# Equitable Partitions into Matchings and Coverings in Mixed Graphs

Matchings and coverings are central topics in graph theory. The close relationship between these two has been key to many fundamental algorithmic and polyhedral results. For mixed graphs, the notion of matching forest was proposed as a common generalization of matchings and branchings. In this paper, we propose the notion of mixed edge cover as a covering counterpart of matching forest, and extend the matching-covering framework to mixed graphs. While algorithmic and polyhedral results extend fairly easily, partition problems are considerably more difficult in the mixed case. We address the problems of partitioning a mixed graph into matching forests or mixed edge covers, so that all parts are equal with respect to some criterion, such as edge/arc numbers or total sizes. Moreover, we provide the best possible multicriteria equalization.

## Authors

• 6 publications
• 9 publications
• ### Notes on Equitable Partition into Matching Forests in Mixed Graphs and into b-branchings in Digraphs

An equitable partition into branchings in a digraph is a partition of th...
03/24/2020 ∙ by Kenjiro Takazawa, et al. ∙ 0

• ### Colourings of (m, n)-coloured mixed graphs

A mixed graph is, informally, an object obtained from a simple undirecte...
10/18/2020 ∙ by Gary MacGillivray, et al. ∙ 0

• ### Fractional hypergraph isomorphism and fractional invariants

Fractional graph isomorphism is the linear relaxation of an integer prog...
08/19/2020 ∙ by Flavia Bonomo-Braberman, et al. ∙ 0

• ### The Simple Chromatic Number of (m,n)-Mixed Graphs

An (m,n)-mixed graph generalizes the notions of oriented graphs and edge...
09/12/2018 ∙ by Christopher Duffy, et al. ∙ 0

• ### On relative clique number of colored mixed graphs

An (m, n)-colored mixed graph is a graph having arcs of m different colo...
10/12/2018 ∙ by Sandip Das, et al. ∙ 0

• ### Computational Complexity of Covering Two-vertex Multigraphs with Semi-edges

We initiate the study of computational complexity of graph coverings, ak...
03/28/2021 ∙ by Jan Bok, et al. ∙ 0

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## 1 Introduction

Let be a mixed graph with undirected edges and directed arcs . In this paper, graphs have no loops (edge/arc), but may have parallel edges and arcs. Each arc has one head and we regard both endpoints of an edge as heads. We say that is covered by an edge/arc if is a head of . A matching forest, introduced by Giles [5, 6, 7], is a subset such that (i) the underlying undirected graph has no cycle and (ii) every vertex is covered at most once in . This is a common generalization of the notion of matching in undirected graphs and the notion of branching in directed graphs. Matching forests have been studied in order to unify fundamental theorems about matchings and branchings. In particular, unifying results were given on total dual integrality by Schrijver [15], on Vizing-type theorems by Keijsper [10], and on the delta-matroid property of degree-sequences by Takazawa[17].

In undirected graphs, as shown by Gallai’s theorem [4] and other results, matching is closely related to edge cover, a set of edges covering all vertices. In this paper, we offer a covering counterpart of the notion of matching forest, that can be regarded as a common generalization of edge covers and bibranchings. We present two natural ways to define covering structures in mixed graphs; later we will show that these two are in some sense equivalent. First, suppose that, in a mixed graph , each edge can extend its endpoints via directed paths. Then we can conceive the following notion as a mixed graph version of edge cover.

• A mixed edge cover is a subset such that every vertex is reachable from some edge. That is, for any , there is a directed path (which can be of length ) in from an endpoint of some to .

This is exactly an edge cover for an undirected graph. Also, bibranchings in a partitionable directed graph can be represented as mixed edge covers in an associated mixed graph (see Section 5.3). Thus, mixed edge cover generalizes both edge cover and bibranching. Alternatively, the following notion may also be considered as a covering counterpart of matching forest.

• A mixed covering forest is a subset such that (i) the underlying graph has no cycle and (ii) every vertex is covered at least once in .

These two notions coincide under the minimality condition. That is, a minimal mixed edge cover is also a minimal mixed covering forest and vice versa (see Proposition 3). In case of nonnegative weight minimization or packing problems, where the optimal solutions can be assumed to be minimal, the terms are interchangeable. This is however not true for partitioning problems. In this paper we mainly work with mixed edge covers, and obtain results on mixed covering forests as consequences.

Our results can be divided into the following two parts. While the first one shows that results on matching and edge-cover naturally extend to mixed graphs, the second one deals with new problems which arise from the heterogeneous feature of mixed graphs.

### Structure of Mixed Edge Covers

In undirected graphs, matching and edge cover are closely related, and for both of them, polyhedral and algorithmic results are known. For mixed graphs, however, only matching forests have been investigated. In Sections 2 and 3, we show connections between matching forest and mixed edge cover, and use these connections to derive polyhedral and algorithmic results on mixed edge covers.

We first generalize Gallai’s theorem [4], whose original statement is as follows: For any undirected graph without isolated vertices, the sum of the cardinalities of a maximum matching and a minimum edge cover is . For a mixed graph, define the mix-size of a subset by . With this mix-size, the statement of Gallai’s theorem holds for matching forests and mixed edge covers.

We also show that the optimization problem on mixed edge covers can be reduced to optimization on perfect matching forests in an auxiliary graph. This fact immediately implies a polynomial time algorithm to find a minimum weight mixed edge cover. Furthermore, using this relation we can provide a polyhedral description of the mixed edge cover polytope and show its total dual integrality, obtaining a covering counterpart of the result of Schrijver [15].

These algorithmic and structural results show that mixed edge covers exhibit similar properties in mixed graphs as edge covers do in undirected graphs.

### Equitable Partitions in Mixed Graphs

Recall that matching forest is a common generalization of matching in undirected graph and branching in directed graph. These structures are known to have the following equitable partition property [16]: if the edge set of an undirected graph (resp., the arc set of a directed graph) can be partitioned into matchings (resp., branchings) , then we can re-partition (resp., ) into matchings (resp., branchings) such that for any (where ). Note that bounding the difference of cardinality by is the best possible equalization.

Equitable partition problems have been studied extensively for various combinatorial structures, the most famous being the equitable coloring theorem of Hajnal and Szemerédi [8] and the stronger conjecture of Meyer [12], which is still open. The equitable partition property of matchings implies that the equitable chromatic number of any line graph equals its chromatic number. Edge/arc partitioning problems with equality or other cardinality constraints have also been studied for other graph structures [1, 2, 18, 19]. In this paper, we consider equitable partitioning into matching forests and into mixed edge covers.

#### Partitioning into Matching Forests.

Since mixed graphs have two different types of edges, there are several possible criteria for equalization: the number of edges, the number of arcs, and the total cardinality. (We call them edge-size, arc-size, and total size, respectively.) We study equalization with respect to each of these criteria, as well as the possibility of “multicriteria equalization.”

It turns out that the coexistence of edges and arcs makes equalization more difficult. See the graph in Fig. 1, which consists of two edges and two arcs. Here, a completely equalized partition would be a pair of matching forests, each with one edge and one arc, but there is no such partition.

In this example, the two arcs are in the same part in any partition. Thus, unlike in the case of branchings, the difference of in arc-size is unavoidable in some instances. The example also shows the impossibility of equalizing edge-size and total size simultaneously.

We show that equalization is possible separately for edge-size and total-size. Also, simultaneous equalization is possible by relaxing one criterion just by . These results are summarized in the following two theorems. We sometimes identify a mixed graph with (e.g., we say “ is partitionable” to mean “ is partitionable.”) For a set of matching forests , we write for their edge parts and for their arc parts.

###### Theorem 1.1

Let be a mixed graph that can be partitioned into matching forests. Then can be partitioned into matching forests in such a way that, for every , we have , , and .

###### Theorem 1.2

Let be a mixed graph that can be partitioned into matching forests. Then can be partitioned into matching forests in such a way that, for every , we have , , and .

We remark again that, even if we consider a single criterion, the minimum differences in , , can be respectively. These theorems say that relaxing one criterion just by is sufficient for simultaneous equalization.

#### Partitioning into Mixed Edge Covers.

Next, we consider equitable partitioning into mixed edge covers. In contrast to the first part, where polyhedral and algorithmic results on mixed edge covers were obtained via reduction to matching forests, there seems to be no easy way to adapt these reductions to equalization problems. The reason is that the correspondence between matching forest and mixed edge cover presumes maximality/minimality, but these cannot be assumed in equitable partitioning problems.

That said, equalization faces similar difficulties as in the case of matching forests. See the graph in Fig. 2, which has two components. Each component has a unique partition into two mixed edge covers, so the whole graph has only two possible partitions (one is shown in Fig. 2, while the other is obtained by flipping the colors in one component.)

This example shows that the difference of in arc-size is unavoidable, and simultaneous equalization of edge-size and total size is impossible. Fortunately, this is the worst case. Similarly to matching forests, we can obtain the following theorems for mixed edge covers. For mixed edge covers , we use the notation for their edge parts and for their arc parts.

###### Theorem 1.3

Let be a mixed graph that can be partitioned into mixed edge covers. Then can be partitioned into mixed edge covers in such a way that, for every , we have , , and .

###### Theorem 1.4

Let be a mixed graph that can be partitioned into mixed edge covers. Then can be partitioned into mixed edge covers in such a way that, for every , we have , , and .

We now mention equitable partitioning into mixed covering forests, the other type of structure we introduced as a covering counterpart of matching forests. Unlike mixed edge covers, mixed covering forests require acyclicity, which makes partitioning even harder. The graph in Fig. 3 has a unique partition into two mixed covering forests, where edge-size is not equalized. However, if we consider packing rather than partitioning, then we can show that the corresponding versions of Theorems 1.3 and 1.4 hold for mixed covering forests. The formal statements are given in Section 5.3 as Corollaries 2 and 3.

We add two more remarks about the results. First, our multicriteria equalization result is new even for bibranchings. We describe the consequences for bibranchings in Section 5.3.

Second, our results are costructive in the sense that if an initial partition is given, then our proof gives rise to a polynomial-time algorithm to obtain the desired partition in Theorems 1.1, 1.2, 1.3, and 1.4. Note however that it is NP-complete to decide if a mixed graph can be partitioned into matching forests or mixed edge covers, even in the undirected case.

The rest of the paper is organized as follows. Section 2 describes basic properties of matching forests and mixed edge covers, incuding a new extension of Gallai’s theorem. In Section 3, we show that a minimum weight mixed edge cover can be found in polynomial time, and we give a TDI description of the mixed edge cover polytope. Sections 4 and 5 contain our results on equitable partitioning of matching forests and mixed edge covers, respectively. In the last subsection, we describe the corollaries for mixed covering forests and bibranchings.

## 2 Matching Forests and Mixed Edge Covers

We review some basic properties of matching forests and mixed edge covers. Let be a mixed graph. For a subset , we say that is covered in if is an endpoint of some edge or is the head of some arc . We denote by the set of vertices covered in .

An edge set is a matching (resp., edge cover) if each vertex is covered at most once (resp., at least once) in . An arc set is a branching if each vertex is covered at most once in and there is no directed cycle in . For a branching , we call the root set of . Note that, in a branching , any vertex is reachable from some root via a unique directed path (which can be of length ).

We provide characterizations of matching forests and mixed edge covers, where the first one is clear from the definition.

###### Proposition 1

A subset is a matching forest if and only if is a branching and is a matching such that .

###### Proposition 2

A subset is a mixed edge cover if and only if contains a branching such that .

###### Proof

The “if” part is clear because every is covered by an edge and every is reachable from in . For the “only if” part, suppose that is a mixed edge cover. By definition, for any , there is a directed path from to . This means that, if we contract to a new vertex , then there exists an -arborescence. In the original graph, this arborescence corresponds to a branching such that , and hence .∎

As mentioned in the Introduction, mixed edge covers and mixed covering forests have the following relationship.

###### Proposition 3

Every mixed covering forest is a mixed edge cover. Moreover, a subset is a minimal mixed edge cover if and only if it is a minimal mixed covering forest.

###### Proof

For the first claim, suppose for contradiction that a mixed covering forest is not a mixed edge cover. Then, some vertex is unreachable from . Let be the set of vertices from which is reachable; then no is incident to edges. As is a covering forest, every is covered by some arc , whose tail is also in by the definition of . Therefore, there are at least arcs whose head and tail are both in , which contradicts the acyclicity of .

For the “if” part of the second claim, take a minimal mixed covering forest . This is a mixed edge cover as just shown. The minimality of implies that any proper subset of has some uncovered vertex, and hence is not a mixed edge cover. So is a minimal edge cover.

For the “only if” part, let be a minimal mixed edge cover. By the first claim, it suffices to show that this is a mixed covering forest. Clearly, all vertices are covered at least once because they are reachable from , so we have to show acyclicity. Observe that the minimality of implies that the head of any arc is covered only by (otherwise we can remove or another arc whose head is ). Suppose, to the contrary, that is a cycle in the underlying graph. If all elements of are edges, then we can remove at least one edge, which contradicts minimality. Therefore, contains some arc . By the above observation, the head of is covered only by , so the other element in incident to should be an arc whose tail is . By repeating this argument, we see that all elements of are arcs. Then all vertices in are only covered by arcs in , which means that they are unreachable from , a contradiction. ∎

Recall that we define for any . Using this, we can generalize Gallai’s well known theorem on the relation between maximum matching and minimum edge cover to mixed graphs.

###### Theorem 2.1

For a mixed graph that admits a mixed edge cover, define and . Then we have .

###### Proof

For any vertex , we denote by the minimum length of a directed path from to . If admits a mixed edge cover, then is finite for every . For any , we have if and only if .

###### Claim

Among matching forests satisfying , let minimize

 D(F):=∑\setdistG(v)v∈V∖∂(F).

Then , and hence .

Suppose, to the contrary, , i.e.  for some . Take a shortest directed path from to and let be the arc whose head is . Since is uncovered in , every vertex is covered at most once in , which is not a matching forest by the maximality of . This means that there exists a directed cycle with . Let be the arc preceeding in and let be the head of (which is also the tail of ). Then is a matching forest and satisfies . Because , we have . As is on the shortest path to , we see , and hence , which contradicts the choice of . The claim is proved.

By this claim, every is incident to some edge.

###### Claim

.

Let be a superset of obtained by adding an arbitrary incident edge for each . Then is a mixed edge cover. To see this, we show that any is reachable from in . By Proposition 1, forms a branching. Let be the root of the component containing (which can be itself). Then is reachable from in . Because is not covered by any arc, or . Both of them imply by the definition of , and hence is reachable from . Thus, is a mixed edge cover, and we have .

Because has heads, we have , and by the construction of . Hence, we obtain .

###### Claim

.

Take a mixed edge cover with and let be an inclusion-wise maximal matching forest in . By the minimality of , the head of any arc is covered only by in . Also, Proposition 3 implies that the underlying graph of has no cycle. Thus includes , and hence and . By the maximality of , any edge has at most one endpoint in , while . Then, , which implies . Thus, . ∎

## 3 Algorithms and Polyhedral Descriptions

### 3.1 Previous Results on Matching Forests

We introduce some known results on matching forests that will be used in our proofs for mixed edge covers in Section 3.2. Giles [6] showed that the maximum weight matching forest problem is tractable.

###### Theorem 3.1 (Giles [6])

There is a strongly polynomial-time algorithm to find a maximum weight matching forest or a maximum weight perfect matching forest, for any weight function .

Giles also gave a linear description of the matching forest polytope and characterized its facets [6, 7]. It was later shown by Schrijver that this system is totally dual integral (TDI). To state the result, we call a collection of subpartitions laminar if for any and , one of the following is true:

• for every , there exists such that ,

• for every , there exists such that ,

• for every and .

###### Theorem 3.2 (Schrijver [15])

For a mixed graph and a vertex , let denote the union of the set of edges in incident to and the set of arcs in with head . The following is a TDI description of the convex hull of matching forests in a mixed graph .

 xe ≥0  for every e∈E∪A (1) x(δhead(v)) ≤1  for every v∈V (2) x(E[∪S])+∑Z∈Sx(A[Z]) ≤|∪S|−⌈|S|/2⌉  for % every subpartition S of V. (3)

For any cost function , there is an integer optimal dual solution such that the support of the dual variables corresponding to (3

) is laminar and consists of odd subpartitions.

In general, it is known that a TDI description remains TDI when some inequalities are replaced by equalities [16]. By this fact, Theorem 3.2 implies the following TDI description of perfect matching forests, where (6) is obtained by subtracting (3) from the summation of (5) on .

###### Corollary 1

For a mixed graph , the following is a TDI description of the convex hull of perfect matching forests.

 xe ≥0  for every e∈E∪A (4) x(δhead(v)) =1  for every v∈V (5) x(E[∪S])+x(δE(∪S))+∑Z∈Sx(δinA(Z)) ≥⌈|S|/2⌉  for every subpartition S of V. (6)

For any cost function , there is an integer optimal dual solution such that the support of the dual variables corresponding to (6) is laminar and consists of odd subpartitions.

### 3.2 Algorithmic and Polyhedral Properties of Mixed Edge Covers

We first show that there is a close relationship between mixed edge covers and perfect matching forests in a modified graph. This allows us to find a minimum weight mixed edge cover in strongly polynomial time, and to give a TDI description of the convex hull of mixed edge covers.

Given a mixed graph with weights , we construct an auxiliary mixed graph with costs on . Let be a copy of , and let be the perfect matching between corresponding vertices of and , with costs (the cost is infinite if there is no such edge). For , let . Finally, let consist of arcs for every and , with cost .

###### Lemma 1

If has a mixed edge cover, then the minimum weight of a mixed edge cover in equals the minimum cost of a perfect matching forest in .

###### Proof

Let be a minimum weight mixed edge cover in . We may assume that is a disjoint union of stars and is a branching whose roots are exactly the endpoints of . Let be a star component of with center of degree at least 2. Remove all but one edges of from , and for every removed edge , add to . Do this for every star component of with at least 2 edges, and then add arbitrary incoming arcs to the remaining isolated vertices in . The resulting is a perfect matching forest and .

Conversely, let be a minimum weight perfect matching forest in . For every edge , replace by a minimum weight edge in incident to . Remove all arcs in . The resulting edge set is a mixed edge cover in such that .∎

Combining Lemma 1 with Theorem 3.1 yields the following.

###### Theorem 3.3

There is a strongly polynomial-time algorithm to find a minimum weight mixed edge cover.

Using the same auxiliary graph and Corollary 1, we can obtain the following TDI description of mixed edge covers. The proof is provided in Section 3.3.

###### Theorem 3.4

The following is a TDI description of mixed edge covers:

 1≥xe ≥0  for every e∈E∪A x(E[∪S])+x(δE(∪S))+∑Z∈Sx(δinA(Z)) ≥⌈|S|/2⌉  for every subpartition S of V.

### 3.3 Proof of TDIness of the Mixed Edge Cover System

Let be a mixed graph with edge weights . We assume that has a mixed edge cover. We construct the auxiliary mixed graph and cost function as in Section 3.2

. Consider the dual of the linear program (

4)–(6) for the auxiliary graph and the cost function :

 πv ≥0  for every v∈V (7) yS ≥0  for every subpartition S of V (8) −πu−πv+∑{yS:{u,v}∩∪S≠∅} ≤c(uv)  for every uv∈E (9) −πv+∑{yS:uv∈δin(Z) for some% Z∈S} ≤c(uv)  for every uv∈A (10) max∑{⌈|S|/2⌉yS:S is a subpartition of V}−∑v∈Vπv.

By Corollary 1, there is an integral optimal dual solution such that the support of is laminar and consists of odd subpartitions.

###### Lemma 2

The dual linear program for has an integral optimal solution such that the support of is laminar, it consists of subpartitions disjoint from , and .

###### Proof

Consider an integral optimal dual solution where the support of is laminar and the value is minimal. Let us call a subpartition positive if . Since the support of is laminar, each is either uncovered by positive subpartitions, or there is a minimal positive subpartition such that . In the latter case, is called the minimal positive subpartition covering . An edge is called tight if (9) for is satisfied with equality.

for every .

###### Proof

Suppose for contradiction that , and consider the following cases.

• If neither nor is covered by a positive subpartition, then we can decrease by 1.

• Suppose that is not covered by a positive subpartition, and the minimal positive subpartition covering is . Let be the class of containing , and let be the subpartition obtained from by removing the class . We decrease and by 1, and increase by 1. This is still a feasible dual solution, because still holds for , and (10) holds for any arc since is not covered by a positive subpartition. The objective value does not decrease but decreases.

• Let be the minimal positive subpartition covering is , and let be the class of containing . Suppose that or . Let be the subpartition obtained from by removing the class . We can decrease and by 1, and increase by 1.

• Let by the minimal positive subpartition covering is , let be the class of containing , and let be the class containing . Let be the subpartition obtained from by removing the classes and . We get a feasible dual solution by decreasing and by 1, and increasing by 1. The objective value remains the same.

In all cases, we obtained an optimal dual solution where is smaller, contradicting the choice of .

for every .

###### Proof

First, we consider the case when no positive subpartition covers . Since for every by the previous Claim, (10) for the arcs implies that positive subpartitions are disjoint from . If there is no tight edge , then we can just decrease by 1. Suppose that there is a tight edge , i.e. . Since , for implies that . Thus implies , contradicting the previous Claim.

Let now be the minimal positive subpartition covering , and let be the class of containing . If , then , otherwise (10) would be violated for the arc . Let be the subpartition obtained from by removing the class . If there is no tight edge with , then we can decrease and by 1, and increase by 1.

Suppose that there is a tight edge with . Every positive subpartition covering also covers , so tightness implies . Since , for implies . Thus implies , contradicting the previous Claim.

The two Claims together show that , as required. To show that positive subpartitions can be assumed to be disjoint from , observe that if is covered by a positive subpartition, then the class containing must be a superset of , otherwise (10) is violated for some arc . We can replace this class by and still get a feasible dual solution.∎

###### Proof (of Theorem 3.4)

Let denote the minimum weight of a mixed edge cover in for weight function . First, we prove dual integrality for nonnegative integer weights. Given a mixed edge cover problem instance with edge weights , we construct the auxiliary mixed graph and cost function as above. By Lemma 1, equals the minimum cost of a perfect matching forest in . By Lemma 2, the latter problem has an integer optimal dual solution where and every positive subpartition is disjoint from . Since is a feasible dual solution to the mixed edge cover system for and its objective value equals , it is an optimal dual solution.

Consider now the case when has some negative values. Write as , where is the positive part of and is the negative part. Clearly, . Let be the optimal integer dual solution for , obtained as above. For , let denote the dual variable corresponding to the condition . If we set , then is a feasible integer dual solution for and its objective value equals , so it is an optimal dual solution.∎

## 4 Equitable Partitions into Matching Forests

In this section, we consider equalization of matching forests. We provide specific construction methods for the partitions required in Theorems 1.1 and 1.2.

Our construction is based on repeated application of operations that equalize a pair of matching forests. Recall that a matching forest consists of a branching and a matching such that (see Proposition 1). For equalization of edge-size, we want to perform exchanges along alternating paths on edges, but at the same time we have to modify the arc parts so that the resulting root sets and edge sets satisfy again. To cope with this issue, we invoke the following result of Schrijver on root exchange of branchings.

###### Lemma 3 (Schrijver [15])

Let and be branchings and let and denote their root sets. Let and be vertex sets satisfying and . Then, can be re-partitioned into branchings and with and if and only if each strong component without entering arc (i.e., each source component of ) intersects both and .

This will also be used for the equalization of mixed edge covers in Section 5.

### 4.1 Operations for a Pair of Matching Forests

The following two lemmas are the key to the proof of Theorems 1.1 and 1.2. As in those theorems, for a matching forest , we use the notations and .

###### Lemma 4

Let be a mixed graph that is the disjoint union of two matching forests , . Then can be partitioned into two matching forests , such that and .

###### Lemma 5

Let be a mixed graph that is the disjoint union of two matching forests , . Then can be partitioned into two matching forests , such that and .

In the following, we give a combined proof of the two lemmas.

###### Proof

To construct the required matching forests, we introduce four equalizing operations.

###### Claim

It is possible to implement the following four operations on disjoint matching forests , that repartition into matching forests with the properties below.

1. If and , it returns such that and .

2. If and , it returns such that and .

3. If and , it returns such that and .

4. If and , it returns such that and .

We postpone the proof of this claim and complete the proof of the lemmas relying on it. Note that we also have Operations 1’,2’,3’,4’ by switching the role of and . To prove Lemma 4, we repeat updating in the following manner:

• If , apply Operation 1, 1’, 2, or 2’ depending on the signs of and , and update with , .

• If and , apply Operation 3, 3’, 4, or 4’ depending on the signs of and , and update .

Note that decreases when Operation 1, 1’, 2, or 2’ is applied. Also, when Operation 3, 3’, 4, or 4’ is applied, decreases while is preserved. Therefore, we finally obtain and . Then if or , while otherwise we can apply Operation 1, 1’, 2, or 2’ to update and so that and . Thus, we have , and Lemma 4 is proved. Lemma 5 can be shown similarly by swapping the roles of and and of Operations 1–2 and 3–4.∎

Here we prove the postponed claim.

#### Proof of the Claim.

Let for each . Note that

 |Fi|=|Mi|+|Bi|=|Mi|+|V|−|Ri|=|V|−|Mi|−|Ri∖∂(Mi)|.

We construct an auxiliary undirected graph . For every node , we add a new node . The edge set consists of two disjoint matchings and , where

 M∗i =Mi∪M∙i, M∙i =\setv∙vv∈Ri∖∂(Mi).

By definition, is a matching, and . If a node is covered by both and , then , so is a singleton source component in .

For each source component of in the original graph, if there are such that and , take such a pair and contract and in . Let be the resulting node set. After this operation, and are still matchings, and hence