The Edmonds-Gallai Decomposition for the k-Piece Packing Problem Marek Janata Dept. of Applied Mathematics and Institute of Theoretical Computer Science (ITI), Charles University, Malostranske n. 25, 118 00 Praha 1, Czech Republic. janata@kam.mff.cuni.cz, Martin Loebl Dept. of Applied Mathematics and Institute of Theoretical Computer Science (ITI), Charles University, Malostranske n. 25, 118 00 Praha 1, Czech Republic. loebl@kam.mff.cuni.cz, and Jácint Szabó∗ Dept. of Operations Research, Eötvös University, Pázmány Péter sétány 1/C, Budapest, Hungary H-1117. jacint@cs.elte.hu Submitted: Feb 4, 2004; Accepted: Feb 4, 2005; Published: Feb 14, 2005 MR Subject Classifications: 05C70 Abstract Generalizing Kaneko’s long path packing problem, Hartvigsen, Hell and Szabó consider a new type of undirected graph packing problem, called the k-piece packing problem. A k-piece is a simple, connected graph with highest degree exactly k so in the case k = 1 we get the classical matching problem. They give a polynomial algorithm, a Tutte-type characterization and a Berge-type minimax formula for the k-piece packing problem. However, they leave open the question of an Edmonds-Gallai type decomposition. This paper fills this gap by describing such a decomposition. We also prove that the vertex sets coverable by k-piece packings have a certain matroidal structure. Research is supported by OTKA grants T 037547, N 034040 and by the Egerváry Research Group of the Hungarian Academy of Sciences and by European MCRTN Adonet, Contract Grant No. 504438. ∗ the electronic journal of combinatorics 12 (2005), #R8 1 1 Introduction In this paper all graphs are simple and undirected. Given a set F of graphs, an F -packing of a graph G is a subgraph P of G such that each connected component of P is isomorphic to a member of F . An F -packing P is called maximal if there is no F -packing P ′ with V (P ) ( V (P ′ ). An F -packing is maximum if it covers a maximum number of vertices of G and it is perfect if it covers every vertex of G. The F -packing problem is to describe the properties of the F -packings of G. Finally, the F -packing problem is polynomial if for all input graphs G the size of the maximum F -packings of G can be determined in time polynomial in the size of G. (The size of a graph is the number of its vertices.) Several polynomial F -packing problems are known in the case K2 ∈ F . For instance, we get a polynomial packing problem if F consists of K2 and a finite set of hypomatchable graphs [2, 3, 4, 6]. A complete classification of the {K2 , F }-packing problems for graphs F is given in [10]. In all known polynomial F -packing problems with K2 ∈ F it holds that each maximal F -packing is maximum too; those vertex sets which can be covered by an F -packing form a matroid (this is the matroidal property); and the analogue of the Edmonds-Gallai structure theorem holds. The first polynomial F -packing problem with K2 ∈ / F was considered by Kaneko [7], who presented a Tutte-type characterization of graphs having a perfect packing by long paths, ie. by paths of length at least 2. A shorter proof for Kaneko’s theorem and a minmax formula was subsequently found by Kano, Katona and Király [8] but polynomiality remained open. The long path packing problem was generalized by Hartvigsen, Hell and Szabó [5] by introducing the k-piece packing problem, ie. the F -packing problem where F consists of all connected graphs with highest degree exactly k. Such a graph is called a k-piece. Note that a 1-piece is just K2 , thus the 1-piece packing problem is the classical matching problem. The 2-piece packing problem is equivalent to the long path packing problem because a 2-piece is either a long path or a circuit C of length at least 3 so deleting an edge from C results in a long path. The main result of [5] is a polynomial algorithm for finding a maximum k-piece packing. From this algorithm a characterization for graphs having a perfect k-piece packing and a min-max result for the size of a maximum k-piece packing are derived. Neither the Edmonds-Gallai decomposition nor the matroidal property of packings is considered in [5]. This paper fills this gap by giving a canonical Edmonds-Gallai type decomposition for the k-piece packing problem. We also show that the vertex sets coverable by maximal k-piece packings have a certain matroidal structure, see Section 2. It turns out that in the k-piece packing problem maximal and maximum packings do not coincide and the maximal packings are of more interest than the maximum ones. In Section 5 we present some results on barriers related to k-piece packings, for instance we prove that the intersection of two barriers is a barrier. The number of connected components of a graph G is denoted by c(G) and the highest degree of G by ∆(G). For X ⊆ V (G) the subgraph induced by X is denoted by G[X], and the set of vertices in V (G) − X which are adjacent to a vertex in X is denoted by Γ(X). We say that an edge e enters X if exactly one end-vertex of e is contained in X. the electronic journal of combinatorics 12 (2005), #R8 2 For a subgraph P of G let G − P = G[V (G) − V (P )]. Finally, we say that an F -packing P of G misses a vertex set X ⊆ V (G) if X ∩ V (P ) = ∅ and that P covers X if X ⊆ V (P ). 2 The theorems In this section we state the main theorems of the paper. The proofs are contained in Sections 4 and 7. Till Section 8, k is a fixed positive integer. Definition 2.1. A k-piece is a connected graph G with ∆(G) = k. Definition 2.2. For a graph G we denote IG = G[{v ∈ V (G) : degG (v) ≥ k}]. Definition 2.3. A graph G is hypomatchable if G − v has a perfect matching for all v ∈ V (G). In [5] it was revealed that galaxies play a central role in the k-piece packing problem. Definition 2.4. [5] For an integer k ≥ 1 the connected graph H is a k-galaxy if it satisfies the following properties: • each component of IH is a hypomatchable graph, • for each v ∈ V (IH ) there exist exactly k − 1 edges between v and V (H) − V (IH ), each being a cut edge in H. A hypomatchable graph has no vertex of degree 1 so a k-galaxy has no vertex of degree k. Furthermore, each component of IH is a hypomatchable graph on at least 3 vertices. Since k is fixed, we shall call a k-galaxy simply a galaxy. Galaxies generalize hypomatchable graphs because the 1-galaxies are exactly the hypomatchable graphs. The 2-galaxies were introduced by Kaneko under the name ‘sun’ [7]. See Fig. 1 for some galaxies. The vertices of IH are drawn as big dots and the edges of IH as thick lines. 11 00 11 00 11 00 1 0 0 1 1 0 1 0 1 0 0 1 11 00 11 00 11 00 00 11 1 0 1 0 11 00 11 00 11 00 11 00 11 00 00 11 11 00 00 11 a 1-galaxy 11 00 11 00 1 0 1 0 1 0 1 0 0 1 11 00 11 00 11 00 2-galaxies IH : 11 00 11 00 11 00 11 00 11 00 11 00 1 0 1 0 11 00 11 00 11 00 11 00 11 00 1 0 1 0 1 0 1 0 1 0 11 00 00 11 11 00 11 00 11 00 11 00 11 00 11 00 11 00 1 0 0 1 1 0 1 0 1 0 11 00 11 00 11 00 11 00 11 00 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 11 00 00 11 11 00 00 1 11 0 00 1 11 0 1 0 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 1 0 1 0 1 0 1 0 0 1 1 0 0 1 11 00 00 11 11 00 11 00 11 00 11 00 11 00 1 0 1 0 11 00 11 00 11 00 1 0 1 0 1 0 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 11 00 11 00 11 00 11 00 11 00 1 0 0 1 11 00 11 00 11 00 1 0 1 0 1 0 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 11 00 11 00 11 00 11 00 11 00 11 00 1 0 1 0 11 00 11 00 11 00 1 0 1 0 11 00 11 00 1 0 1 0 1 0 a 4-galaxy tips: Fig. 1. Galaxies The following important property of galaxies was proved in [5]. the electronic journal of combinatorics 12 (2005), #R8 3 Lemma 2.5. [5] A k-galaxy has no perfect k-piece packing. Now we introduce special subgraphs of galaxies, called tips. Each tip is circled by a thin line in Fig. 1 (except in the 4-galaxy of Fig. 1 where not all tips are circled). Definition 2.6. [5] If k ≥ 2 then for a k-galaxy H the connected components of H −V (IH ) are called tips. In the case k = 1 we call each vertex of H a tip. The union of vertex sets of the tips is denoted by WH ⊆ V (H). So WH = V (H) if k = 1 and WH = V (H) − V (IH ) if k ≥ 2. In the case k ≥ 2 a k-galaxy may consist of only a single tip (a graph with highest degree at most k − 1), but must always contain at least one tip. The Edmonds-Gallai structure theorem can be formulated for the k-piece packing problem as follows. The classical Edmonds-Gallai theorem first defines the vertex set D to consist of those vertices which can be missed by a maximal matching. In the k-piece packing problem we have to use a different formulation. This causes the fact that Theorem 2.8 is not a direct generalization of the classical Edmonds-Gallai theorem. Definition 2.7. For a graph G let UG = {v ∈ V (G) : there exists a maximal k-piece packing P of G with v ∈ / V (P ) }. Theorem 2.8. For a graph G let D = {v : |UG−v | < |UG |}, A = Γ(D) and C = V (G) − (D ∪ A). Now 1. the connected components of G[D] are k-galaxies, 2. for all ∅ = 6 A′ ⊆ A the number of those k-galaxy components of G[D] which are adjacent to A′ is at least k|A′ | + 1, 3. G[C] has a perfect k-piece packing, 4. a k-piece packing P of G is maximal if and only if (a) exactly k|A| connected components of G[D] are entered by an edge of P and these components are completely covered by P , (b) if H is a component of G[D] not entered by P then P [H] is a maximal k-piece packing of H, (c) P [C] is a perfect k-piece packing of G[C], 5. for each maximal k-piece packing P of G, the graph G−P has exactly c(G[D])−k|A| connected components. For proof, see Section 4. We could also choose D = {v : UG−v ( UG } by Theorem 4.19. It is a well known fact in matching theory that those vertex sets which can be covered by a matching form a matroid. In the k-piece packing problem this property holds only in the following weaker form. The proof is contained in Section 7. the electronic journal of combinatorics 12 (2005), #R8 4 Theorem 2.9. There exists a partition π on V (G) and a matroid M on π such that the of the maximal k-piece packings are exactly the vertex sets of the form S vertex sets ′ {X : X ∈ π } where π ′ is a base of M. 3 Preliminaries In this section we summarize the results and notions of [5] which are needed to prove the main theorems of the paper. First we introduce two other classes of graphs which are near to galaxies. Definition 3.1. For an integer k ≥ 2 the connected graph H is an almost k-galaxy of type 1 if it satisfies the following properties: • one of the components of IH has a perfect matching and the others are hypomatchable, • for each v ∈ V (IH ) there exist exactly k − 1 edges between v and V (H) − V (IH ), each being a cut edge in H. Definition 3.2. For an integer k ≥ 2 the connected graph H is an almost k-galaxy of type 2 if it satisfies the following properties: • each component of IH is a hypomatchable graph, • there is a distinguished vertex w ∈ V (IH ) such that for each v ∈ V (IH ) each edge between v and V (H) − V (IH ) is a cut edge in H, and the number of these edges is k − 1 for v 6= w and k − 2 for w. w almost k-galaxy of type 1 almost k-galaxy of type 2 Fig. 2. Almost galaxies, k = 4 Fig. 2 shows some almost 4-galaxies. Just like in the case of galaxies, we define tips for almost galaxies. Some tips are circled by a thin line in Fig. 2. Definition 3.3. For an almost galaxy H the connected components of H − IH are called tips. Many properties of the galaxies are explained by the following lemma, which is implicit in [5]. the electronic journal of combinatorics 12 (2005), #R8 5 Lemma 3.4. Each almost k-galaxy has a perfect k-piece packing. Proof. First we prove the statement for almost galaxies of type 2. Let H be an almost k-galaxy of type 2. We proceed by induction on |V (H)|. Let K be the component of IH containing the specified vertex w. K is a hypomatchable graph on at least 3 vertices so it is easy to see that w has two neighbors w ′ , w ′′ ∈ V (K) such that K − {w ′ , w, w ′′ } has a perfect matching M. For each edge uv ∈ M let Puv be the subgraph of H induced by the vertex set [ {u, v} ∪ {V (T ) : T is a tip of H adjacent to {u, v}}. Furthermore, let Pw be the subgraph of H induced by the vertex set [ {w ′ , w, w ′′ } ∪ {V (T ) : T is a tip of H adjacent to {w ′ , w, w ′′ }} , with the deletion of the edge w ′w ′′ (if any). Clearly Puv (uv ∈ M) and Pw are disjoint k-piece subgraphs of H. Deleting these k-pieces from H, each connected component of the remaining graph is an almost k-galaxy of type 2 so we are done by induction. Now let H be an almost k-galaxy of type 1. Denote by K the perfectly matchable component of IH . For each edge uv of a perfect matching of K let Puv be the k-piece subgraph of H induced by the vertex set [ {u, v} ∪ {V (T ) : T is a tip of H adjacent to {u, v}}. Deleting these k-pieces from H, each connected component of the remaining graph is an almost k-galaxy of type 2 so we are done by the first part of the proof. Lemma 3.5. [5] If T is a tip of a k-galaxy H then H − T has a perfect k-piece packing. Proof. The statement holds for k = 1 by definition. Let k ≥ 2. It is easy to see that each component of H − T is an almost k-galaxy of type 2, which has a perfect k-piece packing by Lemma 3.4. For the proof of the following lemma see [5]. Lemma 3.6. [5] If P is a k-piece packing of the k-galaxy H then there exists a tip T of H such that V (P ) ∩ V (T ) = ∅. The maximal matchings of a hypomatchable graph H are exactly the perfect matchings of H − v for the vertices v ∈ V (H). The characterization of the maximal k-piece packings of a k-galaxy can be stated by means of the tips. Lemma 3.7. [5] The maximal k-piece packings of a k-galaxy H are exactly the perfect k-piece packings of H − T where T is a tip of H. Proof. By Lemmas 3.5 and 3.6. the electronic journal of combinatorics 12 (2005), #R8 6 The next lemma is another generalization of the defining property 2.3 of hypomatchable graphs. This lemma is only implicit in [5]. Lemma 3.8. If H is a k-galaxy and v ∈ V (H) then there exists a vertex set v ∈ X ⊆ V (H) such that H[X] is connected, ∆(H[X]) ≤ k − 1 and H − X has a perfect k-piece packing. Proof. The statement is trivial for k = 1 so assume that k ≥ 2. If v is contained in a tip T then let X = V (T ). Now H − X has a perfect k-piece packing by Lemma 3.5 so we are done. If v ∈ V (IH ) then let [ X = {v} ∪ {V (T ) : T is a tip of H adjacent to v}. Clearly ∆(H[X]) = k − 1. It is easy to check that each component of H − X is an almost k-galaxy of type 1 or 2. Hence H − X has a perfect k-piece packing by Lemma 3.4. Definition 3.9. A connected graph G is a k-solar-system (see Fig. 3) if it has a vertex y, called center, such that degG (y) = k and G − y has k connected components, each being a k-galaxy. v1 vk k-galaxies Hk H1 H2 v2 . y . . . Fig. 3. A k-solar system Lemma 3.10. Each k-solar-system has a perfect k-piece packing. Proof. Let G be a k-solar-system with center y. Denote the neighbors of y by vi (1 ≤ i ≤ k) and denote the k-galaxy component of G − y containing vi by Hi . Lemma 3.8 implies that for all 1 ≤ i ≤ k there exists a vertex set vi ∈ Xi ⊆ V (Hi ) such that Hi − Xi has a perfect k-piece packing and Hi [Xi ] is a connected graph S with highest degree at most k − 1. The latter condition on Hi [Xi ] implies that G[{y} ∪ 1≤i≤k Xi ] is a k-piece. [5] describes a polynomial algorithm finding a maximum k-piece packing in the input graph G. The algorithm consists of two phases and already the first phase obtains a maximal k-piece packing of G which is further refined in the second phase (called ’Re-Rooting procedure’) to become a maximum k-piece packing. Now we are interested only in the first phase of the algorithm of [5] to which we simply refer as the algorithm. This algorithm is a direct generalization of the alternating forest matching algorithm of Edmonds. the electronic journal of combinatorics 12 (2005), #R8 7 It builds certain alternating forests and it outputs a decomposition V (G) = D ∪ A ∪ C where the sets D, A, C are pairwise disjoint. It also outputs a maximal k-piece packing P of G but we are not interested in it now. The algorithm may have different runs on the same graph G depending on the actual implementation. We refer to the outputs of these runs as decomposition outputs. In the next section we prove that the decomposition output is unique for all runs of the algorithm and it is canonical for the k-piece packing problem in a certain way. The following proposition is implicit in the description of the algorithm in [5], see Fig. 4. Proposition 3.11. [5] Each run of the algorithm outputs a decomposition V (G) = D ∪ A ∪ C where D, A, C are pairwise disjoint and 1. the connected components of G[D] are k-galaxies, 2. G contains no edge joining D to C, 3. for all ∅ = 6 A′ ⊆ A the number of those k-galaxy components of G[D] which are adjacent to A′ is at least k|A′ | + 1, 4. G[C] has a perfect k-piece packing. k-galaxy components D: A: C: G[C] has a perfect k-piece packing Fig. 4. A decomposition output of the algorithm, k = 2 Any decomposition output of the algorithm implies the Tutte-type existence theorem 3.13 for the k-piece packing problem, proved in [5]. Definition 3.12. Let k-gal(G) denote the number of those connected components of the graph G that are k-galaxies. Theorem 3.13. [5] A graph G has a perfect k-piece packing if and only if k-gal(G − A) ≤ k|A| for all set of vertices A ⊆ V (G). the electronic journal of combinatorics 12 (2005), #R8 8 Proof. The “only if” part is straightforward using that a k-galaxy has no k-piece packing by Lemma 2.5. On the other hand, if G has no perfect k-piece packing then A in any decomposition output of the algorithm will do. 4 The Edmonds-Gallai decomposition In this section we prove that the decomposition output is unique for all runs of the algorithm and that this decomposition has the properties described in Theorem 2.8. Definition 4.1. For A ⊆ V (G) let [ D A = {V (H) : H is a k-galaxy component of G − A}. A We use the notation DG if confusion may arise. Moreover, let C A = V (G) − (D A ∪ A) (or A CG ). Definition 4.2. The vertex set A ⊆ V (G) has k-surplus if for all ∅ = 6 A′ ⊆ A the number A ′ ′ of k-galaxy components of G[D ] adjacent to A is at least k|A | + 1. The vertex set A is perfect if C A has a perfect k-piece packing. Definition 4.3. We say that a vertex set A ⊆ V (G) can be k-matched into X ⊆ V (G)−A by M if M is a subgraph of G with k|A| edges such that degM (v) = k for all v ∈ A and exactly k|A| connected components of G[X] are entered by an edge of M (each by one edge). The vertex set A can be k-matched into X ⊆ V (G) − A if there exists a subgraph M of G such that A can be k-matched into X by M. The following property (in fact, characterization) of the vertex sets with k-surplus is implied by Hall’s theorem. Lemma 4.4. If A ⊆ V (G) has k-surplus then A can be k-matched into D A − V (H) for each connected component H of G[D A ]. Using these definitions we can reformulate Proposition 3.11. Proposition 4.5. For any decomposition output V (G) = D ∪ A ∪ C of the algorithm the set A is perfect with k-surplus. Proof. A k-galaxy has no perfect k-piece packing so D A = D and C A = C. So Proposition 3.11, 3. is tantamount to that A has k-surplus and 4. to that A is perfect. The next lemma describes an important property of the galaxies. Lemma 4.6. If H is a k-galaxy and ∅ = 6 X ⊆ V (H) then k-gal(H − X) ≤ k|X| − 1. Proof. The statement is well-known for k = 1. Indeed, otherwise for x ∈ X the number of hypomatchable components of (H − x) − (X − x) is more than |X − x| implying that H − x has no perfect matching, a contradiction. For k ≥ 2 it is easier to prove the lemma for a broader set of graphs, called pseudo galaxies. the electronic journal of combinatorics 12 (2005), #R8 9 Definition. For an integer k ≥ 2 the connected graph G is a pseudo k-galaxy if for each v ∈ V (IG ) there exist exactly k − 1 edges between v and V (G) − V (IG ), each being a cut edge in G. Note, that this is just the definition of the k-galaxies with the relaxation that the connected components of IG need not be hypomatchable. What we actually prove is Lemma 4.7 which immediately implies Lemma 4.6. Lemma 4.7. If G is a pseudo k-galaxy and ∅ = 6 X ⊆ V (G) is a vertex set with the property that each vertex of X ∩ V (IG ) is contained in a hypomatchable component of IG then k-gal(G − X) ≤ k|X| − 1 holds. Proof. Suppose that G is a pseudo galaxy of minimum size for which a vertex set ∅ 6= X ⊆ V (G) fails Lemma 4.7, ie. k-gal(G − X) ≥ k|X| holds. degG (v) ≤ k − 1 for vertices v∈ / V (IG ) so clearly X ∩ V (IG ) 6= ∅. Let F be a hypomatchable component of IG with XF = X ∩ V (F ) 6= ∅. Assume that the number of k-galaxy components of G − XF is s and denote these components by H1 , . . . , Hs . It is easy to see that the other components of G − XF are pseudo k-galaxies. Let their number be t and denote them by G1 , . . . , Gt . Note that each component K of G − XF satisfies the condition of Lemma 4.7, ie. each vertex of (X ∩ V (K)) ∩ V (IK ) is contained in a hypomatchable component of IK . Let h (resp. g) denote the number of vertices x ∈ X contained in a k-galaxy (resp. pseudo k-galaxy) component of G − XF . Clearly |X| = |XF | + h + g. Let Xi = X ∩ Gi for 1 ≤ i ≤ t. By induction, k-gal(Gi − Xi ) ≤ k|Xi | for 1 ≤ i ≤ t independently of the emptiness of Xi . So the number of k-galaxy components of G − X contained in a component Gi for 1 ≤ i ≤ t is at most kg. Now we bound s. Let Hi be a k-galaxy component of G − XF such that Y = V (Hi ) ∩ V (F ) 6= ∅. It is easy to see that F [Y ] is connected. This implies that F [Y ] is a component of IHi so it is hypomatchable. The number of such hypomatchable components F [Y ] is at most k|XF | − 1 by the already proved case k = 1 of Lemma 4.6. Thus the number of k-galaxy components of G − XF which intersect V (F ) is at most k|XF | − 1. On the other hand, the number of components of G − XF which do not intersect V (F ) is exactly (k − 1)|XF | because each vertex v ∈ XF ⊆ V (F ) is incident with exactly k − 1 cut edges in G. So s ≤ |XF | − 1 + (k − 1)|XF | = k|XF | − 1. Let s′ be the number of those k-galaxy components Hi of G − XF for which X i = X ∩V (Hi ) 6= ∅. For such a component k-gal(Hi −X i ) ≤ k|X i | −1 holds by the minimality of G. So these components contain altogether at most kh − s′ of the k-galaxy components of G − X. Finally, it is trivial that the number of k-galaxy components Hi of G − XF for which X ∩ V (Hi ) = ∅ is s − s′ . Summarizing, k-gal(G − X) ≤ kg + (kh − s′ ) + (s − s′ ) ≤ k(h + g) + s ≤ k(|XF | + h + g) − 1 = k|X| − 1. Theorem 4.8. If A1 , A2 ⊆ V (G) are perfect vertex sets with k-surplus then A1 = A2 . the electronic journal of combinatorics 12 (2005), #R8 10 Proof. Let Di = D Ai and Ci = C Ai for i = 1, 2. Denote by gi the number of components of G[Di ] intersecting A3−i for i = 1, 2. We prove that g1 = g2 = 0. Suppose that g1 ≥ g2 and that A′2 = A2 ∩ D1 6= ∅. By the k-surplus of A2 , the vertex set A′2 is adjacent to at least k|A′2 | + 1 k-galaxy components of G[D2 ]. Let K be a k-galaxy component of G[D2 ] which is adjacent to A′2 . If V (K) ∩ A1 = ∅ then V (K) ⊆ D1 because A′2 ⊆ D1 so K is contained in a k-galaxy component of G[D1 ]. Thus the number of such components K with V (K) ∩ A1 = ∅ is at most k|A′2 | − g1 by Lemma 4.6. So the number of components of G[D2 ] which are adjacent to A′2 and intersect A1 is at least g1 + 1. Thus g2 ≥ g1 + 1, a contradiction. This implies g1 = g2 = 0. Suppose that A1 \ A2 6= ∅. By the k-surplus of A1 the number of components of G[D1 ] which are adjacent to A1 \ A2 is at least k|A1 \ A2 | + 1. These components do not intersect A2 because g1 = 0. Hence k-gal(G[C2 ] − (A1 \ A2 )) ≥ k|A1 \ A2 | + 1 implying that G[C2 ] has no perfect k-piece packing by Theorem 3.13, a contradiction. So A1 ⊆ A2 and by symmetry, A1 = A2 . Theorem 4.9. The decomposition output is unique for all runs of the algorithm. Proof. Let V (G) = D ∪ A ∪ C be any decomposition output of the algorithm. Proposition 4.5 implies that A is perfect with k-surplus hence it is unique by Theorem 4.8. Finally, a k-galaxy has no perfect k-piece packing so D = D A and C = C A . Hence the following definition is sound: Definition 4.10. The unique decomposition output of the algorithm is denoted by V (G) = DG ∪ AG ∪ CG and called the canonical decomposition of G with respect to the k-piece packing problem. Proposition 4.5 and Theorem 4.8 imply Corollary 4.11. If A ⊆ V (G) is perfect and has k-surplus then A = AG . Now we investigate the structure of maximal k-piece packings of G. Lemma 4.12. Each maximal k-piece packing P of G has the following structure: 1. exactly k|AG | connected components of G[DG ] are entered by an edge of P and these components are completely covered by P , 2. if H is a component of G[D] not entered by P then P [H] is a maximal k-piece packing of H, ie. there exists a tip T of H such that P [H] is a perfect k-piece packing of H − T , and 3. P [CG ] is a perfect k-piece packing of G[CG ]. Proof. Let P be a maximal k-piece packing of G. We construct a k-piece packing P ′ with V (P ′) ⊇ V (P ) such that if P fails any of properties 1.-3. then V (P ′ ) ) V (P ) would hold. We need the theorem of Mendelsohn and Dulmage (see 1.4.3 in [11]). the electronic journal of combinatorics 12 (2005), #R8 11 Theorem 4.13. (Mendelsohn, Dulmage) Let B be a bipartite graph with color classes U and V . If B has a matching covering U ′ ⊆ U and another matching covering V ′ ⊆ V then it has a matching covering U ′ ∪ V ′ . We apply Theorem 4.13 to the bipartite graph BA defined as follows. Definition 4.14. We denote kAG = {v i : v ∈ AG , 1 ≤ i ≤ k}. Let V (BA ) = kAG ∪ {H : H is a component of G[DG ]} and E(BA ) = {v i H : 1 ≤ i ≤ k, v is adjacent to H in G}. BA has a matching covering kAG by the k-surplus of AG . Moreover, P shows that BA has a matching covering HP = {H : H is a component of G[DG ] entered by an edge of P }. So Theorem 4.13 implies that BA has a matching M with vertex set kAG ∪ HM where HP ⊆ HM . Using Lemma 3.10, this matching gives rise to a perfect k-piece packing P1 in the subgraph induced by [ AG ∪ {V (H) : H ∈ HM }. Let H be a component of G[DG ] such that H ∈ / HM . By Lemma 3.6 there exists a tip T of H such that V (P ) ∩ V (T ) = ∅. Take a perfect k-piece packing of H − T guaranteed by Lemma 3.5 and denote the union of these k-pieces by P2 . Finally, let P3 be a perfect k-piece packing of G[CG ]. With P ′ = P1 ∪ P2 ∪ P3 we get that V (P ′) ⊇ V (P ). Trivially |HP | ≤ k|AG |. In fact, |HP | = k|AG | holds here because otherwise the matching M of BA would enter strictly more components of G[DG ] than P , resulting in V (P ′) ) V (P ), a contradiction. Properties 1. and 2. are straightforward by the maximality of P and by Lemmas 3.7 and 3.10. For 3. observe that P has no edge joining AG to CG because otherwise |HP | < k|AG | would hold. Observe that Lemma 4.12 holds also by replacing AG by A, DG by D A and CG by C A where A ⊆ V (G) is a perfect vertex set which can be k-matched into D A . This observation will be needed in the proof of Theorem 4.19. Lemma 4.15. If P is a k-piece packing satisfying properties 1., 2. and 3. of Lemma 4.12 then P is maximal. Proof. Properties 1., 2. and 3. imply that c(G − P ) = c(G[DG ]) − k|AG | and that each component of G − P is a tip of some galaxy component of G[DG ]. Let HP = {H : H is a component of G[DG ] entered by an edge of P }. Suppose that P ′ is a k-piece packing covering V (P ) and one more vertex v ∈ / V (P ). Now v is contained in a tip of a galaxy H ∈ / HP . So Property 2. implies that P ′ intersects each tip of H thus P ′ enters H by Lemma 3.6. Moreover, P ′ enters each component in HP by Lemma 3.6. So P ′ enters at least k|AG | + 1 components of G[DG ] which is impossible because degP ′ (v) ≤ k for v ∈ AG . For characterizing DG in the canonical decomposition first we need to characterize the union of the vertex sets of tips in G[DG ]. Recall that UG was introduced in Definition 2.7. S Definition 4.16. Let WG = {WH : H is a k-galaxy component of G[DG ]}. the electronic journal of combinatorics 12 (2005), #R8 12 Lemma 4.17. WG = UG . Proof. Lemma 4.12 implies that UG ⊆ WG . On the other hand, let v ∈ WG be a vertex contained in a tip T of a k-galaxy component H0 of G[DG ]. AG has k-surplus so AG can be k-matched into DG − V (H0 ) by a subgraph M of G. Let HM = {H : H is a component of G[DG ] entered by an edge of M}. Using Lemma S 3.10, M gives rise to a perfect k-piece packing P1 in the subgraph induced by AG ∪ {V (H) : H ∈ HM }. By Lemma 3.7, for each component H ∈ / HM of G[DG ] we can take a perfect k-piece packing of H −TH where TH is any tip of H. Take care to choose TH0 = T . The union of these k-pieces is denoted by P2 . Finally, let P3 be a perfect k-piece packing of G[CG ]. By Lemma 4.15, the k-piece packing P1 ∪ P2 ∪ P3 is maximal and it misses v ∈ WG . In the matching case (ie. in the case k = 1) it holds that WG = DG thus Lemma 4.17 itself characterizes the canonical DG . In the general case only WG ⊆ DG holds so we have to go one step further in order to characterize DG in Theorem 4.19. First we need the following lemma. Lemma 4.18. If H is a k-galaxy and v ∈ V (H) then each component of H − v is either a k-galaxy or has a perfect k-piece packing. Moreover, [ {WK : K is a k-galaxy component of H − v} ( WH . Proof. The statement is well-known for k = 1 so assume k ≥ 2. If v is contained in a tip then clearly each component of H − v is either a k-galaxy or an almost k-galaxy of type 2. Each almost k-galaxy component has a perfect k-piece packing by Lemma 3.4. Furthermore, [ {V (T ) : T is a tip in a component of H − v} = WH − v so we are done. If v ∈ IH then H − v consists of k-galaxy components (the number of which is exactly k − 1), and almost galaxy components of type 1, the number of which is at least 1. Each almost k-galaxy component has a perfect k-piece packing by Lemma 3.4. Moreover, [ {V (T ) : T is a tip in a component of H − v} = WH , but each almost galaxy component contains at least one tip of H, yielding that [ {V (T ) : T is a tip in an almost k-galaxy component of H − v} = 6 ∅. Theorem 4.19. DG = {v : UG−v ( UG } = {v : |UG−v | < |UG |} holds for all graphs G. Proof. We investigate the canonical decomposition of the graph G − v. 1. Let v ∈ CG . Denote the graph G[CG − v] by G′ . Observe that in the graph G − v the set AG ∪ AG′ is perfect with k-surplus. So AG−v = AG ∪ AG′ by Corollary 4.11, yielding that WG−v ⊇ WG , ie. UG−v ⊇ UG by Lemma 4.17. the electronic journal of combinatorics 12 (2005), #R8 13 2. Let v ∈ AG . In the graph G − v the set AG − v is perfect with k-surplus so AG−v = AG − v by Corollary 4.11. Hence WG−v = WG or equivalently, UG−v = UG by Lemma 4.17. 3. Finally, suppose that v ∈ V (H) for a k-galaxy component H of G[DG ]. ∅ is perfect and has k-surplus in the graph H − v by Lemma 4.18 so AH−v = ∅ by Corollary 4.11, yielding that DH−v = {V (K) : K is a k-galaxy component of H − v} and CH−v = {V (K) : K is a component of H − v with a perfect k-piece packing}. G−v Let D ′ = DA = (DG \ V (H)) ∪ DH−v , C ′ = CAG−v = CG ∪ CH−v and W ′ = G G {V (T ) : T is a tip in a component of G[D ′ ]}. Lemma 4.18 implies that W ′ ( WG . In the graph G − v the set AG is perfect because G[C ′ ] has a perfect k-piece packing. Moreover, AG can be k-matched into D ′ in G − v because AG has k-surplus in G. So the statement of Lemma 4.12 holds for AG in the graph G − v, as we mentioned after the proof of 4.12. This especially implies that each maximal k-piece packing of G − v misses only vertices in W ′ . So UG−v ⊆ W ′ ( WG = UG and we are done. At this point the proof of Theorem 2.8 is straightforward using the results of this section. Proof of Theorem 2.8. D = DG , A = AG and C = CG by Theorem 4.19. Now Property 1. holds by definition. AG is perfect with k-surplus which is just tantamount to Properties 2. and 3. Property 4. is equivalent to Lemmas 4.12 and 4.15. Finally, 5. follows from Property 4. By Theorem 2.8 the graph G has a canonical decomposition V (G) = Dk ∪ Ak ∪ Ck for each k ≥ 1. Here D1 ∪ A1 ∪ C1 is the classical Edmonds-Gallai decomposition. Observe that Ak = Ck = ∅ if k ≥ ∆(G) + 1 and Dk = Ak = ∅ if k = ∆(G). Nevertheless, there does not seem to be any nice relation between the decompositions for different k’s. 5 The calculus of barriers In this section we prove some properties of barriers which we define to be those vertex sets A which maximize k-gal(G − A) − k|A|. Not all of the following results generalize the theory of barriers described by Lovász and Plummer [11] because they count the odd size components instead of the hypomatchable components as we do. Definition 5.1. For A ⊆ V (G) the deficiency of A is def(A) = k-gal(G − A) − k|A|. The deficiency of G is def(G) = max{def(A) : A ⊆ V (G)}. Finally, A ⊆ V (G) is a barrier if def(A) = def(G). the electronic journal of combinatorics 12 (2005), #R8 14 Theorem 3.13 is tantamount to saying that G has a perfect k-piece packing if and only if def(G) = 0. In this case ∅ is a barrier with deficiency 0. Proposition 5.2. AG is a barrier of G. Proof. Let P be a maximal k-piece packing of G. Lemma 4.12 implies that c(G − P ) = k-gal(G−AG )−k|AG | = def(AG ). On the other hand, let A be a barrier of G. The number of components of G[D A ] which are not entered by P is clearly at least k-gal(G−A)−k|A| = def(A). Thus c(G − P ) ≥ def(A) by Lemma 2.5. This implies that def(AG ) ≥ def(A) and so that AG is a barrier. In the matching case (ie. when k = 1) each maximum (and so each maximal) matching misses def(G) vertices of G. This property fails for general k because a maximal k-piece packing of a galaxy may miss an arbitrary number of vertices instead of only one (namely, the vertices of a tip). What is salvaged, is that c(G − P ) = def(G) for each maximal k-piece packing P by Lemma 4.12 and Proposition 5.2. Lemma 5.3. Each barrier is perfect. Proof. Let A be a barrier of G. Assume that G[C A ] has no perfect k-piece packing. Then by Theorem 3.13 there exists a set X ⊆ C A such that k-gal(G[C A ] − X) − k|X| > 0. But then def(A ∪ X) > def(G) would hold, a contradiction. Theorem 5.4. If A is a barrier then AG ⊆ A and DG ⊆ D A . Proof. Let A be a barrier of G and let H = {H : H is a component of G[D A ]}. For J ⊆ H let n o [ Γ(J ) = v ∈ A : v is adjacent to {V (H) : H ∈ J } . Consider the following function f on H: for J ⊆ H let f (J ) = |J | − k|Γ(J )|. Clearly f (J ) ≤ def(G) for J ⊆ H and f is a supermodular function. Suppose that f (J1) = f (J2 ) = def(G) for J1 , J2 ⊆ H. Now 2·def(G) = f (J1 )+f (J2) ≤ f (J1 ∩J2 )+f (J1∪J2 ) ≤ 2 · def(G) implying that f (J1 ∩ J2 ) = def(G). f (H) = def(G) thus there exists an inclusion-wise minimum set H0 ⊆ H with f (H0 ) = def(G). Let A0 = Γ(H0 ). The set A0 has k-surplus S because H0 is minimum. ′ Let D = {V (H) : H ∈ H − H0 }. We state that A − A0 can be k-matched into D ′ by a subgraph M of G. This is due to Hall’s theorem: if Y ⊆ A − A0 was adjacent to less than k|Y | components of G[D ′ ] then def(A − Y ) > def(A) = def(G) would hold because Y is not adjacent to any component H ∈ H0 . Moreover, k|A − A0 | = |H − H0 | so M gives rise to a perfect k-piece packing in D ′ ∪ (A − A0 ) using Lemma 3.10. Moreover, by Lemma 5.3, G[CA ] has a perfect k-piece packing so A0 is perfect. Summarizing, A0 is perfect with k-surplus so AG = A0 ⊆ A by Corollary 4.11. Moreover, clearly DG = D A0 = D A − D ′ . Note that in this proof, A − A0 is adjacent to at most k|A − A0 | components in G[D A ] hence if A has k-surplus then A − A0 = ∅. This implies that AG is the only barrier with k-surplus. the electronic journal of combinatorics 12 (2005), #R8 15 Theorem 5.5. The intersection of two barriers is a barrier. Proof. Let A1 , A2 be barriers of G. We let Di = D Ai and Ci = C Ai for i = 1, 2. Denote by gi the number of components of G[Di ] intersecting A3−i . Wlog. we may assume that g1 ≤ g2 . Furthermore, • gC is the number of components of G[D1 ] contained in C2 , • gD is the number of components of G[D1 ] contained in D2 and not adjacent to A1 ∩ D2 , ′ is the number of components of G[D1 ] contained in D2 and adjacent to A1 ∩ D2 . • gD Now ′ k|A1 | + def(G) = k-gal(G − A1 ) = gC + g1 + gD + gD . The graph G[C2 ] has a perfect k-piece packing by Lemma 5.3 so gC ≤ k|A1 ∩ C2 |. The components of G[D1 ] which are contained in D2 but which are not adjacent to A1 ∩D2 are connected components of G − (A1 ∩ A2 ) as well so gD ≤ k-gal(G − (A1 ∩ A2 )). Each component of G[D1 ] which is contained in D2 and which is adjacent to A1 ∩ D2 is contained in some component H of G[D2 ]. The number of such components H was denoted by g2 . Hence Lemma 4.6 implies that ′ gD ≤ k|A1 ∩ D2 | − g2 . Summarizing, k|A1 | + def(G) ≤ k|A1 ∩ C2 | + k|A1 ∩ D2 | + g1 − g2 + k-gal(G − (A1 ∩ A2 )) ≤ ≤ k|A1 | + k-gal(G − (A1 ∩ A2 )) − k|A1 ∩ A2 |. So def(G) ≤ def(A1 ∩ A2 ), ie. A1 ∩ A2 is a barrier. Theorem 5.6. If A1 and A2 are barriers such that there is no edge between A1 ∩ D A2 and A2 ∩ D A1 then A1 ∪ A2 is a barrier. Proof. Let Di = D Ai and Ci = C Ai for i = 1, 2. We prove that A1 ∩ D2 and A2 ∩ D1 are empty. Assume that A1 ∩ D2 6= ∅ and let K be a component of G[D2 ] such that X = A1 ∩ V (K) 6= ∅. X ⊆ A1 is adjacent to at least k|X| components of G[D1 ] since otherwise def(A1 − X) > def(G) would hold. Let v ∈ D1 be a vertex adjacent to x ∈ X. v∈ / C2 since G contains no edge between D2 and C2 . v ∈ / A2 either by the condition of the theorem. Hence v is contained in the same component of G[D2 ] than x, ie. v ∈ V (K). But then Lemma 4.6 implies that X can have at most k|X| − 1 neighbors among the components of G[D1 ], a contradiction. So A1 ∩ D2 = ∅ and by symmetry A2 ∩ D1 = ∅. Let the electronic journal of combinatorics 12 (2005), #R8 16 • gC1 be the number of components of G[D1 ] contained in C2 , • gC2 be the number of components of G[D2 ] contained in C1 and • gD = c(G[D1 ∩ D2 ]). Clearly k · |A1 | + def(G) = k-gal(G − A1 ) = gC1 + gD , k · |A2 | + def(G) = k-gal(G − A2 ) = gC2 + gD and k · |A1 ∩ A2 | + def(G) = k-gal(G − (A1 ∩ A2 )) ≥ gD . These inequalities sum up to gD + gC1 + gC2 ≥ k · |A1 ∪ A2 | + def(G). It is easy to see that k-gal(G − (A1 ∪ A2 )) ≥ gD + gC1 + gC2 and so A1 ∪ A2 is a barrier. Theorem 5.6 fails for arbitrary barriers. For example, let k = 2 and P3 be the path of length 3 with vertices v1 , v2 , v3 , v4 in this order. P3 has a perfect 2-piece packing so CP3 = V (P3 ). The barriers of P3 are AP3 = ∅, {v2 } and {v3 } but {v2 , v3 } is not a barrier. In the matching theory, the deficiency is usually defined as q(G − A) − |A| where q(G − A) is the number of odd size components of G − A. For this ’odd-deficiency’ it holds that AG ∪ CG is the union of inclusion-wise maximal barriers. This property fails for our deficiency, see P3 defined in the previous paragraph. For the odd-deficiency it also holds that AG is the intersection of the inclusion-wise maximal barriers. This property fails in our case as well. For example, let P2 be the path of length 2 with vertices v1 , v2 , v3 in this order. P2 has a perfect 2-piece packing and its barriers are AP2 = ∅ and {v2 }. Nevertheless, Theorem 5.5 fails for the classical odd deficiency. 6 Two more properties of galaxies First we show a characterization of k-galaxies which is a direct generalization of the defining property 2.3 of the hypomatchable graphs. Theorem 6.1. A graph G satisfies properties 1. and 2. if and only if G is a k-galaxy. 1. G has no perfect k-piece packing. 2. For each v ∈ V (G) there exists a vertex set v ∈ X ⊆ V (G) such that G[X] is connected, ∆(G[X]) ≤ k − 1 and G − X has a perfect k-piece packing. Proof. If G is a k-galaxy then 1. follows from Lemma 2.5 and 2. from Lemma 3.8. For the reverse direction, suppose that G satisfies the above two properties. First, if AG = ∅ then either CG = V (G) which contradicts to 1. by Theorem 2.8 property 3., or DG = V (G). In this latter case each component of G is a k-galaxy. However, G cannot have more than one component since then 2. would yield a perfect k-piece packing of G contradicting to 1. Second, assume that AG 6= ∅. Choose a vertex v ∈ AG and let X be the electronic journal of combinatorics 12 (2005), #R8 17 the vertex set guaranteed by 2. Now degG[X] (v) ≤ k − 1 since ∆(G[X]) ≤ k − 1. Adjoin k − degG[X] (v) new isolated vertices to G and join each new vertex to v by an edge. The new graph is denoted by G′ . Now X and the set of new vertices induce a k-piece in G′ . This k-piece together with the perfect k-piece packing of G − X gives a perfect k-piece packing of G′ . However, k-gal(G′ − AG ) ≥ k|AG | + 1 by Theorem 2.8, property 2., which is a contradiction by Theorem 3.13. In the case k = 1 Theorem 6.1 2. is equivalent to the defining property 2.3 of hypomatchable graphs. This implies property 1. as well by parity arguments when k = 1. However, parity has no consequence in the case k ≥ 2. Another easy characterization of galaxies is the following corollary of Theorem 4.19. Proposition 6.2. The following statements are equivalent for a connected graph G. 1. G is a k-galaxy. 2. |UG−v | < |UG | for all v ∈ V (G). 3. UG−v ( UG for all v ∈ V (G). Proof. 1. ⇒2. and 1. ⇒3.: ∅ is a perfect set with k-surplus so AG = ∅ by Corollary 4.11. So DG = V (G) and both 2. and 3. are implied by Theorem 4.19. 2. ⇒1. and 3. ⇒1.: Theorem 2.8 yields that DG = V (G) hence G is a k-galaxy by Theorem 2.8 property 1. and by the connectivity of G. 7 The matroidal property and maximum packings Definition 7.1. We say that the F -packing problem is matroidal if for all graphs G those vertex sets X ⊆ V (G) which can be covered by an F -packing of G form a matroid. Loebl and Poljak conjecture [9] that for graph sets F with K2 ∈ F the F -packing problem is polynomial if and only if it is matroidal. This conjecture is still open. In [5] it was shown that the k-piece packing problem is not matroidal in the case k ≥ 2. For an example, let k = 2 and G be a claw (ie. a 3-star) with one of its edges subdivided by a new vertex. Still, the k-piece packing problem has the matroidal property in a somewhat weaker form. So Theorem 2.9 gives another support for the validity of the conjecture of Loebl and Poljak. Theorem. 2.9. There exists a partition π on V (G) and a matroid M on π such that the of the maximal k-piece packings are exactly the vertex sets of the form S vertex sets ′ {X : X ∈ π } where π ′ is a base of M. Proof. Lemmas 4.12, 4.15 and the k-surplus of AG imply that the following considerations hold. π = {{v} : v ∈ / WG } ∪ {V (T ) : T is a tip of a k-galaxy component of G[DG ]} . the electronic journal of combinatorics 12 (2005), #R8 18 Denote by N the matroid with ground set H = {H : H is a component of G[DG ]} such that a set H′ ⊆ H of size k|AG | is a base in N if and only if AG can be k-matched into S {V (H) : H ∈ H′ }. Observe that N is indeed a matroid, it is the transversal matroid of the bipartite graph BA , see Definition 4.14. Now for each component H of G[DG ] replace H in N by TH = {V (T ) : T is a tip of H} ⊆ π such that the elements of TH are in series with each other. The resulting matroid is N ′ with ground set {V (T ) : T is a tip of a k-galaxy component of G[DG ]} ⊆ π. Add as a direct sum to N ′ the elements {v} as a bridge for v ∈ / WG . The resulting matroid is M. The co-rank of N and N ′ are def(G) thus the co-rank of M is def(G) too. Note that for each maximal k-piece packing P of G, every vertex set of π is either fully covered or fully missed by P and the number of the fully missed sets is def(G). In the case k = 1 a tip has exactly one element so π is the partition into singletons. In the case k = 2 a tip has one or two elements so the vertex sets of π are of size one or two. Finally, for k ≥ 3 a tip may be of arbitrary size thus a vertex set of π can be of arbitrary size as well. Because the ground set of the matroid M is a partition into different size sets, in the k-piece packing problem a maximal packing is not necessarily maximum, as it is the case in the polynomial packing problems with K2 ∈ F . Still, the vertex sets which can be covered by maximum k-piece packings admit a similar matroid: take the maximum weight bases of M with the weight function X 7→ |X| for X ∈ π. This weighted matroidal approach yields a proof for the Berge-type formula of [5] on the size of a maximum k-piece packing. Indeed, the maximum weight bases of M correspond to the minimum weight bases of N (defined in the proof of Theorem 2.9) with the weight function H 7→ (the minimum size of a tip of H). So one can apply the greedy method to the k-galaxy components of G[DG ]. In fact, a little additional work is needed for proving Theorem 7.2 since it is stated in a more compact form in [5]. Let k-gali (G) denote the number of k-galaxy components H of the graph G with the property that each tip of H has size at least i. Theorem 7.2. [5] If G is a graph of size n then the size of the maximum k-piece packings of G is n X n − max (k-gali (G − Ai ) − k|Ai |) , i=1 taken over all sequences of vertex sets V (G) ⊇ A1 ⊇ A2 ⊇ . . . ⊇ An . A1 can be chosen to be the canonical barrier AG . The sequence of vertex sets is related to the structure of the minimum weight bases of the transversal matroid N . We do not go into details. In the case k = 1 we get the Berge-Tutte theorem on maximum matchings [1]. The case k = 2 was proved by Kano, Katona and Király [8]. 8 The (l, u)-piece packing problem As a generalization of the k-piece packing problem, the (l, u)-piece packing problem is introduced in [5]. It turns out that all the above results hold with the straightforward the electronic journal of combinatorics 12 (2005), #R8 19 modifications. We do not go into details, only illustrate this relation using the reduction to the k-piece packing problem shown in [5]. Let two integer bounds u(v) ≥ l(v) ≥ 0 be given for each vertex v ∈ V (G). A connected subgraph P of G is an (l, u)-piece if degP (v) ≤ u(v) holds for each v ∈ V (P ) and there exists at least one vertex w ∈ V (P ) with degP (w) ≥ l(w). Note that l ≡ u ≡ k gives the k-piece packing problem. Galaxies and tips change in the following way. Definition 8.1. Given the bounds l, u : V (H) → N, the graph H is an (l, u)-galaxy if it satisfies the following properties: • denoting by IH the graph induced by the vertices v with degG (v) ≥ l(v), each component of IH is a hypomatchable graph, • l(v) = u(v) ≥ 1 for v ∈ V (IH ), • for each v ∈ V (IH ) there exist exactly l(v) − 1 edges between v and V (H) − V (IH ), each being a cut edge in H. The tips are the connected components of H − V (IH ) together with the vertices v ∈ V (IH ) with l(v) = u(v) = 1 as single vertex subgraphs. The difference in the definition of the galaxies and tips can be explained by the following reduction to the k-piece packing problem, described in [5]. Let k = 1 + max{u(v) : v ∈ V (G)}. For each vertex v ∈ V (G) let Mv and Nv be disjoint sets of new vertices with |Mv | = u(v) − l(v) + 1 and |Nv | = k − u(v) − 1. Now for each v ∈ V (G) take a complete graph on Mv and join the vertices of Mv ∪ Nv to v. Denote the new graph by Gk . It is easy to see that Gk has a perfect k-piece packing if and only if G has a perfect (l, u)-piece packing, and that G is an (l, u)-galaxy if and only if Gk is a k-galaxy. With the help of this reduction one can see that all the above considerations for the k-piece packings hold for the (l, u)-piece packings as well, with the necessary modifications. For illustrating this, we briefly describe how to get the canonical decomposition of G related to the (l, u)-piece packing problem. Let V (Gk ) = Dk ∪˙ Ak ∪˙ Ck be the canonical decomposition of Gk related to the k-piece packing problem. Due to the k-surplus of Ak , each vertex of Ak has degree at least k + 1 in Gk . Because the new vertices of Gk (ie. the vertices in V (Gk ) − V (G)) have degree at most u(v) − l(v) + 1 ≤ k, we get that Ak ⊆ V (G). So the deletion of the new vertices yields a partition V (G) = D ∪˙ A ∪˙ C where D = Dk ∩ V (G), A = Ak and C = Ck ∩ V (G). This canonical partition has all the properties listed in Theorem 2.8, for example the connected components of G[D] are (l, u)-galaxies, for all ∅ = 6 A′ ⊆ A the number of those ′ (l, u)-galaxy components of G[D] which are adjacent to A is at least u(A′ ) + 1, and C has a perfect (l, u)-piece packing. This partition is unique, because if V (G) = D ′ ∪˙ A′ ∪˙ C ′ is another partition with these properties then in Gk the set A′ is a perfect barrier with k-surplus, hence by Corollary 4.11 it equals to Ak . The analogue of Theorem 4.19 also holds. This Edmonds-Gallai type theorem for the (l, u)-piece packing problem becomes quite compact in the case l(v) = l < u = u(v) for all v ∈ V (G), so we include this. Here an the electronic journal of combinatorics 12 (2005), #R8 20 (l, u)-piece packing is a packing with connected graphs F with l ≤ ∆(F ) ≤ u. Call such a packing an (l < u)-packing. The simplicity of this structure theorem comparing to the general case is due to the fact that here an (l, u)-galaxy is just a graph with highest degree at most l − 1. So it always consists of only one tip. Theorem 8.2. For a graph G let D = {v ∈ V (G) : v can be missed by a maximal (l < u)packing of G}. Let A = Γ(D) and C = V (G) − (D ∪ A). Now 1. ∆(G[D]) ≤ l − 1, 2. for all ∅ 6= A′ ⊆ A the number of those components of G[D] which are adjacent to A′ is at least u|A′| + 1, 3. G[C] has a perfect (l < u)-packing, and 4. for each maximal (l < u)-packing P of G, the graph G−P has exactly c(G[D])−u|A| connected components. References [1] C. Berge, Sur le couplage maximum d’un graphe. C.R. Acad. Sci. Paris Ser. I Math. 247, (1958) 258–259. [2] G. Cornuéjols, D. Hartvigsen, An extension of matching theory. J. Combin. Theory Ser. B 40, (1986) 285–296. [3] G. Cornuéjols, D. Hartvigsen, W. R. Pulleyblank, Packing subgraphs in a graph. OR Letters 1, (1982) 139–143. [4] G. Cornuéjols, W. R. 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North-Holland Publishing, Amsterdam the electronic journal of combinatorics 12 (2005), #R8 21