graphs and so-called K k, -pseudo-covers, which originate from topological graph theory and have applications in the area of distributed computing [4,5]. A homomorphism f from a graph G to a graph H is a pseudo-covering from G to H if there exists a spanning subgraph G of G such that f is a covering from G to H. In that case G is called an H-pseudo-cover and we write G P − → H. The computational complexity classification of the H-Pseudo-Cover problem, which is to test for a fixed graph H (i.e., not being part of the input) whether G P − → H for some given G is still open, and our paper can also be seen as a first investigation into this question. We explain the exact relationship between factors and pseudo-coverings in detail later on; we refer to Figure 1 for some examples that illustrate the notions introduced.
Our Results and Paper Organization. Section 2 contains additional terminology, notations and some basic observations. In Section 3 we pinpoint the relationship between factors and pseudocoverings. In Section 4 we completely classify the computational complexity of the S(K k, )-Factor problem for bipartite graphs. Recall that S(K 1,1 )-Factor is polynomial-time solvable on general graphs. We first prove that S(K 1, )-Factor is NP-complete on bipartite graphs for all fixed ≥ 2. By applying our result of Section 3, we then show that NP-completeness of every remaining case can be shown by proving NP-completeness of the corresponding K k, -Pseudo-Cover problem. We classify the complexity of K k, -Pseudo-Cover in Section 5. We show that it is indeed NP-complete on bipartite graphs for all fixed pairs k, ≥ 2 by adapting the hardness construction of Hell, Kirkpatrick, Kratochvíl and Kříž [14] for restricted 2-factors. In contrast to S(K k, )-Factor, we show that K k, -Pseudo-Cover is polynomial-time solvable for all k, ≥ 1 with min{k, } = 1. In Section 6 we further discuss the relationships between pseudocoverings and locally constrained homomorphisms, such as the aforementioned coverings. We shall see that as a matter of fact the NP-completeness result for K k, -Pseudo-Cover for fixed k, ≥ 3 also follows from a result of Kratochvíl, Proskurowski and Telle [15] who proved that K k, -Cover is NP-complete for k, ≥ 3. This problem is to test whether G B − → K k, for a given graph G. However, the same authors [15] showed that K k, -Cover is polynomial-time solvable when k = 2 or = 2. Hence, for those pairs (k, ) we can only use our hardness proof in Section 5.

Preliminaries
From now on let X = {x 1 , . . . , x k } and Y = {y 1 , . . . , y } denote the partition classes of K k, . If k = 1 then we say that x 1 is the center of K 1, . If = 1 and k ≥ 2, then y 1 is called the center. We denote the degree of a vertex u in a graph G by deg G (u).
Recall that a homomorphism f from a graph G to a graph H is a pseudocovering from G to H if there exists a spanning subgraph G of G such that f is a covering from G to H. We would like to stress that this is not the same as saying that f is a vertex mapping from V G to V H such that f restricted to some spanning subgraph G of G becomes a covering. The reason is that in the latter setting it may well happen that f is not a homomorphism from G to H.
For instance, f might map two adjacent vertices of G to the same vertex of H. However, there is an alternative definition which turns out to be very useful for us. In order to present it we need the following notations.
We let f −1 (x) denote the set {u ∈ V G | f (u) = x}. For a subset S ⊆ V G , G[S] denotes the induced subgraph of G by S, i.e., the graph with vertex set S and edges uv whenever uv ∈ E G . For Because f is a homomorphism, G[x, y] is a bipartite graph with partition classes f −1 (x) and f −1 (y). We can now state the alternative definition of pseudo-coverings.

Proposition 1 ([4]).
A homomorphism f from a graph G to a graph H is a pseudo-covering if and only if G[x, y] contains a perfect matching for all x, y ∈ Let f be a pseudo-covering from a graph G to a graph H. We then sometimes call the vertices of H colors of vertices of G. Due to Proposition 1, G[x, y] must contain a perfect matching M xy . Let uv ∈ M xy for xy ∈ E H . Then we say that v is a matched neighbor of u, and we call the set of matched neighbors of u the matched neighborhood of u.

How Factors Relate to Pseudo-Covers
Our next result shows how S(K k, )-factors relate to K k, -pseudo-covers.
Theorem 1. Let G be a graph on n vertices. Then G is a K k, -pseudo-cover if and only if G has an S(K k, )-factor and G is bipartite with partition classes A and B such that |A| = kn k+ and |B| = n k+ .
Proof. First suppose that G = (V, E) is a K k, -pseudo-cover. Let f be a pseudocovering from G to K k, . Then f is a homomorphism from G to K k, , which is a bipartite graph. Consequently, G must be bipartite as well. Let A and B denote the partition classes of G. Then we may assume without loss of generality that f (A) = X and f (B) = Y . Due to Proposition 1 we then find that |A| = kn k+ and |B| = n k+ . By the same proposition we find that each G[x i , y j ] contains a perfect matching M ij . We define the spanning subgraph G = (V, ij M ij ) of G and observe that every component in G is a K k, -cover. Hence G has an S(K k, )-factor. Now suppose that G has an S(K k, )-factor {F 1 , . . . , F p }. Also suppose that G is bipartite with partition classes A and B such that |A| = kn k+ and |B| = n k+ .
Let A X be the set of vertices of A that are mapped to a vertex in X and let A Y be the set of vertices of A that are mapped to a vertex in Y . We define subsets B X and B Y of B in the same way. This leads to the following equalities: Suppose that = k. Then this set of equalities has a unique solution, namely, |A X | = kn k+ = |A|, |A Y | = |B X | = 0, and |B Y | = n k+ = |B|. Hence, we find that f maps all vertices of A to vertices of X and all vertices of B to Y . This means that f is a homomorphism from G to K k, that becomes a covering when restricted to the spanning subgraph obtained by taken the disjoint union of the subgraphs {F 1 , . . . , F p }. In other words, f is a pseudo-covering from G to K k, , as desired.
Suppose that = k. In this case we have that In the second case, we can exchange the roles of X and Y and find another covering This completes the proof of Theorem 1.

Classifying the S(K k, )-Factor Problem
Here is the main theorem of this section.
Theorem 2. The S(K k, )-Factor problem is solvable in polynomial time for k = = 1. Otherwise it is NP-complete, even for the class of bipartite graphs. Proof. We may assume without loss of generality that k ≤ . First we consider the case when k = = 1. Due to Observation 1, the S(K 1,1 )-Factor problem is equivalent to the problem of finding a perfect matching, which can be solved in polynomial time. We deal with the case when k = 1 and ≥ 2 in Proposition 2. Finally, for all k ≥ 2 and all ≥ 2, we show in Proposition 3 that if the K k, -Pseudo-Cover problem is NP-complete, then so is the S(K k, )-Factor problem for the class of bipartite graphs. Then the result for this case follows from Theorem 4, in which we show that K k, -Pseudo-Cover is NP-complete for all k ≥ 2 and all ≥ 2.
The proof of Theorem 2 is conditional upon proving Propositions 2 and 3, and Theorem 4. We prove Theorem 4 in Section 5, and show Propositions 2 and 3 in this section.
Proposition 2 deals with the case k = 1 and ≥ 2. Recall that for general graphs the NP-completeness of this case immediately follows from Observation 1 and the aforementioned result of Hell and Kirkpatrick [13]. However, we consider bipartite graphs. For this purpose, a result by Monnot and Toulouse [18] is of importance for us. Here, P k denotes a path on k vertices.

Theorem 3 ([18]).
For any fixed k ≥ 3, the P k -Factor problem is NP-complete for the class of bipartite graphs.
We use Theorem 3 to prove Proposition 2.
Proposition 2. For any fixed ≥ 2, S(K 1, )-Factor and K 1, -Factor are NP-complete, even for the class of bipartite graphs. Proof. By Observation 1, S(K 1, ) = {K 1, } for all ≥ 2. Hence we may restrict ourselves to K 1, -Factor. Clearly, K 1, -Factor is in NP for all ≥ 2. Note that P 3 = K 1,2 . Hence the case = 2 follows from Theorem 3. Let = 3. We prove that K 1,3 -Factor is NP-complete by reduction from K 1,2 -Factor. Let G = (V, E) be a bipartite graph with partition classes A and B. We will construct a bipartite graph G from G such that G has an K 1,2 -factor if and only if G has a K 1,3 -factor.
First we make a key observation, namely that all K 1,2 -factors of G (if there are any) have the same number α of centers in A and the same number β of centers in B. This is so, because the following two equalities that count the number of vertices in A and B, respectively, have a unique solution. In order to obtain G we do as follows. Let A = {a 1 , . . . , a p } and Finally we add 2p + α new vertices x 1 , . . . , x 2p+α and add edges such that the subgraph induced by the w-vertices and the x-vertices is complete bipartite. We denote the set of s-vertices by S, the set of t-vertices by T , the set of u-vertices by U , the set of w-vertices by W , and the set of x-vertices by X. We repeat the above process with respect to B. For clarity we denote the new vertices with respect to B by s , t , u , w , x , and corresponding sets by S , T , U , W , X , respectively. This yields the graph G which is bipartite with partition classes Figure 2.
FACTOR. Let be a bipartite graph with partition classes and . We will construct a bipartite graph from such that has an -factor if and only if has a -factor. First we make a key observation, namely that all -factors of (if there are any) have the same number of centers in and the same number of centers in . This is so, because the following two equalities that count the number of vertices in and , respectively, have a unique solution. In order to obtain we do as follows. Let  Finally we add new vertices and add edges such that the subgraph induced by the -vertices and the -vertices is complete bipartite. We denote the set of -vertices by , the set of -vertices by , the set of -vertices by , the set ofvertices by , and the set of -vertices by . We repeat the above process with respect to . For clarity we denote the new vertices with respect to by , and corresponding sets by , respectively. This yields the graph which is bipartite with partition classes and . Also see Figure 6.  We are now ready to prove our claim that G has a K 1,2 -factor if and only if G has a K 1,3 -factor.
Suppose that G has a K 1,2 -factor. We first extend the three-vertex stars in this factor to four-vertex stars by adding the edge a i s i for every star center a i and the edge b i s i for every star center b i . As we argued above, A contains α centers and B contains β centers. This means that we can add: • p − α stars with center in T , one leaf in S and two leaves in U ; • α stars with center in T and three leaves in U ; • p − α stars with center in W , one leaf in U and two leaves in X; • α stars with center in W and three leaves in X. This is possible because |S| = p, |T | = p, |U | = 3p, |W | = p and |X| = 2(p − α) + 3α = 2p + α. With respect to B we can proceed in the same way. Hence, we obtained a K 1,3 -factor of G .
Suppose that G has a K 1,3 -factor. Let γ be the number of star centers in A that belong to stars with one leaf in S and two leafs in B. Let δ be the number of star centers in B that belong to stars with one leaf in S and two leafs in A. We first show that γ ≥ α.
In order to obtain a contradiction, suppose that γ < α. Because every svertex (resp. u-vertex) has degree two, no vertex in S (resp. U ) is a star center. Let p 1 be the number of star centers in T that belong to stars with a leaf in S (and two leafs in U ) and let p 2 be the number of star centers in T that belong to stars with all three leafs in U . By our construction, every star center in W belongs to a star that either has one leaf in U and two leafs in X, or else has three leafs in X. Let q 1 be the number of star centers in W of the first type, and let q 2 be the number of star centers in W of the second type. Finally, let r be the number of star centers in X (centers of stars with all leafs in W ). Then by using counting arguments in combination with the equalities |S| = |T | = |W | = p, |U | = 3p and |X| = 2p + α, we derive the following equalities: The last two equalities imply that q 2 = α + 5r. Equality γ + p 1 = p and our assumption γ < α implies that p 1 > p − α. Equalities p 1 + p 2 = p and 2p 1 + 3p 2 + q 1 = 3p lead to p 1 = q 1 . Hence, we find that q 1 > p − α. Substituting q 1 > p − α and q 2 = α + 5r into equality q 1 + q 2 + 3r = p yields 8r < 0 and this is not possible. Hence γ ≥ α.
By the same reasoning as above we find that δ ≥ β holds. This has the following consequence. Let γ * denote the number of star centers in A that belong to stars with three leaves in B and let δ * denote the number of star centers in B that belong to stars with three leaves in A. Then we find that p = γ + 2δ + γ * + 3δ * ≥ α + 2β + γ * + 3δ * .
Recall that α + 2β = p. If we substitute this in the above equation, we find that p ≥ p + γ * + 3δ * . Hence γ = α, δ = β and γ * = δ * = 0. This means that the restriction of the K 1,3 -factor to G is a K 1,2 -factor of G, which is what we had to show.
For ≥ 4 we can proceed in a similar way as for the case = 3 (or use induction). This completes the proof of Proposition 2.
Here is Proposition 3, which allows us to consider the K k, -Pseudo-Cover problem for all k ≥ 2 and all ≥ 2.
Proposition 3. Fix arbitrary integers k, ≥ 2. If the K k, -Pseudo-Cover problem is NP-complete, then so is the S(K k, )-Factor problem for the class of bipartite graphs.
Proof. Let k, ≥ 2. Let G = (V, E) be an input graph on n vertices of the K k, -Pseudo-Cover problem. By Theorem 1, we may assume without loss of generality that G is bipartite with partition classes A and B such that |A| = kn k+ and |B| = n k+ . Then, by Theorem 1, we find that G P − → K k, holds if and only if G has an S(K k, )-factor. This finishes the proof of Proposition 3.

Classifying the K k, -Pseudo-Cover Problem
Here is the main theorem of this section.  Proof. Let k = 1, ≥ 1, and G be a graph. We show that deciding whether G is a K 1, -pseudo-cover comes down to solving the problem of finding a perfect matching in a graph of size at most |V G |. Because the latter can be done in polynomial time, this means that we have proven the proposition. If = 1, then deciding whether G is a K 1, -pseudo-cover is readily seen to be equivalent to finding a perfect matching in G. Now suppose that ≥ 2. We first check in polynomial time whether G is bipartite with partition classes A and B, such that |A| = n 1+ and |B| = n 1+ . If not, then Theorem 1 tells us that G is a no-instance. Otherwise we continue as follows. Because k = 1 and ≥ 2, we can distinguish between A and B. We replace each vertex a ∈ A by copies a 1 , . . . , a and make each a i adjacent to all neighbors of a. This leads to a bipartite graph G , the partition classes of which have the same size. We claim that G is a K 1, -pseudo-cover if and only if G has a perfect matching.
First suppose that G is a K 1, -pseudo-cover. Then there exists a pseudocovering f from G to K 1, . Because k = 1 and ≥ 2, we find that f (a) = x 1 for all a ∈ A and f (B) = Y . Consider a vertex a ∈ A. Let b 1 , . . . , b be its matched neighbors. In G we select the edges a i b i for i = 1, . . . , . After having done this for all vertices in A, we obtain a perfect matching of G . Now suppose that G has a perfect matching. We define a mapping f by f (a) = x 1 for all a ∈ A and f (b) = y i if and only if a i b is a matching edge in G , where a i is the ith copy of a. Then f is a pseudo-covering from G to K 1, . Hence, G is a K 1, -pseudo-cover. This completes the proof of Proposition 4.
We now prove that K k, -Pseudo-Cover is NP-complete for all k, ≥ 2 (Proposition 5). Our proof is inspired by the proof of Hell, Kirkpatrick, Kratochvíl, and Kríẑ [14]. They consider the problem of testing if a graph has an S Lfactor for any set S L of cycles, the length of which belongs to some specified set L. This is useful for our purposes because of the following. If L = {4, 8, 12, . . . , }, then an S L -factor of a bipartite graph G with partition classes A and B of size n 2 is an S(K 2,2 )-factor of G that is also a K 2,2 -pseudo-cover of G by Theorem 1. However, for k = ≥ 3, this is not longer true, and when k = the problem is not even "symmetric" anymore. Below we show how to deal with these issues. We refer to Section 6 for an alternative proof for the case k, ≥ 3. However, our construction for k, ≥ 2 does not become simpler when we restrict ourselves to k, ≥ 2 with k = 2 or = 2. Therefore, we decided to present our NP-completeness result for all k, with k, ≥ 2.
Recall that we denote the partition classes of K k, by X = {x 1 , . . . , x k } and Y = {y 1 , . . . , y }. We first state a number of useful lemmas. Hereby, we use the alternative definition in terms of perfect matchings, as provided by Proposition 1, when we argue on pseudo-coverings.
Let G 1 (k, ) be the graph in . We first deduce a number of useful lemmas, the proof of which can be found in Appendices B -H. Hereby, we use the alternative definition in terms of perfect matchings, as provided by Proposition 1, when we argue on pseudo-coverings.
Let be the graph in Figure 2. It contains a vertex with neighbors and a vertex with neighbors . For any , , it contains an edge . Finally, it contains a vertex which is only adjacent to .    Lemma 3. Let G be a bipartite graph that contains G 1 (k, ) as an induced subgraph, such that only a and e have neighbors outside G 1 (k, ) and such that a and e have no common neighbor. Let G be the graph obtained from G by removing all vertices of G 1 (k, ) and by adding a new vertex u that is adjacent to every vertex of G that is a neighbor of a or e outside G 1 (k, ). Let f be a pseudo-covering from G to K k, , such that f (u) ∈ X and such that u has exactly one neighbor v of a in its matched neighborhood. Then G is a K k, -pseudo-cover. Proof. We may assume without loss of generality that f (u) =  Proof. Because all v-vertices have degree and all w-vertices have degree k, all edges of G 2 (k, ) must be in perfect matchings. If k = , this means that every vvertex must get an x-color, whereas every u-vertex and every w-vertex must get a y-color. Moreover, if k = , then we may assume this without loss of generality. As all v-vertices have degree , the vertices in any {u i , w h,1 , . . . , w h, −1 } have different x-colors. Moreover, the way we defined the edges between the u-vertices and the v-vertices implies that every u-vertex must have the same y-color, i.e., |f ({u 1 , . . . , u k })| = 1. Because all edges of G 2 (k, ) are perfect matching edges and every u-vertex has degree k − 1 in G 2 (k, ), we find that every u i has exactly one matched neighbor t i outside G 2 (k, ). In the (matched) neighborhood of {u 1 , u 2 , . . . , u k } in G 2 (k, ), each color x i appears exactly k − 1 times. Consequently, in the matched neighborhood of {u 1 , u 2 , . . . , u k } outside G 2 (k, ), each x i appears once and thus |f ({t 1 , . . . , t k })| = k.
Lemma 5. Let G be a bipartite graph that has G 2 (k, ) as an induced subgraph, such that only u-vertices have neighbors outside G 2 (k, ) and such that no two u-vertices have a common neighbor. Let G be the graph obtained from G by removing all vertices of G 2 (k, ) and by adding a new vertex s that is adjacent to every vertex of G that is a neighbor of some u-vertex outside G 2 (k, ). Let f be a pseudo-covering from G to K k, , such that f (s) ∈ Y and such that s has exactly one neighbor t i of every u i in its matched neighborhood. Then G is a K k, -pseudo-cover. Proof. We may assume without loss of generality that f (s) = y and f (t i ) = x i for i = 1, . . . , k. We modify f as follows.
In this way we find a pseudo-covering from G 2 (k, ) to K k, .
Let G 3 (k, ) be the graph defined in Figure 5. It contains k copies of G 1 (k, ), where we denote the a-vertex and e-vertex of the ith copy by a i and e i , respectively. It also contains a copy of G 2 (k, ) with edges e i u i and a i u i+1 for i = 1, . . . , k (where u k+1 = u 1 ). The construction is completed by adding a vertex p adjacent to all a-vertices and by adding vertices q, r 1 , . . . , r −2 that are adjacent to all e-vertices. Here we assume that there is no r-vertex in case = 2.
Lemma 6. Let G be a bipartite graph that has G 3 (k, ) as an induced subgraph, such that only p and q have neighbors outside G 3 (k, ). Let f be a pseudo-covering from G to K k, . Then either every a i is a matched neighbor of p and no e i is a matched neighbor of q, or else every e i is a matched neighbor of q and no a i is a matched neighbor of p.
Proof. We first show the claim below.

Claim.
Either every e i u i is in a perfect matching and no a i u i+1 is in a perfect matching, or every a i u i+1 is in a perfect matching and no e i u i is in a perfect matching.
We prove this claim as follows. Every u i is missing exactly one color in its matched neighborhood in G 2 (k, ) by Lemma 4. This means that, for any i, either a i−1 u i is in a perfect matching, or else e i u i is in a perfect matching. We Let be the graph in Figure 5. It is constructed as follows. We take copies of . We denote the -vertex and the -vertex of the th copy by and , respectively. We take copies of . We denote the -vertex and the -vertex of the th copy by and , respectively. We add an edge between any and . We are now ready to present our NP-completeness reduction. This finishes the proof of Theorem 4.

Proposition 5. The
-PSEUDO-COVER problem is NP-complete for any fixed with .
Proof. We reduce from the problem -DIMENSIONAL MATCHING, which is NPcomplete as (see [10]). In this problem, we are given mutually disjoint sets , all of equal size , and a set of hyperedges . The question is whether contains a -dimensional matching, i.e., a subset of size such that for any distinct pairs show that in the first case e i−1 u i−1 is not in a perfect matching, and that in the second case a i u i+1 is not in a perfect matching. Suppose that a i−1 u i is in a perfect matching. By Lemma 4, u i−1 and u i have the same color. By Lemma 2, d i−1 is a matched neighbor of e i−1 with f (d i−1 ) = f (u i−1 ). Hence, e i−1 u i−1 is not in a perfect matching. Suppose that e i u i is in a perfect matching. Then by the same reasoning, a i u i+1 is not in a perfect matching.
Suppose that e 1 u 1 is in a perfect matching. Then a 1 u 2 is not in a perfect matching, and consequently e 2 u 2 is in a perfect matching, and so on, until we deduce that every e i u i is in a perfect matching and no a i u i+1 is in a perfect matching. Suppose that e 1 u 1 is not in a perfect matching. Then by the same reasoning we can show the opposite. This proves the claim.
Note that every e i r j must be in a perfect matching due to the degree of r j . Thus, every e i has exactly one matched neighbor in {q, u i }. Moreover, each a i has exactly one matched neighbor in {p, u i+1 }. Applying the claim then yields the desired result.
Lemma 7. Let G be a graph that has G 3 (k, ) as an induced subgraph such that only p and q have neighbors outside G 3 (k, ) and such that p and q do not have a common neighbor. Let G be the graph obtained from G by removing all vertices of G 3 (k, ) and by adding a new vertex r * that is adjacent to every vertex of G that is a neighbor of p or q outside G 3 (k, ). Let f be a pseudo-covering from G to K k, such that f (r * ) ∈ Y and such that either all vertices in the matched neighborhood of r * in G are all neighbors of p in G, or else are all neighbors of q in G. Then G is a K k, -pseudo-cover. Proof. We may assume without loss of generality that f (r * ) = y . We show how to modify f . Let First suppose that the matched neighborhood of r * in G is in the neighborhood of p in G. We define perfect matching edges as follows: the matched neighbor of each a i outside the ith copy of G 1 (k, ) is u i+1 ; the matched neighbors of each e i outside the ith copy of G 1 (k, ) are q and the r-vertices. By Lemmas 3 and 5, we can extend f to all other vertices of G 3 (k, ). Hence, we find that G is a K k, -pseudo-cover. Now suppose that the matched neighborhood of r * in G is in the neighborhood of q in G. We define perfect matching edges as follows: the matched neighbor of each a i outside the ith copy of G 1 (k, ) is p; the matched neighbors of each e i outside the ith copy of G 1 (k, ) are u i and the r-vertices. By Lemmas 3 and 5, we can extend f to all other vertices of G 3 (k, ). Hence, also in this case, G is a K k, -pseudo-cover.
Let G 4 (k, ) be the graph in Figure 6. It is constructed as follows. We take k copies of G 3 ( , k). We denote the p-vertex and the q-vertex of the ith copy by p 1,i and q 1,i , respectively. We take copies of G 3 (k, ). We denote the p-vertex and the q-vertex of the jth copy by p 2,j and q 2,j , respectively. We add an edge between any p 1,i and p 2,j . and in we have for and for . Given such an instance, we construct a bipartite graph with partition classes and . First we put all elements in in , and all elements in in . Then we introduce an extra copy of for each hyperedge by adding the missing vertices and edges of this copy to . We observe that indeed is bipartite. We also observe that has polynomial size.
We  Lemma 8. Let G be a bipartite graph that has G 4 (k, ) as an induced subgraph such that only the q-vertices have neighbors outside G 4 (k, ). Let f be a pseudocovering from G to K k, . Then either every p 1,i p 2,j is in a perfect matching and all matched neighbors of every q-vertex are in G 4 (k, ), or else no edge p 1,i p 2,j Due to Proposition 6, the NP-completeness of K k, -Pseudo-Cover for k, ≥ 3 also follows from the NP-completeness of K k, -Cover for these values of k, . The latter is shown by Kratochvíl, Proskurowski and Telle [15]. However, these authors show in the same paper [15] that K k, -Cover is solvable in polynomial time for the cases k, with min{k, } ≤ 2. Hence for these cases we have to rely on our proof in Section 5.
Another consequence of Proposition 6 is that H-Pseudo-Cover is NPcomplete for all k-regular graphs H for any k ≥ 3 due to a hardness result for the corresponding H-Cover [6]. However, a complete complexity classification of H-Pseudo-Cover is still open, just as dichotomy results for H-Partial Cover and H-Cover are not known, whereas for the locally surjective case a complete complexity classification has been given [8]. So far, we could obtain some partial results but a complete classification of the complexity of H-Pseudo-Cover seems already difficult for trees (we found many polynomial-time solvable and NP-complete cases).