Extremal Subsystems of Monotonic Systems. III

Abstract

An attempt is made to find parts of a given graph that are more “saturated” than any other part parts with “small” graphs of the same type. On the basis of such a formulation, constructing a monotonic system from structural elements of graphs (arcs or vertices) solves this problem. The scheme of producing a monotonic system from a given graph is presented in general form, and the necessary constructions are illustrated by examples. This paper is a continuation of [1] and [2]; it has the purpose of illustrating the procedures (developed in the first two parts) of finding extremal subsystems for solving certain problems arising in tournaments, a-cyclic graphs, and undirected and directed trees.

Full text

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SIMULATION OF BEHAVIOR AND INTELLIGENCE
Russian version: http://www.datalaundering.com /download/extrem 03-ru.pdf
Extremal Subsystems of Monotonic Systems. III
J. E. Mullat


Department of Economics, Tallinn Technical University (1973 – 1980)
Translated from Avtomatica i Telemekhanika, No. 1, pp. 109 – 119, January, 1977.
Original article submitted February 23, 1976.

 1977 Plenum Publishing Corporation, 227 West 17th Street, New York, 10011.

We alert the readers’ obligation with respect to copyrighted material.


Correspondence Address: Byvej 269
2650 Hvidovre
Copenhagen Denmark
Phone (45) 36 49 33 05
E-mail: jm@email-telia.dk


Abstract

An attempt is made to find parts of a given graph that are more “saturated” than any other
part parts with “small” graphs of the same type. On the basis of such a formulation,
constructing a monotonic system from structural elements of graphs (arcs or vertices) solves
this problem. The scheme of producing a monotonic system from a given graph is presented
in general form, and the necessary constructions are illustrated by examples. This paper is a
continuation of [1] and [2]; it has the purpose of illustrating the procedures (developed in the
first two parts) of finding extremal subsystems for solving certain problems arising in
tournaments, a-cyclic graphs, and undirected and directed trees.
Keywords: monotonic, system, matrix, graph, cluster

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Introduction
Among the items that are at present of interest to investigators of complex systems, let us
mention graphs [3]. On the other hand a graph is a mathematical object, and on the other
hand it is a conventional means for describing and analyzing the relationship between the
elements of a system. In the case of systems with small number of elements, the analysis of
graphs does not present any difficulties, but in case of a large number of elements we have
problems.
In this paper it is proposed to replace the analysis of a graph by a successive analysis of
parts of this graph. In graph theory there exist many methods of selecting of subgraphs,
parts, etc.; however, in the analysis of large graphs it is not always possible to adapt the
conventional methods to the actual requirements of the investigator. For example, it is well
known that experimental graphs are fairly empty, and therefore contain many maximal
complete subgraphs whose individual selection makes no sense.
From our point of view it is convenient to select the parts of a graph by a method based on
the concept of monotonic system [1]. As a matter of fact, from a graph it is possible to
construct not one, but a whole set of monotonic systems. The investigator of a graph must
select on the basis of its own intuition an admissible class of solutions, and only after that
will he be able to use the formal method developed here.
In Section 4 we give some recommendations how to select the classes of solutions in actual
cases, by using the example of tournaments and a-cyclic graphs that occur in the technique
of modular programming, as well as trees. The other sections deal with the construction of a
general model of the required procedure of selection of parts that is illustrated by examples.
The terminology of graph theory has been adopted from [4 – 6].

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1. Formulation of Problem of Selection of Extremal Subsystems (kernels in graphs)
Let us consider the following problem on graphs. We are given a “large” graph G and a
“small” graph g . From the graph G it is required to select a part (i.e., a set of arcs or edges)
in such a way that this part is “saturated” with small graphs g . The saturation part of a graph
with small graphs g can have different interpretations. For example, it can be assumed that
one part of a graph is more saturated than another part if the first part contains a large
number of graphs g as compared to the second. The definition of saturation can be also
obtained in the following “complex” manner. Let us consider a set of arcs or vertices of a
graph G that occur only in the part of interest to us. That then we can calculate not the total
number of small graphs g located there, but only the “individual” graphs located “near”
each arc or vertex. The individual number of small graphs g located near an arc or vertex is
defined as a number of such graphs containing this vertex or arc; hence this number is
expressed by an integer. By proceeding in this way, we obtain precisely as many integers
specifying the part of interest to us, as there are arcs or vertices in it, and each integer
represents a “local” saturation of the graph G by small graphs g .
On the basis of these integers there can be many ways of defining the saturation of part of
a graph. It is possible to calculate their mean value, their variance, etc. Here we shall
consider the simplest characteristic, namely the least of all the local numbers of small graphs
g located in the selected part of a large graph G . Figuratively speaking we can say that this
is the number of subgraphs of G in the “emptiest” place.
Below we present an exact formulation of the problem of determination of the parts of a
large graph G that have greatest saturation with small graphs g . This problem can be
formulated as follows: among all possible parts of a graph G (or among the largest number
of such parts), find the part in which the least of all the local numbers of a small graphs g
that are entirely contained in it is maximal.

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It is natural to expect that in the thus-obtained part it is possible to locate in the usual
manner a large number of small graphs g . Indeed, at each vertex or arc the number of small
subgraphs g is not less than at the vertex or arc at which this number is minimal. On the
other hand in an external part this minimal number is nevertheless sufficiently large; we
especially selected this part in such a way that the condition of global maximum of the
minimum local number of graphs g is satisfied.
In the same way it is possible to formulate the problem of determination of the least
saturated part of a graph G by small graphs g . In this case each part will be characterized
by a number of subgraphs of g at the vertex or arc at which this number is maximal. Instead
of seeking the graph part in which the minimal local number of graphs g is maximal, we
seek on the contrary the part in which the maximal local number is minimal. In this case the
number of subgraphs of g at each vertex or arc will not be larger than their number at the
“maximal” vertex or arc, this number being small by virtue of the condition of global
minimum.
Let us note yet another advantage of the above-defined external parts of graphs. As a rule,
the saturation or non-saturation of these parts by small graphs is “uniform.” Usually a
saturated extremal part cannot have an especially least number of graphs g at any vertex or
arc, since the part of the graph G without this vertex or arc is apparently more saturated with
subgraphs of g in the above-mentioned “complex” sense. Conversely, for the same reason
an unsaturated extremal part cannot have an especially large number of subgraphs of g at
any one arc or vertex.
The procedure of selection of parts of graph developed in this paper is based on the
concept of a monotonic system. In considering actual applications of this technique, we must
be able to calculate the number of distinct subgraphs of g located at any given part of the
large graph G . This is not a simple problem, but many investigators dealing with the theory
of graphs have considered the calculation of distinct parts of a graph, such as Euler circuits,
regular trees [5], simple chains (paths) [7], and simple circuits [8]. Hence we possess a
highly developed technique of calculation that can be used for finding the extremal parts of
graphs as defined above.

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Among the meaningful problems that can be solved with the aid of the method developed
here, let us note the problem of selection, from family of n object that have to be ordered, of
the most unordered (unmatched), or of the most (ordered) (matched) sets of objects. As a
matter of fact, in the same way as in [9] we can take as a measure of compatibility the
number of transitive triples, and as a matter of incompatibility the number of cyclic triples.
In our terminology, cyclic and transitive triples are certain small graphs.
Such a development of monotonic systems on graphs can be used, for example, in finding
the “bottlenecks” of operational systems described in the language of modules [10]. In such
large systems it is not so easy to orient oneself in the hierarchy of mutually generating
modules, and to understand the principal manners of construction of working programs.
2. General Model of Finding Kernels on Graphs
For a given graph G let us denote by
)G(V
or V the set of its vertices. The set of arcs of
a directed graph G will be denoted by
)G(U
or U , and the set of edges of an undirected
graph will be denoted by
)G(E
or E .
In graph theory we use the concept of a subgraph of a graph G . A graph ' G is a subgraph
 )G(U ),G(VG 
if
G(V) ' G(V

of a graph
)
and
) ' G(U
is the set of those and only
those arcs of G that connect pairs of vertices belonging to
) ' G(V
. The definition of a set of
subgraphs of an undirected graph has the same form. Instead of an arc, we must consider in
this case an edge of G . Sometimes one uses the concept of part of a graph G . A part "
 )G(U ),G(VG 
is a graph such that
G of
a graph
)G(V)"G(V

and
)G(U)"G(U

. In "G
some of the arcs of the graph G are simply absent. In the same way we can define a part of
 )G(E),G(VG 
. Let us note that one of the most important concepts
an undirected graph
in this paper is the isomorphism of graphs [6].
The construction described in [1] begins with the specification of the elements of a system
W . In graphs there exists two structural units  vertices and arcs. First of all let us consider
the case that as an element of the system W we take a vertex of a graph G .

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In accordance with the construction proposed in [1] it is necessary to define the concept of
⊕ and ⊖ actions over vertices (elements) of a system. The definition of ⊕ action and ⊖
action requires the assignment of special significance function  of the vertices of G . As a
result of ⊕ actions the significance of vertex in a system must increase, whereas the ⊖
actions decreases the significance.
The construction carried out in [1] requires numerical arrays (weights) on each subset H
of elements of the system W . In [1] we have shown that for this purpose we need an initial
weight array on W and a method of realization of ⊕ and ⊖ actions. The initial weight array

V)(


can be defined, for example, as follows. In addition to a “large” graph G , let
us consider also a “small” graph g . Let us calculate the number of distinct subgraphs of G
that are isomorphic to a graph g whose set of vertices contains the vertex  . Let us take
this integer as the initial significance level
)(
. For emphasizing the dependence of the
just-introduced level
)(
of “small” graphs, we shall also use the expression “the weight
)(
of a vertex  in the graph G with respect to g .”
Below we present two operations of generation new graphs from a graph G ; they are
denoted by ⊕ and ⊖. Let us consider a graph G and let  be en empty graph, i.e., a graph
that does not contain any arc, but which has

V

)G(
1 vertices. It is assumed that
)(V  is
an exact copy of
)G(V
, and in referring to a vertex  we have in mind a vertex of graph
G , through it apparently can be of two sorts, namely as a vertex of G and as a vertex of  .
An operation of type ⊖ with a vertex  in the graph G consists of removal of all the arcs
leading to a vertex  of G .
An operation of type ⊕ with a vertex  in the graph G consists of restoring on an empty
graph all the arcs leading to a vertex  of G .

1 M is the number of elements of the set M

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It is easy to see that as a result of the ⊖ operation on any vertex  , the weights of all the
other vertices with respect to a selected small graph g are either decreasing, or the at least
remain at the previous level. In realizing the ⊕ operation, there naturally arises the question
of what can be regarded as a weight of vertex after its realization.
This problem can be solved by the following construction. On the graph  we calculate
the proper weights of the vertices with respect to a small graph g and we add them together
with the weights on the vertices of G . The thus-obtained sum is taken as a total weight of
the vertices. In this case we can observe the opposite effect; i.e., as a result of ⊕ operation
the total weights either increase or they remain (as in case of ⊖ weights) at the previous
level. In general the initial array of weights 



V)(



(i.e., the array of weights prior to
any operations) on the vertices of the graph G can be taken as a total array of weights, since
the contribution of the graph  is zero. Below we shall consider only the total weights
)(
of graph vertices that are called weights in the above sense.
Summing up, we can say that a ⊖ operation is equivalent to defining a ⊖ action on
elements of the system W , whereas ⊕ operation is equivalent to ⊕ action if we take the
above-defined total weights as significance levels of the vertices of the graph G . Thus the
monotonicity inequalities are satisfied in the above scheme, this being the principal property
of monotonic systems [1].
In constructing the sets of weight arrays of the system W it is necessary to indicate in
which manner the above-calculated array of initial weights 
V)(




is redistributed as
a result of ⊕ and ⊖ actions.
Suppose we have specified a sequence of vertices
,...,,
321

forming the set
VH 
2.
Let us successively perform ⊕ actions on the vertices of the graph G in accordance with
this sequence. As a result we obtain on the set
)(V  a part of the graph G . At each vertex
belonging to
)(V  in this part it is possible to calculate the number of subgraphs of the part
that are isomorphic to a small graph g , and obtain the weights on the elements of the set H .

2 In contrast to the general model described in [1], we do not allow here the repetition of elements
H is the complement of H .
i  . The set

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Following the notation used in [1], we can write that a new significance function has been
defined on H that has the form

...

321


(1)
and which has been constructed from the initial array of weights 
V)(




.
Thus by specifying a sequence of vertices
,...,
21
forming the set H , we obtain on H
a weight array specified by the function (1). This array denoted by
H


and called a
weight array on the set of vertices H . The weight arrays form a collection of weight arrays

VHH




. Sometimes it is convenient to use the expression “collection of ⊕ arrays
with respect to a small graph g .”
The collection of weight arrays 
VHH



can be defined in a similar way. As
above, the array of weights
H


is defined by the function

...

321


(2)
and specified on the part of the graph G left over after applying a sequence of ⊖ actions to
the vertices
,...,,
321

, forming the set H . Let us only note that the array of weights on
each subset
VH 
is actually a proper array of the remaining part, and not the total array,
since in this case the contribution yielded by the graph  is equal to zero.
Let us continue the construction of the procedure (needed below) of finding extremal
subsystems (kernels). In contrast to the foregoing, we shall take an arc as an element of the
system. The system W will be defined as an interrelated set of arcs
)G(U
of the graph G .
Following [1], it is necessary to specify ⊕ and ⊖ actions on the arcs of the graph G ; as in
the case of a system of vertices, this requires the determination of the initial significance
function  of arcs in the graph G .
Let us consider a small graph g . We shall calculate the number of distinct subgraphs of
the graph G that are isomorphic to a graph g whose set of arcs contains the arc  . This
integer is taken as the initial significance level
)(
of the arc  in the graph G , and it is
called the weight of the arc  with respect to the graph g .

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The concept of ⊕ and ⊖ actions on arcs of the graph G can be defined constructively and
exactly according to scheme similar to the one used for the vertices of the graph G .
Let us consider a graph G and let  be an empty graph with

)G(V
vertices. We shall
assume that the set of vertices
)(V  is an exact copy of
)G(V
.
An operation of type ⊖ on an arc  of a graph is called an operation of removal of this arc
on the graph G .
An operation of type ⊕ on an arc  is called an operation of restoration of this arc on an
empty graph  .
At the first let us consider the ⊖ operation. It is evident that as a result of removing the arc
 , the initial array of weights with respect to a small graph g can either decrease or remain
at the previous level. A decrease in significance (weights) proves that the ⊖ operation is
equivalent to the definition of a ⊖ action on an element of the system W .
Let us specify a sequence
,...,,
321

of distinct arcs of the graph G that form a set
)G(UH 
. Let us perform ⊖ actions on the arcs of the graph G in accordance with this
sequence. As a result, a certain part of the graph G is left over on the set of vertices
)G(V
;
the elements of this part are the arcs of the set H ,
)G(UH 
. For each arc
H


let us
calculate the number of subgraphs that are isomorphic to g ; this number is assumed to be
the value of the weight of the element  with respect to the set H . In accordance with our
notations, this method of determination of weights specifies a function
...

321


on the
elements (arcs) of the set H .
Thus, just as in case of assignment of collections of weight arrays on vertices of a graph,
we obtain on the arcs belonging to
)G(UH 
a weight array 
)


G(UHH


on the
arcs of the graph G . We shall use also the expression “⊖ collection of weight arrays of ⊖
actions on arcs with respect to a small graph g .”

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The determination of  actions on the basis of  operations over an empty graph 
requires a more detailed analysis. Suppose we have again specified a sequence
,...,,
321


of arcs of the graph G that form a set H . Let us successively perform  operations on arcs
of the set H . As a result we obtain on the set of vertices
)(V  a part of the graph G with a
string of arcs equal to H . Previously we calculated with the aid of a model at the vertices
the total weight of each vertex
)G(V


. In the present case we try to proceed in the same
way and calculate the total weight of the arcs forming the set H . The arcs of the set H are
not drawn on the graph  , and there naturally arises the question of how to calculate the
number of subgraphs that are isomorphic to a graph g and that contain an arc  , which is
absent on the graph  . We shall proceed as follows: we shall assume that this arc has been
fictitiously drawn only at the instant of calculation of subgraphs. Thus we obtain on the set
of arcs H certain integers that depend both on the graph G and on the part of the graph G
that appears on the empty graph  . These numbers are the sum of two arrays of numbers,
i.e., of the initial array of weights on the arcs of the graph G with respect to the small graph
g , and the array of weights with respect to this same graph g , but calculated only on the
just-mentioned part.
In the manner described above we determine on the set H a function

H



...

321


that
specifies a weight ⊕ array
can determine a collection of weights arrays of ⊕ actions with respect to a small graph g . It
is justified to use the expression “⊕ action,” since the total weights of elements not yet
subjected to ⊕ action can either increase or remain at the same level.

H)(





. Thus also in case of ⊕ operations we
3. Illustrative Examples on Directed Graphs
A graph G of partial ordering is defined as a binary relation G with the following
properties:
a) reflexivity, i.e., if
transitivity; if there exists an arc 

,
, or from

G
)G(V


, then
G
. The graph G has a loop at the vertex  .

,

,
, then the graph G has an arc
it follows that

G
.
b) and 

and



G

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A complete order is defined as a graph of partial ordering in which any pair of vertices 
and  is connected by an arc.
It is possible to formulate the following problem: in a given directed graph it is required to
find the (in certain sense) most “saturated” regions that are “close” to a graph of partial
ordering or to graphs of complete ordering. This problem will be solved by a method of
organization (on a graph) of a monotonic system with subsequent determination of kernels.
In accordance with the scheme of organization of a monotonic system on graphs described
in the previous section, it is necessary to assign a small graph g . Suppose that this graph
consists of three vertices
z , y , x
, and it is such that

y , x

,

,



z , xz , y )g(U

. The graph
has a total of three arcs (a transitive triple).
Now let us consider the assignment of collection of weights arrays at the vertices of a
graph shown in Fig.1. The loops on this graph have been omitted.
According to the scheme of assignment of collections of weight arrays at the vertices of a
graph, it is required to determine an initial array of weights 
)


(
, where
73
,
2
,
1
,...,


.
According to the method of calculation of the values
)(
with respect to the graph g (a
transitive triple), we obtain
31 
)(

,
22 
)(

,
23 
)(

,
74 
)(

,
45 
)(

,
36 
)(

,
37 
)

(

. As an example, let us determine a weight array on a subset of vertices
 543
,,,
2
,
1
H 
. By successively performing ⊖ actions on the set

)(

,
 76,H 
, we obtain on
the set H a new weight array
1544232231

)()()()(

,

,

, .

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11
The values of the function

7

6
can be obtained in a similar way, but for this purpose it is
necessary to use the assignment of collections of total ⊕ arrays with respect to a transitive
triple. According to Fig.2, the values of this function in their order at the vertices 
5
,

4
,
3
,
2
,
1

are as follows:
4584232231


7
,

)()

()()()(

,

,

,

,

. In exactly the same way we
can determine on any subset H of vertices

6
,
5
,
4
,
3
,
2
,
1
V 
a proper weight array of  or
⊖ actions with respect to a transitive triple.
Now let us consider a construction that is assigned not on vertices, but on the arcs of the
graph presented on Fig.1. In this case the set of elements of the system W will be

m , n,..., c , b , a)G(U

. As the small graph g we shall take the same graph as above, with a

z , x,z , y, y , x)g(U

.
set
By analogy with the foregoing, we realize the construction in the same succession. We
determine an initial weight array 

U)(



on the arcs of the graph G in accordance
with the general scheme. We find that

. , , , , , ,
,
3
,

, , ,
2
,
22122
3

21

1

11


)

)

)

)

)p(v()m(n(k(h()g(
)f(

)e(

)d(

)c()b()a(



As an example, let us now perform ⊖ actions on the arcs

m, k , fH 
. On the set H we hence obtain
k , f
and m , i.e., on the set

. , , , ,
, ,
2
,
0
,
0
,
02
2

1101


)

)

)

)

p(v(n(h()g(
)e()d()c()b()a(



In accordance with the adopted system of notations this array of numbers will be denoted
by
lines in Fig.3 represent the arcs of graph  that experience the effect of ⊕ actions
performed on the arcs
k , f
and m .
H


. For obtaining a
H


array, we must calculate the total weights. The dashed

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12
According to Fig.3, the total weight array will be as follows:
11

)h()g(


. , , , ,
, ,
2
,
3
,
2
,
32
2

1

1


)

)

)p(v(n(
)e(

)d(

)c()b()a(



Thus on any subset H of arcs of the graph shown in Fig.1 we can construct the weight
arrays
H


and
H


.
Next we describe the procedures of construction of determining sequences of ⊕ or ⊖
actions, at first for vertices, and then for arcs of the graph shown in Fig.1. The construction is
carried out for the purpose of illustrating the concepts of ⊕ or ⊖ kernels of the monotonic
system [1], and also for ascertaining the effect of the duality theorem formulated in [2].
Let us consider an example in which ⊖ weight arrays are assigned at vertices with respect
to a transitive triple. According to the scheme prescribed in [2], the procedure of
construction of a determining ⊖ sequence of vertices of a graph on the basis of ⊖ actions
(the kernel-finding procedure KFP) consists of two steps (the zero-th and the first step) for
the graph shown in Fig.1; it yields two subsets
)G(V,


10
, where

7

3
,
2
,
1
0
,...,)G(V



,

7
,

6
,
5
,
4
1



, and the thresholds
2
0
u
,
3
1
u
.
The determining sequence of vertices constructed with the aid of ⊖ actions is as follows:
7654123
,,,, ,,



. Thus on the basis of Theorems 1 and 3 of [1], and of Theorem 1 on KFP
in [2], it can be asserted that the set 
shown in Fig.1, and hence this set is the largest K⊖ kernel.

7
,

6
,
5
,
4
is a definable set of vertices of the graph

Page 14

13
Now let apply the KFP for constructing a ⊕-determining sequence. We find that

3217654
, , ,,,,



. The procedure terminates at the third step, and it consists of four steps,
namely the zero-th, the first, the second and the third. According to the construction of ⊕
sequences prescribed in the KFP, we produce the sets

321765
1
,, ,,,


,
32176
2
, , ,,


, 3
3


. As in the case of a ⊖ sequence, we conclude on the basis of
Theorems 2 and 3 of [1], and of Theorem 1 of [2], that 

j
 :

3
,

217
,
6
,
5
,
4
0
, ,



,




2
,

and a sequence of thresholds
7
0
u
,
4
1
u
,
3
2
u
,
2
3
u
 32,
is the largest K⊕ kernel of the
system of vertices of the graph in Fig.1.
A careful analysis of Fig.1 shows that the K⊖ kernel is in fact completely ordered, i.e.,
7
,
6
,
5
,
4
. On the other hand the K⊕ indicates from the point of view of the “structure” of a
graph the region in which the vertices are least ordered. This is in agreement with the our
formulation of the problem of finding kernels as representatives of “saturated” or
“unsaturated” regions (parts of a graph) with small graphs g .
Now let us use the KFP for constructing determining sequences of arcs of the graph in
Fig.1. The graph has a total of 13 arcs. After applying the KFP, we obtain on the basis of ⊖
actions the following sequence:

g , h ,m, n , k , f , p , e , v ,d , c , b , a



.
The procedure terminates at first step and it consists of two steps, namely the zero-th step
and the first step. At the zero-th step we have

g , h ,m, n , k , f

1

, with the thresholds
u
)G(U


0

, and at the first step we have

1
0
and
2
1
u
respectively. Summing up, we
can assert on the basis of the results of [1] and [2] that this is a definable set and at the same
time the largest K⊖ kernel in the system of arcs.
From the point of view of the graph structure, the application of the KFP to arcs in the
construction of a ⊖-determining sequence does not yield anything new compared to the
application of the KFP to vertices. We obtain the same complete order
7
,
6
,
5
,
4
represented
in the form of a string of arcs, and it also corroborates our assertions concerning the
saturation of a K⊖ kernel by transitive triples. On the other hand the use of KFP for
constructing ⊕-determining sequence of arcs yields a K⊕ kernel


, k

d , c , a , b , p , e , h , g , n ,m


1

,
whose meaning with regard to “non-saturation” with transitive triples cannot be ascertained.

Page 15

14
Below we shall illustrate the peculiar features of using the duality theorem from [2] for
finding K⊖ and K⊕ kernels of a monotonic system specified by vertices or arcs of a directed
graph.
At first let us consider the monotonic system of vertices of the graph in Fig.1. The
sequence of sets
j

  4

\
V
, 54
2
,V


,
V



u)(F 
. From the construction of a determining sequence

37654



,,,F
. Hence by virtue of Corollary 1 of Theorem 1 of [2] we

specified by the KFP on the basis of ⊕ actions uniquely determines

\

6541
3
,,,

the sets
1


7
,

\
. Above we have found that
3
22


 of vertices of a
graph we know that
can assert already after the second step of construction of an

76541
,,,,

 sequence that the set
contains the largest K⊖ kernel. Thus we have shown that the sufficient conditions
of the duality theorem of [2] are satisfied in the example of the graph represented in Fig.1.


21
1
,,V


\
exists a set
 32
3
,


such that
2
3


)(F 
Corollary 4 of the duality theorem we can assert that set 
Now let us consider the set

3

. As was shown above, inside this set there

. On the other hand,
3
1



)(F 

31 ,,
. By virtue of

2
contains the largest K⊕
kernel of the system of vertices of the graph (Fig.1); this likewise confirms that existence of
the conditions governing the theorem.
At last let us consider a collection of weight arrays on the arcs of the graph. The
determining

 sequence of arcs specifies a set
see that inside the set
1

\
U
there does not exist a set H as required by the conditions of

, kd , c , a , b , p , e , h , g , n ,m


1

. It is easy to

Corollaries 1 and 2 of the duality theorem in [2]. This shows that in comparison to arrays on
vertices, weight arrays on arcs do not satisfy the duality theorem.
4. Methods of Constructing of Monotonic Systems on a Special Classes of Graphs
In contrast to the previous section, we do not carry out here a detailed construction of
collections of weight arrays and determining sequences and kernels on any illustrative
example. Here we shall show how to select a small graph g and ⊕ and ⊖ actions so as to
match the selection of these elements with the desired “saturation” of the investigated graph.
The desired saturation of a graph can be understood as the saturation desirable for the
investigator who usually has a working hypothesis with respect to the graph structure. In
view of this, we shall consider the following classes of graphs: tournaments, a-cyclic
(directed) graphs, and (directed or undirected) trees.