Coverings and matchings of the vertices of a graph by the edges

COVERINGS AND MATCHINGS OF THE VERTICES OF A GRAPH BY THE EDGES*

G. Ts.AKOPYAN

Moscow

(Received 6 March 1973)

A METHOD of investigating the set of all the c-coverings and the set of all the c-matchings of the

points of a complex by intervals is proposed. Results, obtained by this method, for the problem of finding the minimal c-covering and the maximal c-matching of the vertices of a graph by the edges, are presented.

Many problems of the theory of graphs are problems of coverings, for example, the problems of finding the minimum externally stable or the maximum internally stable set a graph, the minimum covering of the vertices of a graph by the edges or the maximum pair-matching etc. [ 1 ] .

Various algorithms exist for solving covering problems. However, the majority of the algorithms, falling far short of an exhaustive examination of the alternatives, do not in general solve the

problem of finding optimum coverings. Introducing the local method of investigation. Zhuravlev f2,3] proved that there is a fundamental difficulty here, discrete extremal problems on the const~ction of optimal coverings are not solvable in the class of Iocal algorithms. Therefore, in

many cases methods of a more global nature are used for finding optimal coverings and studying their properties. The methods of alternating chains, for example, are such methods [4,5] ,

In this paper we consider the problem of the c-covering (c-matching) of the points of a complex. In the set of all c-coverings (c-matchings) a relationship of equivalence is introduced and

the properties of c-coverings (c-matchings) of the vertices of the graph by the edges, connected with

this equivalence (see [ 1 O]), are investigated.

Let X=(x$,..., z,} be a finite set and U= (IL,, . . . , u, > a class of subjects of the

set X. We call the elements of the set X points, and the elements of U intervals. A pair (X, v) is called a complex. We denote by N(s) the set of all the intervals of N, where NsU, containing

the point 2. Let c (x) be an integer-valued positive function defined on the set X. The N is a

c-covering of the set of points of the complex by intervals, if IN(si) 1 >c(Ici), i=l, 2, . . . , n.

The set N is a c-matching of the set of points of the complex, if ] N (z,) ] Gc (Xi), i=l, 2, . . . , n.

The complexity of a c-covering (c-matching) is put equal to its power.

We call the point z free relative to the c-covering (c-matching) N, if ] N(X) ] -ite (5). An

interval containing only points free with respect to N is called free with respect to N, and a c-covering N, containing even one interval free with respect to N is called simple; we also call a c-matchingN simple, if in U\N even one interval free with respect to N is found. Otherwise

a c-covering (c-matc~ng) is a dead-end (or min~al-essential).

Two intervals Ui and ui, where u,flu,+0, are called dependent with respect to the c-covering (c-matching) N, if a point XEU,fIUj can be found such that 1 [N\ {Q, Uj} ] (x) I

*Zh. vj%hisl. Mat. mat. Fiz., 14,2,461-469, 1974.

184

Coverings and matchings of the vertices of a graph 185

--@I (I NJ{ % ujl 1 t5) 1 >c tz) 1.

Otherwise the intervals ui and ui are independent of N,

For a given complex (X, 0) we will consider the problem of finding the minimum (maximum) c-covering (c-matching), that is, the c-covering (c-matching) of minimum (maximum) power.

De~nition 1. We say that the c-coverings (c-matchings) Ni and Ni are adjacent, if

(NiUNj) \ w-w) ={%r 41,

where ZiGN, and UjeNj.

Definition 2. We say that the c-coverings (c-matchings) Nr and N2 are equivalent, if a finite

chain with the c-coverings (c-matchings)

N,, Iv<*, a w 4 , Ni,

can be found such that Ivi j and Nij+l are adjacent c-coverings (c-matchings), where

y=o, 1,. . . : (s-1)) Ni”~Ni, N,,=Nj,

This equivalence partitions the whole set of c-coverings (c-matchings) of the complex (X, U) into equivalence classes. The c-coverings (c-matchings) of the same class have the same power.

We will construct two graphs for the complex (X, U).

The graph I’ of c-coverings (c-matchings). The set of vertices of this graph is the set of all c-coverings (c-matchings) of the complex; the ccoverings (c-matchings) Ni and Ni are joined by an

edge if and only if (NtCNj) and 1 Nj\Ni 1 =I.

For convenience the vertices of the graph I’ will be arranged on levels. The levels are

numbered from 0 to m and are arranged in the order in which the numbers increase from bottom to top. A vertex corresponding to a c-covering (c-matching) N, is situated on the level numbered / N 1.

We consider some vertex of the graph I’, corresponding to the c-covering (c-matching) N. The subgraph of the graph I, generated by this vertex and the vertices corresponding to all the c-supercoverings (c-submatchings) (supersets (subsets) which are c-coverings (c-matchings)) of the

c-covering (c-matching) N, is an (m- 1 N I ) -dimensional ( 1 N 1 -dimensional) unit cube, tight on the vertices corresponding to the c-coverings N and U (c-matchings N and Cp. It is obvious

from this that the graph r. has a fairly complex structure.

The graph D of equivalent c-coverings (c-matchings).

The set of vertices of the graph D is the set of all equivalence classes. We will call the vertices of the graph L? degrees. Two degrees are connected by an edge if and only if each of them has at least one c-covering (c-matching) connected by an edge in the graph I‘ of c-coverings ~c-matchings~.

The graph of equivalent c-coverings (c-matchings) may have three types of degrees (see Figs. 1 and 2):

186 G. Ts. Akopyan

1) 0 - simple degree, all the c-coverings (c-matchings) occuring in this degree are simple;

2) 8 - semi-dead-end degree, containing at least one simple and at least one dead-end (minimal-essential) c-covering (c-matching);

3) e - dead-end degree, all the c-coverings (c-matchings) occurring in this degree are dead-end ones.

We say that a dead-end degree is minimal (maximal), if it contains a min~al c-covering (maximal c-matching).

Two graphs of equivalent c-coverings FIG. 1. FIG. 2. (c-matchings) are called strongly isomorphic, if

they are isomo~hic and the types of the corresponding degrees are identical.

Theorem 1.

The graph of equivalent c-coverings (c-matchings) is a tree.

PRX$ 1. All the c-supercoverings (c-submatchings) of the c-covering (c-matching) N joined to N by an edge in the graph I’, are adjacent, as follows from the definition of adjacency.

2. If the c-coverings (c-matchings) Ni and Nj are equivalent, their c-supercoverings (c-submatchin~) joined to them by edges in the graph I’, are equivalent, as follows from part 1

of the proof, the definition of equivalence and from the remark that any two adjacent c-coverings (c-matchings) have a common c-covering (c-matching), with which each of them is joined by an edge

in the graph l?. The theorem is proved.

Below we will consider only complexes, all of those intervals consist of two points, that is, graphs whose vertices and edges are points and intervals, respectively.

Let G(n) be the set of all non-isomorphic graphs without loops and parallel edges, possessing n vertices. We consider the graph G = (X, v) from G(n). In the case of finding c-coverings we also assume that G does not contain isolated vertices. We notice that there is imposed on the function

c (2) the natural constraint c (si) <g(z;), where g (xi) is the number of edges incident with

the vertex zi in the graph G.

The structure of the tree of equivalent c-coverings (c-matchings) of vertices of the graph G by edges is given by the following theorem.

Theorem 2.

The tree of equivalent c-coverings (c-matchings) of vertices of the graph by edges has the form shown in Fig. 1 (Fig. 2); the powers of the minimal (maximal) degrees being equal.

The numbers of the corresponding levels are indicated on the right of Figs. 1 and 2.


Proof 1. All the dead-end degrees of the tree of equivalent c-coverings (c-matchings) of

vertices of the graph by edges are minimal (maximal).

We assume that NT is a dead-end c-covering (c-matching) of dead-end degree which is not minimal (maximal), And let NM be a minimal c-covering (maximal c-matching) for which

I(NM, NT) (l(NT, NM) ) assumes the least value, and let Z (K,, Nj) be the number of

edges of Ni not occurring in A’j. We consider the set of edges ;VI,,\N, (NT\NM) . If Z(N,, NT) =0 (Z(~~*, XM) =O), we have Y,=M,. We assume that I(:Vlt, N,) >O (6( if;:,, N,‘r >O), and show that in this case we obtain a contradiction.

a. Among the edges of the graph G incident with the given vertex X, there are not fewer

belonging to N,\NM (N,\:V,), than belonging to Nk,\NT (NT\Nd, that is

f (~=\~~~) (z) 12 (<I I (NA&) (5) I.

Indeed, otherwise we would have

If z= (5, x’) E (N~\~V=) (uE~V,\N,), there exists the edge u’= (2, z”) E (X,\

N,)(zz’~(N,\N,))(otherwise N,\(u) (NMU{u}) would be a c-covering (c-matching) less

(more) complex than NM). And the set .V’= (N,U {T-A'}) \{u} (N'== (il;,,,U {u>) \{u'}) is the minimal c-covering (maximal c-matching) for which 2 (N’: NT) tl (N,,, N,) (I (NT, N') <Z(N,, N,)) contrary to hypothesis.

b. There do not exist simple cycles of even length whose edges would alternately belong to

N,\N, and N,\N,. Let such a cycle w exist. Then the set N’= (N-U o) \ (N,CI 01) is a c-covering (c-matching), and is minim+ (maximal) (because of the equation 1 A” I= 1 N, 1) , which is impossible, since Z(N', A4rT) tl (J'!'~, N,) .

c. Let o=(z&, uz, . . . , U./J be the longest simple chain whose edges belong

alternately to NM\JV, and N,\N,,, and let x0 and XI; be the extreme points of this chain.

If SOPXk, it follows from paragraph a that both extreme edges or and uk belong to

NT\flM (NS\NT), and

that is, the points x0 and xk are free with respect to the c-covering (c-matching) NT. There exists the following sequence of adjacent c-coverings (c-matcliings):

where N,,_, is a simple c-covering (c-matching) (uk is free with respect to Nr,_,) contrary to hypothesis. If zO=xk, we have 1 NT (x,) 1 ~c(~~)i-Z (lNT(a) IGc(~o)-2) andthesame chain of adjacent c-coverings (c-match~gs) leads to a simple c-covering (c-matc~g).

188 G. Ts. Akopyan

2. All c-supercoverings of minimal c-coverings (c-submatchings of maximal c-matchings) of the same complexity, of a graph of c-coverings (c-matchings) are equivalent to one another.

Let NM, be the minimal c-covering (maximal c-matching) of degree d. We consider the minimal (maximal) degree d’, different from d. Let NM, be such a minimal c-covering (maximal c-matching) of degree d’, for which 1 (NM,, N,,) (Z(N,,, Iv,,) ) assumes its minimal value. It is obvious that 1 (N_ NM,) Xl (2 (A’,,, NY*) >O).

a. We have 1 (N,,\N,,) (2) 13 (G) 1 (N,,\N,,) (2) I. The proof repeats the proof of paragraph 1 ,a, with the sole difference that instead of NT and NM we take N,, and NM, respectively. Only the remark that N/Ed’ is added.

b. Let there exist a simple cycle o of even length whose edges belong alternately to N~~\N~* and N,\N,,. We consider that the set N’s (~~~*U~~ \ (NJlw). It is a c-covering

(c-matching) not contained in d’, since 1 (N’, N,,) <I (NM,, NJ (I (NM,, N’) 4 (NM,, NJ ) . It is obvious that the c-supercoverings of the c-coverings (c-submatchings of the c-matchings) N’

and N,, of the adjacent upper (lower) level in the graph ‘r are equivalent.

If N’E~, the c-supercoverings (c-submatchings) N,, of the adjacent upper (lower) level

are also equivalent to the c-supercoverings (c-submatchings) NM,.

If N’&, then WEd” (d” distinct from d and d’), and we repeat all the reasoning in paragraph 2, after replacing d’ by d”. Since there are a finite number of degrees, we finally arrive

at the case where N’Ed.

c. Let o be the longest simple chain whose edges belong alternately to NM,\NM, and

N,,\N,,. Then, just as in paragraph 1 ,c, we arrive at a contradiction.

3. Below (above) a semi-dead~nd degree in the tree of equiv~ent c-coverings (c-match~gs) of

the vertices of a graph by edges, there cannot be a simple degree, which follows from the fact that

for each adjacent c-covering (c-matching) N’ of a dead-end c-covering (c-matching) NT there cannot

be two independent free edges with respect to N’ from N’ (from U\N’) .

4. The powers of the minims (mammal) degrees of the tree of equivalent c-coverings

(c-matchings) of points of a given graph by edges are equal.

Let d and d’ be two minimal (maximal) degrees. Also let NM,4 and NM,Ed’ be the

c-coverings (c-matchings) considered in paragraph 2. It follows from the proof of the statement of

paragraph 2 that the set (~~~\N~~*) U (N~*\N~,) is a set of non-intersect~g cycles of even length whose edges belong alternately to N,,\Nn, and N,,\Nw To each c-covering

(c-matching) ARM of d there corresponds one-to-one a c-covering (c-matching) N’= (NAJ CO) \(NJlm) of d’. The theorem is proved.

It follows from the theorem just proved that the tree of equivalent c-coverings (c-matchings) of the vertices of a graph by the edges is described by the following parameters:

Y (II) =m (v’(D) =O) is the complexity of the most (least) powerful simple c-covering

(c-matching);

~(0) (z’(D)) is th e complexity of the most (least) powerful dead-end c-covering

(c-matching)‘;


CL(D) (cc’(W) isth e complexity of the minimal c-covering (maximal c-matching);

p=m-T(D) ($=7’(D)) is th e number of simple degrees of the tree D;

q=-c(D) -p(D) (q’=p’(D) -T’(D) )is the number of semi-dead-end degrees of the

tree D;

r (r’) is the number of minimal (maximal) degrees of the tree D;

t (t’) is the power of the minimal (maximal) degree of the tree D;

We consider trees of equivalent c-coverings (c-matchings) of the vertices of two graphs by the

edges. Let vi, ‘ti, l-t+ pi, ri, ti, i=l, 2, be the parameters of these trees.

Corollav 1. Two trees of equivalent c-coverings (c-matchings) of the vertices of a graph by

the edges are strongly isomorphic if and only if pr = p2, q, = q2, rl = r2.

We call a c-matching perfect, if 1 N(zi) 1 =c (xi) for any vertex of the graph G.

Let % be the set of all c-coverings (c-matchings), and 3, the set of all the minimal

c-coverings (maximal c-matchings) of the graph G. We consider a partial graph G’ of the graph G:

G'=(X, U’), rge U’= U IV,. NMEEM

Let Gi,‘, . . . , Gt,’ be those connectivity components of the graph G’ which allow

a perfect c-matching. We denote the remaining connectivity by Gi,‘, . . . , Gj, ‘. We

represent the graph G’ in the form G’=G,‘UG,‘, where

Gi’= 0 Gik', Gz'= ; Gjk'. k=i k=i

We call this representation of the graph the partition of the graph G with respect to c-covering

(c-matching). Let rir, . . . , rice , tic,. . . , tia be respectively, the numbers of minimal c-coverings

(maximal c-matchings) of the graphs Gi,‘, . . . , Gt,‘, Gj,‘, . . . , Gjp’. Then the proof of the

theorem implies that

a B

r= rI

rik, t= h,, II

I%l=rt.

k=i k=i

It follows from the partition of G into Gr ’ and GZ’ with respect to c-coverings (c-matchings), that

the minimal c-coverings (maximal c-matchings) of any minimal (maximal) degree of the tree D

are characterized by the minimal c-coverings (maximal c-matchings) of the graph Gz’, and the

different minimal (maximal) degrees by the minimal c-coverings (maximal c-matchings) of the

graph Gr’.

Let there be some c-covering N. We consider all those vertices 2 of the graph G for which

IN(z) /k(z). F or each such vertex we choose 1 N(z) 1 -c (5) edges of the set of edges

belonging to N(s). The set obtained will be a c-matching. It is easy to prove that if we start from

the minimal c-covering, after discarding edges we obtain the maximal c-matching. The converse is

also true. To the c-matching N let us add c (x) - 1 N(s) I edges to those vertices of the graph G

for which I N(s) 1 <c (CT). Th e set obtained will be a c-covering.

190 G. Ts. Akopyan

If we start from the maximal c-matching, after the additional of edges we obtain the minimal c-covering. This implies the validity of the following lemma.

Lemma

An edge of the graph G occurs in some minimal c-covering if and only if it occurs in some

maximal c-matching.

It obviously follows form this lemma that the partition of a graph G with respect to c-covering is identical with its partition with respect to c-matching.

Corol&ry 2. For any graph G the number of minimal degrees of the tree of equivalent

c-coverings equals the number of maximal degrees of the tree of equivalent c-matchings.

Corollary 3. If the graph allows a perfect c-matching, the power of the minimal (maximal) degree of the tree of equivalent c-coverings (c-matchings) equals unity.

Let the function c(z) be identically equal to the constant k, 0, 1, . . . . For convenience we will say “k-covering” instead of “c-covering”.

Corollary 4. If a graph does not have two adjacent pendant edges, the power of minimal degree of the tree of equivalent 1 -coverings of the vertices of the graph by edges is equal to unity if

and only if the graph allows a perfect pair-matching.

Corollary 5. The power of the maximal degree of the tree of equivalent 1 -matchings equals

unity if and only if the graph allows perfect pair-matching.

Corollary 6. If n = 2k f 1 and the graph G has a Hamiltonian cycle, the number of minimal

(maximal} degrees of the tree of equivalent 1 -coverings (1 -matchings) of the vertices of this graph

by edges is equal to unity.

In [7,8] it is proved that the fo~owing inequality holds for almost all graphs of G(n):

In [9] it is proved that y(D) =[ (n-l-1)/2] f or almost all graphs of G(n). It is also proved

that almost all graphs of G(n) possess Hamiltonian cycles. From these results and Corollary 6 we obtain Corollary 7.

Coro~~~~ Z For almost all graphs of G(n) the tree of equivalent c-coverings (c-matchings) of the vertices of the graph by edges is of the form shown in Fig. I (Fig. 2), where

Y(D) =e (D) nz, ‘/*&l(D) <%, 4 (L)) =O,

n-2 (log n+1) gr (D) <n-l, 7’ (D) 3 +( i-0( F))

p(D)= [+I, p’(D)=[ +], r=r’==l npn n=2k+l,

r=r’>l nprr n=2k,

and the powers of minimal (max~al) degrees equal unity for II = 2k.


Corollary 8. For almost all graphs of G(n) the tree of equivalent 2coverings of the vertices of

the graph by edges is of the form shown in Fig. 1, where

Y(D) =e(D)n2, ‘/iOa3(D) e/5, cl(D) =n, v’(D) =o, p’(D) =n

and the powers of minimal degrees equal unity.

It is obvious from Corollaries 7 and 8 that for almost all graphs of G(n) the 2coverings are

l-coverings taken from simple degrees of the tree of equivalent l-coverings.

We consider the gradient algorithm Ad. We give a description of its working.

Initially the c-covering U (c-matching IZI) is applied to the input.

Let the c-covering (c-matching) applied at the input be simple. Then A,j discards (adds) some

free interval from this c-covering (from the complement of the c-matching) and the resulting

c-subcovering (c-supermatching) is applied to the input of Ad.

Let the c-covering (c-matching), applied to the input be dead-end; Ad passes on to the

consideration of equivalent c-coverings (c-matchings) of this c-covering (c-matching). If a simple one

is found among them, it is applied to the input ofAd. Otherwise the algorithm ceases its task, and

issues one of the dead-end c-coverings (c-matchings) obtained. If the dead-end c-covering (c-matching)

obtained as a result of the working of the algorithm, is minimal (maximal) for any choice of the

free interval, we say that the algorithm solves the problem.

CoroZZary 9. The gradient problem Ad solves the problem of finding the minimal c-covering

(the maximal c-matching).

In conclusion we give examples of trees of equivalent 1 -coverings and 1 -matchings of vertices

of a graph by the edges.

1. A chain of length n:

Y=n, r=2]$-[ -Sg(+-)Sg(q), ,=I?[, r=l,

v’=O, ‘tl= [f] +sg ($) ) $= [qq ) +=I.

2. A cycle of length n:

v=n, T=2 [3]+%(-5)Sg(~), p=]+[, r=2-sg(+),

V'=O, a’= [+-I Ssg (+-) , p’= [-+I, f-2-q ($).

3. E3 - a three-dimensional unit cube:

v=12, r=6,

v’=O, -r/=3,

lJ=4,

$=4,

r=9,

r’=9.

192

1.

2.

3.

4.

5.

6.

7.

8.

9.

v’=O, ‘6’=3, p/=4, r/=2.

Translated by J. Berry

REFERENCES

BERGE, C. Theory ofgraphs and its applications (Teoriya grafov i ee primeneniya), Izd-vo in. lit., Moscow, 1962.

ZHURAVLEV, Yu. I., Local algorithms for the calculation of information. I. Kibernetika, No. 1, 12-19, 1966.

ZHURAVLEV, Yu. I., Local algorithms for the calculation of information. IL Kibernetika, No. 2, l-l 1, 1966.

NORMAN, R. Z. and RABIN, M. O., An algorithm for a minimum cover of a graph. Notic. Amer. Math, Sot., $315319, 1958.

RAYCHAUDHURI, D. K., An algorithm for a minimum cover of an abstract compIex. Canadian J. Math., 15,1,1 l-24,1963.

AKOPYAN, G. Ts., On equivalent ~ansfo~ations of coverings. Kibernetika, No. 6,121-127, 1972.

NIGMATULLIN, R. G., On the covering of a graph by edges. In: Problems of cybernetics (Probl. kibernet~i), No. 21,241-248, Fizmatgiz, Moscow, 1969.

NIGMATULLIN, R. G., Pair-coverings of graphs. Uch. zap. Kazanskogo un-ta, 128,2,91-94,1968.

PEREPELITSA, V. A., On two problems in the theory of graphs. Dokl. Akad. Nauk. SSSR, 194,6, 1269-1272, 1970.

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Coverings and matchings of the vertices of a graph by the edges