Frame classification of the reduced labeled blocks

Georgiy I. Kalmykov

doi:10.1515/dma-2016-0001

Article Publicly Available

Frame classification of the reduced labeled blocks

Georgiy I. Kalmykov

Published/Copyright: February 15, 2016

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From the journal Discrete Mathematics and Applications Volume 26 Issue 1

Abstract

We describe a method of partitioning the given set of graphs into non-intersecting subsets each consisting of graphs with common frame. Sets of graphs having such representations are presented. The possibility to apply these representations to the derivation of asymptotic expansions in some problems of statistical mechanics is discussed.

Keywords: classification of graphs; enumeration of graphs; frame cycles

Frame classification of labeled graphs has appeared to be a useful mathematical tool for the solution of some mathematical problems in the classical statistical mechanics [1 – 7]. In a somewhat simplified form it may be described as follows. Suppose we are given for classification a set 𝔊(V) of graphs with one and the same finite set V of labeled vertices.

For the given set 𝔊(V) of graphs with the vertex set V some subset 𝔊˜(V)⊂𝔊(V) is constructed which contains significantly smaller number of graphs (with the exception of simplest cases when 𝔊(V) consists of a small number of graphs only). To each graph G˜∈𝔊˜(V) a set Xad(G˜) of edges is put in correspondence: any edge from the set Xad(G˜) connects two non-adjacent vertices of a graph G˜. Then to each graph G˜∈𝔊˜(V) the set 𝔊(G˜) is put in correspondence: 𝔊(G˜) consists of the graph G˜ and all its supergraphs obtained by adding edges from the set Xad(G˜) to the graph G˜ [8]. If it turns out that any graph from the set 𝔊(V) enters into one and only one set of type 𝔊(G˜), then we obtain a partition of the graph set 𝔊(V) into non-intersecting subsets, that is classes, labeled by graphs from the set 𝔊˜(V). So, the set of graphs 𝔊(V) is represented as the union on non-intersecting subsets:

G(V)=⋃G~∈G~(V)G(G~).

The graph G˜ being a label of the graph class 𝔊(G˜) is called the frame of these graphs, and the set Xad(G˜) is called the set of admissible edges of this frame. Such classification was called frame classification [9 – 12]. In particular, if a set of frames consists of trees, then the frame classification is called a tree classification [12 – 14].

So, to define a frame classification of a given set of graphs it is sufficient to specify a set of frames for graphs in this set, and for each frame — the set of admissible edges. In the case of successful frame classification the number of classes is much smaller than the number of graphs in the classified set, examples may be found in [4 – 6, 9 – 15]. Sometimes it appears more convenient to use instead of set of frames another set which is in one-to-one correspondence with the former. For example, in some cases of frame classification of doubly connected labeled graphs (the labeled blocks) it is convenient to consider a set of ensembles of frame cycles (cf. [12] or Appendix).

The solution of a number of mathematical problems arising in theoretical physics (in particular, in the classical statistical mechanics) is connected with the problem of extending the concept of frame classification of labeled graphs to new types of these graphs. On this way we introduce in the article a new notion of reduced doubly connected labeled graphs and construct their frame classification.

Below we consider only connected n-vertex labeled graphs without multiple edges and loops. For brevity further the word “graph” will denote a connected n-vertex labeled graph without multiple edges and loops, and the word “block” will denote doubly connected n-vertex labeled graph without multiple edges and loops. Without loss of generality we may assume that the vertices of these graphs are labeled by the first n natural numbers. In what follows we identify each vertex of such graph with its label and assume that the set of vertices of a graph is a finite set V_n = {1, 2, …, n} of the first n natural numbers.

Further, we shall always assume that the set V_n of vertices of a graph consists of at least three elements.

Let us denote by 𝔅 (n; {1, 2}) the set of all blocks having the vertex set V_n = {1, 2, …, n} and containing the edge {1,2}.

Definition 1

The graph Red(B) obtained from a block B ∊ 𝔅 (n; {1, 2}) by removing the edge {1, 2} is called the reduced block.

Example 1

Let us consider the graph C with the set of vertices V₄ = {1, 2, 3, 4} and the set of edges X(C) = {{1, 2}, {1, 4}, {2, 3}, {3, 4}}. The graph C is a block containing an edge {1, 2} and therefore belongs to the set of blocks 𝔅 (4; {1, 2}). The reduced block Red(C) is the graph with the set of vertices V₄ = {1, 2, 3, 4} and the set of edges X(Red(C₁(0))) = {{1, 4}, {2, 3}, {3, 4}}. ◄

In what follows we will identify a simple cycle of the complete graph with labeled vertices [8] and labeled graph with the same set of vertices and the set of edges coinciding with the set of edges of this cycle, i.e. we will consider such simple cycle as a graph. Also, we will identify a simple path in a labeled graph and the labeled graph with the same sets of vertices and edges as sets of vertices and edges of this path, i.e. we will consider such simple path as a graph. The simple path connecting vertices v₀ and v_h will be denoted by (v₀ – v_h) (as in [8]).

We proceed now to the definition of frame classification of the reduced blocks. This definition is based on the definition of the frame classification of labeled blocks introduced in the book [12]. For the convenience of readers all definitions from this book necessary for understanding the article are collected in the Appendix.

Following [12] we introduce the following notations:

𝔅 (n) is the set of all doubly connected labeled graphs (blocks) with the vertex set V_n;

𝕮(n) = {C} is the set of ensembles of frame cycles (see [12] or Appendix) such that each ensemble satisfies the following conditions: all vertices of any cycle of the ensemble belong to the set V_n and each vertex of the set V_n belongs to at least one cycle of the ensemble;

S(C) is the union of all cycles from the ensemble of cycles C ∈ 𝕮(n) (see [8]);

𝔅(C) is the set of all blocks having the ensemble of cycles C as the ensemble of their frame cycles.

It was proved in [12] that the set of all blocks with a vertex set V may be represented as a decomposition into non-intersecting sets (that is classes) labeled by ensembles of frame cycles (see [12] or Appendix).

For important special case when the vertex set V coincides with V_n this decomposition takes the form

(1)B(n)=⋃C∈C(n)B(C).

The frame classification of the labeled blocks is introduced by the decomposition (1).

Basing on the frame classification of the labeled blocks we introduce frame classification of the set Red(𝔅 (n;{1, 2})) = {Red(B) : B ∊ 𝔅 (n;{1, 2})} of reduced blocks.

To this aim we denote by 𝕮(n; {1, 2}) the subset of the set 𝕮(n) of ensembles of frame cycles which consists of all ensembles of the set 𝕮(n) satisfying the following condition: there is at least one cycle in the ensemble containing the edge {1, 2}. The definitions of the set 𝕮(n; {1, 2}) and of the union of cycles S(C) implies the following statement.

Corollary 1

If a frame cycle ensembleCbelongs to the set 𝕮(n; {1, 2}), then the graph S(C) contains the edge {1, 2}.

Theorem 1

For each n ≥ 3 the set of blocks 𝔅 (n;{1, 2}) may be represented as the decomposition

(2)B(n;{1,2})=⋃C∈C(n;{1,2})B(C)

into non-intersecting sets, that is classes.

Proof. Let C be an ensemble of cycles from the set 𝕮(n; {1,2}). By the Corollary 1 the union S(C) of all cycles of this ensemble contains the edge {1,2} also. So, any block from the class 𝔅(C) contains the edge {1,2} because it is a supergraph of the graph S(C) according to the theorem 6.3 from [12]. Therefore,

(3)B(n;{1,2})⫆⋃C∈C(n;{1,2})B(C).

Let B be a block from 𝔅(C). Then by the definition of 𝔅(C) the block B contains the edge {1,2}. Then, by the remark 2.2 from [12], its frame cycle C(0) also contains the edge {1,2}. Consequently, the ensemble of frame cycles of this block belongs to the set 𝕮(n; {1,2}) of ensembles of frame cycles by definition of this set, and the block B belongs to a class of blocks labeled by an ensemble of frame cycles from 𝕮(n; {1,2}). Thus

(4)B(n;{1,2})⫋⋃C∈C(n;{1,2})B(C).

The theorem follows from relations (3) and (4).□

Example 2

Let n = 4. Let us find the decomposition of the set 𝔅(4; {1,2}) into non-intersecting sets labeled by ensembles of frame cycles.

The set 𝕮(4) of ensembles of frame cycles was found in [12] (see example 6.12). Three ensembles contain one cycle each. Here are the list of these ensembles: ensemble C₁ = {C₁(0)}, where C₁(0) is the cycle with the set of edges X(C₁(0)) = {{1, 2}, {1, 4}, {2, 3}, {3, 4}}, ensemble C₂ = {C₂(0)}, where C₂(0) is the cycle with a set of edges X(C₂(0)) = {{1, 2}, {1, 3}, {2, 4}, {3, 4}}, ensemble C₃ = {C₃(0)}, where C₃(0) is the cycle with the set of edges X(C₃(0)) = {{1,3}, {1, 4}, {2,3}, {2,4}}. The other two ensembles of frame cycles consist of two cycles each. One of them is the ensemble C₄ = {C₄(0), C₄ (0, 1)}, where C₄(0) is the cycle with the set of edges X(C₄(0)) = {{1, 2}, {1, 3}{2, 3}} and C₄(0,1) is the cycle with the set of edges X(C₄(0, 1)) = {{1, 2}, {1, 4}{2, 4}}, the other is the ensemble C₅ = {C₅(0), C₅(0, 1)}, where C₅(0) coincides with the cycle C₄(0) and C₅(0, 1) coincides with the cycle C₂(0).

Four ensembles of this set contain cycles with the edge {1, 2}, namely: C₁, C₂, C₄, and C₅. Consequently, frames S(C₁), S(C₂), S(C₄) and S(C₅) form the set of frames of blocks belonging to the set 𝔅(n; {1, 2}), and ensembles C₁, C₂, C₄ and C₅ form a set 𝕮(4, {1, 2}). So, the decomposition (2) in the case n = 4 takes the form

(5)B4;1,2=BC1⋃BC2⋃BC4⋃BC5.

Lemma 1

The mappingRedof the set 𝔅 (n; {1, 2}) to the set of graphsRed(𝔅(n; {1, 2})) is one-to-one.

Proof. It follows from the definition of the mapping Red that it is single-valued. Let us prove that for any two different blocks B₁ and B₂ from the set 𝔅(n; {1,2}) their images Red(B₁) and Red(B₂) are different.

Since both blocks belong to a set 𝔅(n; {1, 2}), then each block contains an edge {1, 2}. Since the blocks are different and have the same vertex set, their sets of edges are different. Hence one of these blocks has an edge which is different from the edge {1, 2} and does not belong to another block. Therefore, the set of edges of the graph Red(B₁) does not coincide with the set of edges of the graph Red(B₂). Hence, the graph Red(B₁) does not coincide with the graph Red(B₂). This implies the statement of the lemma. □

Since for any C ∈ 𝕮((n; {1, 2}) the set of blocks 𝔅(C) is a subset of the set of blocks 𝔅(n; {1, 2}), the Theorem 1 and Lemma 1 implies the following statement.

Corollary 2

For any ensemble of cyclesC ∈ 𝕮(n; {1, 2}) the mappingRedis a one-to-one mapping of the set 𝔅(C) to the setRed(𝔅(C)).

It follows from Theorem 1, Lemma 1 and corollary 2 (see, for example, [16]) that the set Red(𝔅(n; {1, 2})) may be represented as the decomposition

(6)Red(Bn;1,2)=⋃C∈Cn;1,2Red(BC)

into non-intersecting subsets, that is classes.

Example 3

In the example 2 we have found the set 𝕮(4, {1, 2}) of ensembles of cycles. By means of this result we may find a form of decomposition (6) for n = 4:

RedB4;{1,2}=RedBC1⋃RedBC2⋃RedBC4⋃RedBC5.

Lemma 2

For any ensemble of cyclesCfrom the set 𝕮(n; {1, 2}) the reduced blockRed(S(C)) is a subgraph of every graph from the setRed(𝔅(C)).

Proof. For any graph G from the set Red(𝔅(C)) there exists a block B in the set 𝔅(C) such that B is the preimage of G for the map Red. It follows from the definition of frame classification of labeled blocks (see [12] or Appendix) that the frame S(C) is a subgraph of the block B and that the set of edges of this frame is a subset of the set of edges of block B. Therefore, the definition 1 of the mapping Red implies that the set of edges of the reduced frame Red(S(C)) is a subset of the set of the edges of reduced block Red(B), and reduced frame Red(S(C)) is a subgraph of the reduced block G = Red(B). The lemma is proved.

Theorem 2

The class of reduced blocks Red(𝔅(C)) labeled by an ensemble of frame cycles C ∈ 𝕮(n; {1, 2}) consists of the reduced frame Red(S(C)) and all graphs obtained from the graph Red(S(C)) by adding edges from the set X_ad(C) of the admissible edges of the frame cycles ensemble C.

Proof. Let B be a block from the set 𝔅(C), where C ∈ 𝕮(n; {1, 2}). From the definition of the labeled blocks frame classification (see [12] or Appendix) it follows that the frame S(C) is a subgraph of the block B and that the set of edges of the block B is the union of the edge set of the frame S(C) and the set X_ad(B) which consists of all edges of the block B belonging to the set X_ad(C) of admissible edges of the ensemble C. From the definition of the set X_ad(C) of admissible edges of the frame cycles ensemble C (see [12] or definitions A.11 and A.14 in Appendix) it follows that each admissible edge from this set does not belong to any cycle from the frame cycles ensemble C and, in particular, is different from the edge {1, 2}. Consequently, the edge sets of the reduced block Red(B) contains all edges from the set X_ad(B). Moreover, by lemma 2 the reduced block Red(S(C)) is a subgraph of the graph Red(B) from the set Red(𝔅(C)). It follows that the graph Red(B) is obtained by adding all admissible edges of the frame cycles ensemble C, belonging to the block B, to the graph Red(S(C)). So, every graph from the class Red(𝔅(C)) is obtained from the reduced block Red(S(C)) by adding the corresponding subset of edges from the set X_ad(C).

To complete the proof of the theorem it remains to prove that if a set X is a subset of the set X_ad(C) of admissible edges, then the graph G obtained by adding X to the reduced block Red(S(C)) belongs to the class Red(𝔅(C)) of reduced blocks. In fact, the graph obtained by adding the set of edges X to the frame S(C) is a block from the set 𝔅(C). Then, as it was shown above, the reduced block Red(B) is the graph from the class Red(𝔅(C)) obtained by adding to the reduced block Red(S(C)) the set of edges X. The theorem is proved. □

By means of the Theorem 2 we may describe the structure of frame classification of a set Red(𝔅(n; {1, 2})) of reduced blocks represented by the decomposition (6). In this classification the graph Red(S(C)) is the frame of the class of reduced blocks Red(𝔅(C; {1, 2})) labeled by the frame cycles ensemble C, and the set of edges X_ad(C) (as well as in the frame classification of labeled blocks) is the set of admissible edges of a class of reduced blocks labeled by the frame cycles ensemble C.

Example 4

Let us find the set of frames of reduced blocks belonging to the set Red(𝔅 (4, {1, 2})). It was found in the example 2 that blocks S(C₁), S(C₂), S(C₄) and S(C₅) constitute a set of frames of blocks belonging to the set 𝔅(4, {1, 2}).

Ensembles of cycles C₁, C₂, C₄ and C₅ were described in example 2. Since ensembles C₁ and C₂ contain one cycle each, then the block S(C₁) coincides with the cycle C₁(0) contained in the ensemble C₁, and the block S(C₂) coincides with the cycle C₂(0) contained in the ensemble C₂. It follows that the reduced block Red(S(C₁)) coincides with the reduced cycle Red(C₁(0)) and that the reduced block Red(S(C₂)) coincides with the reduced cycle Red(C₂(0)). The reduced cycle Red(C₁(0)) was found in example 1. It is the simple path with the vertex set V₄ = {1, 2, 3, 4} and the set of edges X(Red(C₁(0)) = {{1, 4}, {2, 3}, {3, 4}}. Similarly, it may be found that the reduced cycle Red(C₂(0)) is the path with the vertex set V₄ = {1, 2, 3, 4} and the edge set X(Red(C₂(0)) = {{1, 3}, {2, 4}, {3, 4}}. It follows from the description of the ensembles C₄ and C₅ given in example 2 that the block S(C₄) is the graph with the vertex set V₄ = {1, 2, 3, 4} and the edge set X(S(C₄)) = {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}} and that the block S(C₅) is the graph with the vertex set V₄ = {1, 2, 3, 4} and the edge set X(S(C₅)) = {{1, 2}, {1, 3}, {2, 3}, {2, 4}, {3, 4}}. Therefore the reduced block Red(S(C₄)) is the graph with the vertex set V₄ = {1, 2, 3, 4} and the set of edges X(Red(S(C₄))) = {{1, 3}, {1, 4}, {2, 3}, {2, 4}}, and the reduced block Red(S(C₅)) is the graph with the vertex set V₄ = {1, 2, 3, 4} and the set of edges X(Red(S(C₅))) = {{1, 3}, {2, 3}, {2,4}, {3,4}}.

The reduced blocks Red(S(C₁)), Red(S(C₂)), Red(S(C₄)) and Red(S(C₅)) form the set of frames of reduced blocks belonging to the set Red(𝔅(4, {1, 2})).

Note: Originally published in Diskretnaya Matematika (2015) 27, N^o1, 62–76 (in Russian).

Acknowledgment

The author considers as a pleasant duty to express his sincere gratitude to Ivan Ivanchyk for the problem formulation and an indication of possible application of the frame classification obtained to some problems of the classical statistical mechanics connected with the construction of expansion of the pair distribution function in powers of the density which is free from the asymptotic catastrophe [7,14].

1 Appendix

The Appendix contains the definitions of notions of the frame cycles ensemble and of a set of admissible edges of this ensemble used in the article (along with the necessary notations).

For every pair of vertices in the set V_n the vertex with smaller number (label) will be called the lower vertex of the pair, and the other vertex will be called the upper vertex of the pair. If the set V′ is contained in the vertex set V_n, then the vertex of set V′ with the smallest number (label) will be called the lowest vertex of the vertex set V′.

Let C be a simple cycle with vertices from the set V_n. We assume that there are at least 3 vertices in the cycle.

Let us introduce some notations. Let V(C) denotes the set of vertices of the cycle C, X(C) denotes the set of edges of the cycle C, α(C) denotes the lowest vertex of the set V(C), β₁(C) denotes the lower of the two vertices adjacent to the vertex α(C) in the cycle C, β₂(C) denotes the other vertex of the cycle C adjacent to the vertex α(C) in this cycle, d(C) denotes the length of the simple cycle C, and υ_h(C) denotes the vertex that is connected with the vertex α(C) by a simple path of cycle C having the length [8] h and containing the vertex β₁(C). For a vertex w of the cycle C we denote by P(C; w) a simple path in the cycle C connecting the vertices α(C) and w and containing the vertex β₁(C).

Definition A. 1

A vertex υ is called a follower of a vertex u along a simple cycle C if the length of the simple path P(C; υ) is longer than the length of the simple path P(C; u).

The definition 1 implies the following statement.

Corollary A. 1

If a vertex w is follower of a vertex υ along a simple cycle C and a vertex w′ is a follower of the vertex w along the cycle C, then the vertex w′ is a follower of the vertex v along the cycle C.

Proof. The path P(C; w′) is the union of the path P(C; w) and the path in the cycle C which connects the vertices w and w′ and doesn’t contain the vertex υ (see [8]). Therefore, the length of the path P(C; w′) is the sum of lengths of two paths forming this union. This means that the length of the path P(C; w′) is latger than the length of the path P(C; w). By the definition A.1 this implies that the vertex w′ is a follower of the vertex υ along the cycle C. □

Definition A. 2

We say that a vertex v is a direct follower of a vertex u along a simple cycle C if the length of the simple path P(C; v) is exceeds the length of the simple path P(C; u) by 1.

Let us denote by v(C; u) a vertex which is a direct follower of a vertex u along a simple cycle C. Definitions 1 and 2 imply the following statement.

Corollary A. 2

A direct follower of a vertex u along a cycle C is a follower of the vertex u along cycle C.

Example A. 1

Let C be a simple cycle C with the vertex set V(C) = {1, 2, 3, 4, 5, 6} and the set of edges X(C) = {{1, 2}, {1, 4}, {2, 5}, {3, 5}, {3, 6}, {4, 6}}. In this case α(C) = 1, β₁(C) = 2, β₂(C) = 4, d(C) = 6, υ₁(C) = 2, υ₂(C) = 5, υ₃(C) = 3, υ₄(C) = 6, υ₅(C) = 4.

The path P(C; 2) is the edge {1, 2}. The path P(C; 5) represents the path with the vertex set V(P(C; 5)) = {1, 2, 5} and with the set of edges X(P(C; 5)) = {{1, 2}, {2, 5}}.The path P(C;3) has the vertex set V(P(C; 3)) = {1, 2, 3, 5} and the set of edges X(P(C; 3)) = {{1,2}, {2, 5}, {3, 5}}. The path P(C; 6) is the path with the vertex set V(P(C; 6)) = {1, 2, 3, 5, 6} and with the set of edges X(P(C; 6)) = {{1, 2}, {2, 5}, {3, 5}, {3, 6}}. The path P(C; 4) is a path with the vertex set V(P(C; 4)) = {1, 2, 3, 4, 5, 6} and with the set of edges X(P(C; 4)) = {{1, 2}, {2, 5}, {3, 5}, {3, 6}, {4, 6}}.

The vertex 2 is a direct follower of the vertex 1 along the cycle C. The vertex 5 is a follower of the vertex 1 and a direct follower of the vertex 2 along the cycle C. The vertex 3 is a follower of the vertices 1 and 2 and a direct follower of the vertex 5 along the cycle C. The vertex 6 is a follower of the vertices 1, 2, and 5 and a direct follower of the vertex 3 along the cycle C. The vertex 4 is a follower of the vertices 1, 2, 3, 5 and a direct follower of the vertex 6 along the cycle C.

If C and C′ are two simple different cycles with the vertices from the set V, then there are only four cases.

Case 1. The lowest vertex of the vertex set V(C) does not coincide with the lowest vertex of the vertex set V(C′), i.e. α(C) ≠ α(C′).

Case 2. The lowest vertex of the vertex set V(C) coincide with the lowest vertex of the vertex set V(C′), i.e. α(C) = α(C′), and the vertex β₁(C) of the cycle C does not coincide with the vertex β₁(C′)of the cycle C′, i.e. β₁(C) ≠ β₁(C′).

Case 3. The lowest vertex of the vertex set V(C) coincides with the the lowest vertex of the vertex set V(C′), i.e. α(C) = α(C′), and the vertex β(C)of the cycle C coincides with the vertex β₁(C′)of the cycle C′, i.e. β₁(C) = β₁(C′). There exists a natural number h (2 ≤ h < min{d(C), d(C′)}) such that the vertex υ_h(C) does not coincide with the vertex υ_h(C′) (i.e. υ_h(C) ≠ υ_h(C′)), whereas for all i = 1, 2, …, h – 1 the vertex υ_i,(C) coincides with the vertex υ_i,(C′).

Case 4. The simple cycles C and C′ are of different lengths, that is d(C) ≠ d(C′). The lowest vertex of the vertex set V(C) coincides with the the lowest vertex of the vertex set V(C′), i.e. α(C) = α(C′). For all natural numbers h satisfying the condition 1 ≤ h < min{d(C), d(C′)} the vertex υ_h(C) coincides with the vertex υ_h(C′).

Definition A. 3

A simple cycle C precedes a simple cycle C′ (it is denoted by C ≺ C′) in the following four cases: 1) case 1 takes place and α(C) α(C′); 2) case 2 takes place and β₁(C) < β₁(C′); 3) case 3 takes place and υ_h(C) < υ_h(C′); 4) case 4 takes place and d(C) > d(C′).

Example A. 2

Let C be the cycle considered in the Example 1. Let us compare the cycle C with the cycle C₁ having the vertex set V(C₁) = {2, 3, 4, 5} and the set of edges X(C₁) = {{{2, 4}, {2, 5}, {3, 4}, {3, 5}}. It is obvious that α(C₁) = 2. Since α(C) < α(C₁), then the cycle C precedes (in the sense of Definition A.3) the cycle C₁.

Let us now compare the cycle C with the cycle C₂ having the vertex set V(C₂) = {1, 2, 3, 4, 5} and the set of edges X(C₂) = {{1, 3}, {1, 4}, {2, 4}, {2, 5}, {3, 5}}. Obviously in this case α(C) = α(C₂) = 1 and α₁ (C₂) = 3. Since α(C) = α(C₂) and β₁ (C) < β₁ (C₂), then the cycle C precedes (in the sense of Definition A.3) the cycle C₂.

Now we compare the cycle C with the cycle C₃ having the vertex set V(C₃) = {1, 2, 3, 4, 5, 6} and the set of edges X(C₃) = {{1, 2}, {1, 3}, {2, 5}, {4, 5}, {3, 6}, {4, 6}}. Obviously, in this case α(C) = α(C₃) = 1, β₁(C₃) = υ₁(C₃) = β₁(C) = υ₁(C) = 2, υ₂(C₃) = υ₂(C) = 5, and υ₃(C₃) = 4 > υ₃(C) = 3. It follows that the cycle C precedes (in the sense of Definition A.3) the cycle C₃.

Finally, we compare the cycle C with the cycle C₄ having the vertex set V(C₄) = {1, 2, 3, 5, 6} and the set of edges X(C₄) = {{1, 2}, {1, 6}, {2, 5}, {3, 5}, {3, 6}}. Obviously in this case α(C) = α(C₄) = 1, β₁(C₄) = υ₁(C₄) = β₁(C) = υ₁(C) = 2, υ₂(C₄) = υ₂(C) = 5, υ₃(C₄) = υ₃(C) = 3, υ₄(C₄) = υ₄(C) = 6, and d(C) = 6 > d (C₄) = 5. It follows that the cycle C precedes (in the sense of Definition A.3) the cycle C₄.

We denote by I the set consisting of the number 0 and all finite ordered sets of natural numbers beginning with the number 0. Any finite set belonging to I will be called a vector, the ordered elements of vector from the set I will be called its components. The number 0 will be considered as zero one-dimensional vector. To simplify notations the (q + 1)-dimensional vector (0, i₁, i₂, …, i_q) from the set I will be denoted by (i)q and zero one-dimensional vector will be denoted by 0.

Definition A. 4

The vector (i)_p is called preceding in a narrow sense the vector (j)_q (symbolically denoted by (i)_p ≺(j)_q) if p < q and the equalities i_n = j_n hold for any n = 1, 2, …, p. The one-dimensional vector 0 precedes any other vector (i)_q of the set I.

If a vector (i)_p precedes in the narrow sense a vector (j)_q, then the vector (j)_q is called next in the narrow sense for the vector (i)_p (symbolically denoted by (j)_q > (i)_p).

Remark A. 1

It is not difficult to prove that the binary relation on the set of vectors I established by the definition A.4 is transitive.

Definition A. 5

The vector (i)_p is called preceding in a wide sense the vector (j)_q if these vectors satisfy one of the following two conditions: (1) p < q and for all n = 1, 2, …, p the equalities i_n = j_n take place; (2) p = q, i_p < j_p, and for all n = 1,2, …, p – 1 the equalities i_n = j_n take place.

If the vector (i)_p precedes in the wide sense vector (j)_q, then the vector (j)_q is called next in the wide sense for the vector (i).

It follows from the definitions A.4 and A.5 that if the vector (i)_p precedes in the narrow sense the vector (j)_p, then the vector (i)_p precedes the vector (j)_q in the wide sense also.

Remark A. 2

It is not difficult to prove that a binary relation established on the set of vectors I by the definition A.5 is transitive.

For brevity we shall say that the vector (i)_p precedes the vector (j)_q if the vector (i)_p precedes in the narrow sense the vector (j)_q. Also for brevity we shall say that the vector (j)q is next for the vector (i)_p if the vector (j)_q is next for the vector (i)_p in the narrow sense.

Hereinafter the binary relation (i)_p ⪯ (j)_q between the vectors (i)_p and (j)_q will denote that the vector (i)_p precedes the vector (j) or coincides with the vector (j)_q; and the binary relation (j)_q ⪰ (i)_p between the vectors (j) and (i)_p will denote that the vector (j) is next for the vector (i)_p or coincides with the vector (i)_p.

For any vector (i)_q (where q ≥ 1) from the set I we denote by (i)_q–1 the preceding q-dimensional vector in the set I. In particular, (i)₀ denotes a zero-dimensional vector 0 preceding all other vectors of the set I. Further, for any vector (i)_q+p (where p is a natural number) from the set I we denote by (i)_q the preceding (q + 1)-dimensional vector from the set I.

Definition A. 6

The set of simple cycles of the complete graph with the vertex set V_n is called a cycle suite if each cycle from this set is labeled by some vector of the set I.

A cycle labeled by a vector (i)_q ∈ I with (i)_q ≠ 0 we denote by C(i)_q. Cycle labeled by the vector 0 we denote by C(0).

Definition A. 7

The vector (i)_q which is a label of a simple cycle C(i)_q is called a vector-label of this cycle.

Let us denote by I(C) the set of vectors-labels of all cycles of a cycle set C = {C(i)_q}.

Now we recall a definitions of a cycle ensemble and of a frame cycle ensemble from [12]. We give somewhat simplified definitions which are sufficient for this article.

Definition A. 8

A cycle set C ={C(i)_q} is called a cycle ensemble if this set satisfies the following conditions:

(𝒜₁) there exists the vector-label 0 in the set I(C) preceding all other vectors-labels in this set;

(𝒜₂) if a vector (i)_q is contained in the set I(C), then all vectors which precede the vector (i)_q and are next for the vector 0 belong to the set I(C) also;

(𝒜₃) if (i)_q ∈ I(C) and (i)_q ≠ 0, then the cycle C(i)_q contains at least one vertex not belonging to the cycle C(i)_q–1;

(𝒜₄) if vectors (i)_q and (j)_q belong to the set I(C) and the vector (i)_q precedes in the wide sense the vector (j)_q, then the cycle C(i)_q precedes the cycle C(j)_q.

Remark A. 3

Obviously the conditions (𝒜₁) – (𝒜₄) are not restrictive for a set of cycles consisting of only one simple cycle C(0). Therefore, any such set is an ensemble of cycles.

Example A. 3

Let C = {C(0), C(0, 1), C(0, 2), C(0, 2, 1), C(0, 2, 1, 1)} is a set consisting of five cycles. Cycle C(0) has the vertex set

(7)V(C(0)) = {1,2,3,4,5}

and the set of edges X(C(0)) = {{1, 2}, {1, 4}, {2, 5}, {3, 4}, {3, 5}}. The cycle C(0, 1) has the vertex set V(C(0, 1)) = {1, 2, 3, 5, 9} and the edge set X(C(0, 1)) = {{1, 2}, {1, 9}, {2, 5}, {3, 5}, {3, 9}}. The cycle C(0, 2) has the vertex set

(8)V(C(0,2)) = {1,2,3,4,5,6,7}

and the edge set X(C(0, 2)) = {{1, 2}, {1, 4}, {2, 5}, {3, 4}, {3, 7}, {5, 6}, {6, 7}}. The cycle C(0, 2, 1) has the vertex set

(9)V(C(0,2,1)) = {1,2,3,4,7,8,10}

And the edge set X(C(0, 2, 1)) = {{1, 2},{1, 4}, {2, 8}, {3, 4}, {3, 7}, {7, 10}, {8, 10}}. Finally, the cycle C(0, 2, 1, 1) has the vertex set V(C(0, 2, 1, 1)) = {1, 2, 3, 4, 7, 8, 11}and the edge set X(C(0, 2, 1, 1)) = {{1, 2}, {1, 4}, {2, 8}, {3, 4},

In this case, the set of vectors-labels I(C) consists of five vectors: I(C) = {0, (0, 1), (0, 2), (0, 2, 1), (0, 2, 1, 1)}. The vectors-labels from this set satisfy the relations

(10)0≺(0,1);0≺(0,2)0≺(0,2,1)≺(0,2,1,1).

indicating that the vector-label 0 precedes all other vectors-labels from the set I(C). Thus, the set of cycles of C satisfies condition (𝒜₁). From these relations it follow also that the set of cycles C satisfies the condition (𝒜₂).

The cycle C(0, 1) labeled by the vector-label (0.1) (which is next for the one-dimensional null vector 0) contains a vertex 9 not belonging to the cycle C(0). The cycle C(0, 2) labeled by the vector-label (0.2) (which is next for the one-dimensional null vector 0) contains the vertices 6 and 7 not belonging to the cycle C(0). Further, the cycle C(0, 2, 1) labeled by the vector-label (0, 2, 1) (which is next for the two-dimensional vector-label (0.2)) contains a vertices 8 and 10 not belonging to the cycle C(0, 2). Finally, the cycle C(0, 2, 1, 1) labeled by the vector-label (0, 2, 1, 1) (which is next for the three-dimensional vector (0, 2, 1)) contains a vertex 11 not belonging to the cycle C(0, 2, 1). Thus, the cycle set C satisfies the condition (𝒜₃).

It is easy to see that the members of this set of cycles satisfy the relations

(11)C0≺C0,1,C0≺C0,2≺C0,2,1≺C0,2,1,1.

Note that by definition A.5 the vector-label (0.1) precedes (in the wide sense) the vector-label (0.2), and by definition A.3 the cycles C(0, 1) and C(0, 2) satisfy the relation C(0, 1) ≺ C(0, 2). These relations along with (10) and (11) imply that the set of cycles C satisfies the condition (𝒜₄).

Thus, the set of cycles C satisfies (𝒜₁), (𝒜₂), (𝒜₃), (𝒜₄) and so (by the definition A.8) is a cycle ensemble.

To define the ensemble frame cycles we need to introduce additional notations and definitions.

Let a cycle C(i)_q belongs to the cycle ensemble C = {C(i)_q}; we denote by C(i)_q the set of cycles consisting of the cycle C(i)_q and of all cycles in the ensemble C labeled by vectors which are next for the vector (i)_q. Let S(C) be the union of all cycles of the cycle ensemble C[8]. Let S(C(i)_q) denotes the union of all cycles of the ensemble of C(i)_q.

Denote by G – V the graph obtained from the graph G by deleting the set V of vertices.

Let a vector-label (i)_q is different from the vector-label 0. Then the graph

(12)D((i)q;C=S(C(i)q−V(C(i)q−1

is defined correctly. Let Γ((i)_q; C) denotes the set of vertices of the cycle C(i)_q–1 which are adjacent to vertices of the graph D((i)_q; C) in the graph S(C(i)_q–1); let l((i)_q; C) be the cardinality of the set Γ((i)_q; C).

Remark A. 4

The condition (𝓐₃) (see definition A.8) implies that the graph D((i)_q; C) contains at least one vertex.

We denote the vertices of the set Γ((i)_q; C) by

(13)γ((i)q,1;C),γ((i)q,2;C),…,γ((i)q,l(i)q;C);C).

If the case when conditions α(C(i)_q–1) ∉ Γ((i)_q; C) and l((i)_q; C) ≥ 2 are satisfied it will be supposed that for all j and l such that 1 ≤ j < l ≤ l((i)_q; C) the vertex γ((i)_q, j; C) ∈ Γ((i)_q; C) precedes the vertex the γ((i)_q, l; C) along the cycle C(i)_q–1.

In the case when conditions α(C(i)_q–1) ∈ Γ((i)_q; C) and l((i)_q; C) ≥ 2 are satisfied it will be supposed that α(C(i)_q–1) = γ((i)_q, l((i)_q; C); C) and that for all j and l such that 1 ≤ j < l ≤ l((i)_q; C) – 1 the vertex γ((i)_q, j; C) ∈ Γ((i)_q; C) precedes the vertex γ((i)_q, l, C) along the cycle C(i)_q–1.

In cases when l((i)_q; C) ≥ 2 to simplify notations we will denote the vertex γ((i)_q, l((i)_q; C) – 1; C) by γ′((i)_q; C) and the vertex γ((i)_q, l((i)_q;C); C) by γ″((i)_q; C). If (i)_q ∈ I(C), (i)_q ≠ 0 and l((i)_q; C) ≥ 2, then C^t((i)_q–1C) denotes a simple path of the cycle C(i)_q–1 connecting the vertices γ′((i)_q; C) and γ″((i)_q; C) and containing the edge {α(C(i)_q–1), β₁(C(i)_q–1)}.

From the definition of the graph D((i)_q; C) it follows that if an ensemble of cycles C contains a cycle C(i)_q–1 which is not the Hamiltonian cycle of the graph S(C(i)_q–1) [8], then the graph S(C(i)_q–1) – V(C(i)_q–1) may be represented as a union of graphs of the form D((i)_q; C):

(14)S(C(i)q−1−V(C(i)q−1)=⋃(i)qD((i)q;C),

where ⋃(i)_q denotes the union over all vectors (i)_q ∈ I(C) of dimension q + 1 which are next for the vector (i)_q–1.

Definition A. 9

An edge that connects two vertices of a simple cycle and does not belong to this cycle is called its chord.

Definition A. 10

A cycle ensemble C is called an ensemble of frame cycles if it satisfies the following seven conditions:

(ℬ₁) Each cycle of an ensemble C contains the vertex α(C) = 1 which precedes all other vertices of all cycles of the ensemble C.

(ℬ₂)If(i)_q ∈ I(C) and (i)_q ≠ 0, then l((i)_q; C) ≥ 2 and the cycle C(i)_q contains the simple path C^t((i)_q; C) of the cycle C(i)_q–1.

(ℬ₃) If (i)_q ∈ I(C) and (i)_q ≠ 0, then all vertices and edges of the cycle C(i)_q belong to the path C^t((i)_q; C).

(ℬ₄) For any (i)_q ∈ I(C) the graph S(C(i)_q) does not contain simple cycles preceding the cycle C(i)_q.

(ℬ₅) For all (i)_q ∈ I(C) such that (i)_q ≠ 0 the graph D((i)_q; C) is connected.

(ℬ₆) If two different vectors (i)_q and (i′) belonging to the set I(C) are next for one and the same vector (i)_q–1 ∈ I(C), then the graphs D((i)_q; C) and D((i′)_q ; C) do not have common vertices.

(ℬ₇) If an ensemble of cycles C contains a cycle C(i)_q with (i)_q ≠ 0, then the graph (C(i)_q) does not contain edges that are chords of any cycle from the ensemble C and are labeled by a vector preceding to the vector (i)_q.

Theorem A 1. If an ensembleCsatisfies conditions (ℬ₁) and (ℬ₂), then the graph S(C) is a block.

Definition A. 11

A chord x of a cycle C(i)_q ∈ C is called an excludable chord of this cycle with respect to the ensembleC if this chord x is an edge of at least one cycle of the ensemble C. Otherwise the chord x is called a saved chord of this cycle with respect to the ensembleC.

Definition A. 12

We say that a chord {v_h(C), v_h+k(C)} (where 0 ≤ h ≤ d(C) – 3 and 2 ≤ k < d(C) – h) of a simple cycle C satisfies the condition of precedence along this cycle if v_h+1(C) < v_h+k(C).

Let C be a cycle ensemble. If the vector (i)_q belongs to the set I(C) and the vertex γ″((i)_q; C) is different from the vertex α(C) = 1, then we denote by R((i)_q; C) the set consisting of the vertex α(C = 1 and all vertices which are next for the vertex γ″((i)_q; C) along the cycle C(i)_q–1.

Let x is a chord of some cycle from a cycle ensemble C which is a saved chord with respect to this ensemble. We denote by (i(x, C))_{q(x, C)} the vector from the set I(C) satisfying the following condition.

Condition 𝓓(x; C): The edge x is a chord of the cycle C(i(x, C))_{q(x, C)}∈C and is not a chord of any cycle of the ensemble C labeled by a vector preceding the vector (i(x, C))_{q(x, C}).

Definition A. 13

Let C be a cycle ensemble and an edge x be a saved chord of some cycle of this ensemble with respect to C. Then x is called a forbidden edge of this cycle ensemble if this chord satisfies the following conditions:

(i(x,C))_{q(x, C)} ≠ 0;
a vertex α(C) is not contained in the vertex set Γ((i(x, C))_{q(x, C)}; C);
the edge x joins one of the vertices of the set R((i(x, C))_{q(x, C)}; C) with one of the vertices of the cycle C(i(x, C))_{q(x, C)} not belonging to the cycle C(i(x, C))_{q(x, C)–1}.

Definition A. 14

An edge x is called an admissible edge of a cycle ensemble C = {C(i)_q} if it is a saved chord for some cycle of C with respect to the ensemble C, satisfies the condition of precedence along the cycle C(i(x, bfC))_q(x,_C) and is not a forbidden edge of this ensemble.

The set of all admissible edges of a cycle ensemble C = {C(i)_q} is denoted by X_ad(C).

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Received: 2013-5-22

Published Online: 2016-2-15

Published in Print: 2016-1-1

Articles in the same Issue

https://doi.org/10.1515/dma-2016-0001

Keywords for this article

classification of graphs; enumeration of graphs; frame cycles