Characterization Of A Class Of Graphs Related To Pairs Of Disjoint Matchings The current work is the main part of the author’s Master’s thesis defended in May 2007.

Characterization Of A Class Of Graphs Related To Pairs Of Disjoint Matchings 1

Abstract

For a given graph consider a pair of disjoint matchings the union of which contains as many edges as possible. Furthermore, consider the ratio of the cardinalities of a maximum matching and the largest matching in those pairs. It is known that for any graph is the tight upper bound for this ratio. We characterize the class of graphs for which it is precisely . Our characterization implies that these graphs contain a spanning subgraph, every connected component of which is the minimal graph of this class.

Keywords: matching, pair of disjoint matchings, maximum matching.

•  Dedicated to my mother, father and sister Arevik

1 Introduction

In this paper we consider finite undirected graphs without multiple edges, loops, or isolated vertices. Let and be the sets of vertices and edges of a graph , respectively.

We denote by the cardinality of a maximum matching of .

Let be the set of pairs of disjoint matchings of . Set:

 λ(G)≐max{|H|+|H′|:(H,H′)∈B2(G)}.

Furthermore, let us introduce another parameter:

 α(G)≐max{|H|,|H′|:(H,H′)∈B2(G) and |H|+|H′|=λ(G)},

and define a set:

 M2(G)≐{(H,H′)∈B2(G):|H|+|H′|=λ(G) % and |H|=α(G)}.

While working on the problems of constructing a maximum matching of a graph such that is maximized or minimized, Kamalian and Mkrtchyan designed polynomial algorithms for solving these problems for trees [5]. Unfortunately, the problems turned out to be NP-hard already for connected bipartite graphs with maximum degree three [6], thus there is no hope for the polynomial time calculation of even for bipartite graphs , where

 β1(G)=max{β(G∖F):F is a maximum matching of G}.

Note that for any graph

 λ(G)=β(G)+β1(G) if and only if α(G)=β(G).

Thus, can be efficiently calculated for bipartite graphs with since can be calculated for that graphs by using a standard algorithm of finding a maximum flow in a network. Let us also note that the calculation of is NP-hard even for the class of cubic graphs since the chromatic class of a cubic graph is three if and only if (see [4]).

Being interested in the classification of graphs , for which , Mkrtchyan in [8] proved a sufficient condition, which due to [2, 3], can be formulated as: if is a matching covered tree then . Note that a graph is said to be matching covered (see [9]) if its every edge belongs to a maximum matching (not necessarily a perfect matching as it is usually defined, see e.g. [7]).

In contrast with the theory of 2-matchings, where every graph admits a maximum 2-matching that includes a maximum matching [7], there are graphs (even trees) that do not have a “maximum” pair of disjoint matchings (a pair from ) that includes a maximum matching.

The following is the best result that can be stated about the ratio for any graph (see [10]):

 1≤β(G)α(G)≤54.

The aim of the paper is the characterization of the class of graphs , for which the ratio obtains its upper bound, i.e. the equality holds.

Our characterization theorem is formulated in terms of a special graph called spanner (figure 1), which is the minimal graph for which (what is remarkable is that the equality also holds for spanner). This kind of theorems is common in graph theory: see [2] for characterization of planar or line graphs. Another example may be Tutte’s Conjecture (now a beautiful theorem thanks to Robertson, Sanders, Seymour and Tomas) about the chromatic index of bridgeless cubic graphs, which do not contain Petersen graph as a minor.

On the other hand, let us note that in contrast with the examples given above, our theorem does not provide a forbidden/excluded graph characterization. Quite the contrary, the theorem implies that every graph satisfying the mentioned equality admits a spanning subgraph every connected component of which is a spanner.

2 Main Notations and Definitions

Let be a graph and be the degree of a vertex of .

Definition 2.1

A subset of is called a matching if it does not contain adjacent edges.

Definition 2.2

A matching of with maximum number of edges is called maximum.

Definition 2.3

A vertex of is covered (missed) by a matching of , if contains (does not contain) an edge incident to .

Definition 2.4

A sequence is called a trail in if , , , and if , for , .

The number of edges, , is called the length of a trail . Trail is called even (odd) if its length is even (odd).

Trails and are considered equal. Trail is also considered as a subgraph of , and thus, and are used to denote the sets of vertices and edges of , respectively.

Definition 2.5

A trail is called a cycle if .

Similarly, cycles and are considered equal for any .

If is a trail that is not a cycle then , and , are called the end-vertices and end-edges of , respectively.

Definition 2.6

A trail is called a path if for .

Definition 2.7

A cycle is called simple if is a path.

Below we omit -s and write instead of when denoting a trail.

Definition 2.8

For a trail of and , define sets , , and , , , as follows:

 Ebi(T)≐{ej:1≤j≤min{n,i}},
 Eei(T)≐{ej:max{1,n−i+1}≤j≤n},
 Ei(T)≐Ebi(T)∪Eei(T),

and

 V0(T)≐{v∈V(G):v is an end-vertex of T},
 Vbi(T)≐{v∈V(G):v is incident to an edge from Ebi(T)},
 Vei(T)≐{v∈V(G):v is incident to an edge from Eei(T)},
 Vi(T)≐Vbi(T)∪Vei(T).

The same notations are used for sets of trails. For example, for a set of trails , denotes the set of end-vertices of all trails from , that is:

 V0(D)≐⋃T∈DV0(T).

Let and be sets of edges of .

Definition 2.9

A trail is called - alternating if the edges with odd indices belong to and others to , or vice-versa.

If is an - alternating trail then () denotes the graph induced by the set of edges of that belong to ().

The set of - alternating trails of that are not cycles is denoted by . The subsets of containing only even and odd trails are denoted by and , respectively. We use the notation instead of do denote the corresponding sets of - alternating cycles (e.g. is the set of - alternating even cycles).

The set of the trails from starting with an edge from () is denoted by ().

Now, let and be matchings of (not necessarily disjoint). Note that - alternating trail is either a path, or an even simple cycle.

Definition 2.10

An - alternating path is called maximal if there is no other - alternating trail (a path or an even simple cycle) that contains as a proper subtrail.

We use the notation instead of to denote the corresponding sets of maximal - alternating paths (e.g. is the subset of containing only those maximal - alternating paths whose length is odd and which start (and also end) with an edge from ).

Terms and concepts that we do not define can be found in [2, 7, 11].

3 General Properties and Structural Lemmas

Let be a graph, and and be (not necessarily disjoint) matchings of it. The following are properties of - alternating cycles and maximal paths.

First note that all cycles from are simple as and are matchings.

Property 3.1

If the connected components of are paths or even simple cycles, and
, then .

Property 3.2

If and then .

Property 3.3

Every edge 2 lies either on a cycle from or on a path from .

Property 3.4
1. if then and have the same number of edges that lie on ,

2. if then the number of edges from lying on is one more than the number of ones from .

These observations imply:

Property 3.5

.

Berge’s well-known theorem states that a matching of a graph is maximum if and only if does not contain an -augmenting path [2, 7, 11]. This theorem immediately implies:

Property 3.6

If is a maximum matching and is a matching of a graph then

 MPHo(M,H)=∅,

and therefore, .

The proof of the following property is similar to the one of property 3.6:

If then .

Property 3.8

If and for which , then and .

Proof.

Assume that . Denote by and the sets of edges lying on the paths from with odd and even indices, respectively (indices start with ). Set

 H1=(H∖E)∪O,

and

 H′1=(H′∖O)∪E.

Note that and (as ). Also note that and . Due to property 3.5, , i.e. , and therefore , which contradicts the definition of . Thus, , and, due to property 3.5, , which means that . ∎

Now let be a fixed maximum matching of . Over all , consider the pairs for which is maximized. Denote the set of those pairs by :

 M2(G,M)≐{(H,H′)∈M2(G):|M∩(H∪H′)| % is maximum}.

Let be an arbitrarily chosen pair from .

Lemma 3.9

For every path from

1. ;

2. .

Proof.

Let us show that . If then , , and is not adjacent to an edge from as is maximal. Thus, as otherwise we could enlarge by adding to it which would contradict . Thus, suppose that . Let us show that . If then define

 H1≐(H∖{h1})∪{m1}.

Clearly, is a matching, and which means that . But which contradicts . Thus . Similarly, it can be shown that .

Now let us show that . Due to property 3.7, , thus there is , such that , since , and we have . ∎

Lemma 3.10

Each vertex lying on a path from is incident to an edge from .

Proof.

Assume the contrary, and let be a vertex lying on a path from , which is not incident to an edge from . Clearly, is incident to an edge lying on .

If is not incident to an edge from too, then and are disjoint matchings and which contradicts .

On the other hand, if is incident to an edge , then consider the pair , where . Note that and are disjoint matchings and , which means that . But contradicting . ∎

For a path , consider one of its end-edges . Due to statement (a) of lemma 3.9, . By maximality of , is adjacent to only one edge from , thus it is an end-edge of a path from . Moreover, according to property 3.7. Define a set as follows:

 Y(M,H,H′)≐{Pf:P∈MPMo(M,H),f∈E1(P)}.
Lemma 3.11
1. The end-edges of paths of lie on different paths of ;

2. .

3. For every , where , , ,

1. and lie on a path from , but and do not lie on any path from ;

2. .

Proof.

(a) is true as otherwise we would have a path from with both end-edges from contradicting . Furthermore, (a) together with property 3.6 imply (b).

Now, let us prove (i).

By the definition of , is an end-edge of a path from , and therefore lies on too.

does not lie on any path from as otherwise, due to lemma 3.10, both vertices incident to would be incident to edges from , which contradicts the maximality of . Note that is not incident to an inner vertex (not an end-vertex) of a path from as any such vertex is incident to an edge from lying on , and therefore different from . is incident neither to an end-vertex of a path from as it would contradict the maximality of . Thus, is not adjacent to an edge lying on a path from , and therefore does not lie on any path from . The proof of (i) is complete.

Statement (ii) immediately follows from (i). ∎

Taking into account that , , and , we get the following result (also obtained in [10]) as a corollary from the statements (b) and (ii) of lemma 3.11:

, i.e. .

4 Spanner, S-Forest and S-Graph

The graph on figure 1 is called spanner. A vertex of spanner is called -vertex, , if . The -vertex closest to a vertex of spanner is referred as the base of . The two paths of the spanner of length four connecting -vertices are called sides.

For spanner define sets and as follows:

 U(S)≐{e∈E(S):e is incident to a 1-vertex},
 L(S)≐{e∈E(S)∖U(S):e is incident to a 2-vertex}.

Note that for spanner , and for every , the edge connecting the -vertices does not belong to , hence , , and

 β(S)α(S)=54,

as . The pair shown on figure 1 belongs to .

It can be implied from the lemma 3.11 that spanner is the minimal graph for which the parameters and are not equal.

Property 4.1

For spanner and , -vertices and -vertices of are covered by both and .

Property 4.2

For every -vertex of spanner there is such that is missed by ().

Definition 4.3

-forest is a forest whose connected components are spanners.

An -vertex of a connected component (spanner) of an -forest is referred simply as an -vertex of .

If are connected components of -forest then define sets and as follows:

 U(F)≐k⋃i=1U(Si);
 L(F)≐k⋃i=1L(Si).
Property 4.4

If the number of connected components (spanners) of an -forest is , then , and , thus .

Property 4.5

If is an -forest and then .

Definition 4.6

-graph is a graph containing an -forest as a spanning subgraph (below, we will refer to it as a spanning -forest of an -graph).

Note that, spanning -forest of an -graph is not unique in general.

It is easy to see that spanner, -forests, and -graphs contain a perfect matching, and for -forest it is unique.

Let be an -graph with a spanning -forest .

Property 4.7

If has connected components (spanners) then .

Let us define an - edge of as an edge connecting an -vertex to a -vertex of . Also define:

 Δ(G,F)≐{e∈E(G):e connects a 1-vertex of F to its % base},
 B(G,F)≐E(G)∖(L(F)∪U(F)∪Δ(G,F)).
Property 4.8

For any - alternating even cycle the numbers of - and - edges lying on it are equal.

Proof.

Consider an - alternating even cycle

 C:(u1,v1),(v1,u2),(u2,v2),...,(vn−1,un),(un,vn),(vn,u1),

where .

For a vertex of the cycle let be the frequency of appearance of the vertex during the circumference of (the number of indices for which or ). As any vertex lying on is incident to an edge from that lies on before or after during the circumference, and as edges from are - edges, we get:

 ∑w is a 2-vertex lying on Cδ(w)=∑w is a 3-vertex % lying on Cδ(w)=n.

On the other hand, denote by the numbers of -, -, - edges lying on , respectively. As for each vertex lying on , is the number of edges that lie on and are incident to , implies:

 ∑w is a 2-vertex lying on C2δ(w)=2m22+m23,
 ∑w is a 3-vertex lying on C2δ(w)=2m33+m23,

where the left sides of the equalities represent the numbers of edges lying on and incident to -vertices and -vertices of , respectively. Thus, . ∎

5 Main Theorem

• missing For a graph ( does not contain isolated vertices), the equality holds, if and only if is an -graph with a spanning -forest , satisfying the following conditions:

1. -vertices of are not incident to any edge from ;

2. if a -vertex of is incident to an edge from , then the3 -vertex of adjacent to is not incident to any edge from ;

3. for every - alternating even cycle of containing a - edge, the graph is not bipartite.

The proof of the theorem is long, so it is divided into subsections: Necessity and Sufficiency, which, in their turn, are split into numbers of lemmas.

5.1 Necessity

In this subsection, we assume that , and prove that is an -graph. Then, on the contrary assumptions we prove consequently that the conditions (a), (b) and (c) are satisfied for an arbitrary spanning -forest of . As one can see, we prove a statement stronger than the Necessity of the theorem.

Let be a graph, be a fixed maximum matching of it, and be an arbitrarily chosen pair from .

Suppose that for the graph the equality holds.

Due to corollary 3.12, we have:

 α(G)=λ(G)−α(G)=4(β(G)−α(G)). (‡)
Lemma 5.1.1

Each path from is of length four, each path from is of length five, and every edge from lies on a path from .

Proof.

Due to equality (5.1) and statement (b) of lemma 3.11, we get:

 |H|=2|Y(M,H,H′)|.

Therefore, as there are at least two edges from lying on each path of (statement (ii) of the lemma 3.11), the length of each path from is precisely four, and every edge from lies on a path from . Moreover, the length of every path from is precisely five (due to statement (b) of the lemma 3.9 it is at least five for any graph), as otherwise we would have either an edge from not lying on any path from , or a path from with length greater than four. ∎

This lemma implies that each path from together with the two paths from starting from the end-edges of form a spanner (figure 2).

Since there are paths in (property 3.6), we get:

Corollary 5.1.2

There is a subgraph of the graph that is an -forest containing spanners as its connected components.

Now, let be an -forest arbitrarily chosen among the ones described in the corollary 5.1.2. Due to property 4.4, , therefore due to equality (5.1), . This means that .

Let be an arbitrarily chosen pair from . Note that the choice of differs from the one above (we keep this notation as the reader may have already got used with a pair from denoted by ).

Lemma 5.1.3

If , and or is a -vertex of , then is a -vertex of .

Proof.

Due to property 4.2, without loss of generality, we may assume that is missed by .

Clearly, as otherwise would also be missed by and we could “enlarge” by “adding” to it, which contradicts .

Now, let us show that is neither a -vertex nor a -vertex. Suppose for contradiction that it is, and let be the spanner (connected component) of containing . Define matchings as follows (figure 3):

 H1≐(H∖E(S0))∪M0,

where is the perfect matching of ;

 H′1≐(H′∖E(S0))∪J0,

where is a matching of cardinality three satisfying (it always exists). Clearly, , and, since , and , we have and . This contradicts , concluding the proof of the lemma. ∎

Lemma 5.1.4

If , then and if is a -vertex of then is its base.

Proof.

Assume the contrary. Let , where either belongs to , or is a -vertex whose base is not . As satisfies the conditions of the lemma 5.1.3, implies that is a -vertex of . Let be the side of the spanner (connected component of ) containing . It is easy to notice that does not belong to as otherwise it would be a -vertex of whose base is . Due to property 4.2, without loss of generality, we may assume that is missed by . Define matchings and as follows (figure 4):

 H1≐(H∖E(W0))∪{(u,v)}∪(U(S0)∩E(W0)),
 H′1≐(H′∖E(W0))∪{e},

where is an edge from .

Clearly, and are disjoint matchings. Moreover, since , and , which contradicts concluding the proof of the lemma. ∎

Lemma 5.1.5

is an -graph with spanning -forest .

Proof.

Lemma 5.1.4 asserts that there is no edge incident to a vertex from , i.e. all vertices from are isolated. This is a contradiction as we assume that has no isolated vertices (see the beginning of Introduction). Thus, and is a spanning -forest of , which means that is an -graph. ∎

Due to this lemma, (an arbitrarily chosen -forest with spanners) is spanning. Obviously, the converse is also true. So, we may say that is an arbitrary spanning -forest of .

Lemma 5.1.6

The graph with its spanning -forest satisfies the condition (a) of the theorem.

Proof.

Let be a -vertex of . Lemma 5.1.4 asserts that if is an edge from then is the base of , thus . This means that the condition (a) is satisfied. ∎

Lemma 5.1.7

The graph with its spanning -forest satisfies the condition (b) of the theorem.

Proof.

Assume that is a -vertex of incident to an edge from ( is the base of ), and is the -vertex of adjacent to (this -vertex is unique as, due to lemma 5.1.6, can be incident only to edges from and ). On the opposite assumption is incident to an edge from (figure 5a).

Let us construct a subgraph