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Computers, Materials & Continua
DOI:10.32604/cmc.2021.015790
images
Article

On Network Designs with Coding Error Detection and Correction Application

Mahmoud Higazy1,2,*and Taher A. Nofal1

1Department of Mathematics and Statistics, College of Science, Taif University, Taif, 21944, Saudi Arabia
2Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf, 32952, Egypt
*Corresponding Author: Mahmoud Higazy. Email: m.higazy@tu.edu.sa
Received: 07 December 2020; Accepted: 07 January 2021

Abstract: The detection of error and its correction is an important area of mathematics that is vastly constructed in all communication systems. Furthermore, combinatorial design theory has several applications like detecting or correcting errors in communication systems. Network (graph) designs (GDs) are introduced as a generalization of the symmetric balanced incomplete block designs (BIBDs) that are utilized directly in the above mentioned application. The networks (graphs) have been represented by vectors whose entries are the labels of the vertices related to the lengths of edges linked to it. Here, a general method is proposed and applied to construct new networks designs. This method of networks representation has simplified the method of constructing the network designs. In this paper, a novel representation of networks is introduced and used as a technique of constructing the group generated network designs of the complete bipartite networks and certain circulants. A technique of constructing the group generated network designs of the circulants is given with group generated graph designs (GDs) of certain circulants. In addition, the GDs are transformed into an incidence matrices, the rows and the columns of these matrices can be both viewed as a binary nonlinear code. A novel coding error detection and correction application is proposed and examined.

Keywords: Network decomposition; network designs; network edge covering; circulant graphs

1  Introduction

Graph (Network) designs are introduced as a generalization of symmetric balanced incomplete block designs (BIBDs) (see, e.g., [1,2]) which are decompositions of complete graphs (networks) to subgraphs (subnetworks) satisfying certain conditions (see [1]). There are several research papers on the subject of graph decompositions; for more details see [3]. Through the paper we use the word (graph) to mean (network).

As defined in [1], a symmetric graph design, or SGD, with parameters images, where n, images are positive integers and G and F are graphs with n vertices, is a set images of spanning subgraphs of the complete graph Kn such that

a)   Gi images for images;

b)   any edge of Kn is contained in exactly images subgraphs Gi, and

c)   images for images, images.

In [4], Dalibor Fronček and Alex Rosa determined all graphs F and all orders for which there exists an images-SGD where images, the friendship graph on n vertices.

In this paper, a generalization of symmetric BIBDs is investigated and we introduce a new graph representation that will help in constructing new graph designs (GDs).

Definition 1.1 Let H be a r-regular Cayley graph of order n and B be a non-empty set of spanning subgraphs of H. images-GD (Graph Design, GD) is a collection images of spanning subgraphs of H such that

1.    all graphs G in B have the same size images,

2.    any graph of images is isomorphic to one graph of B,

3.    every edge of H belongs to exactly images elements of images,

4.    for any two different subgraphs Gi and Gj of images, we have images.

If images, images, images and images or 0, then the images-GD is equivalent to the sub-orthogonal double covers (SODCs) of the complete bipartite graph by G. SODC’s have been studied by many authors (for SODCs of Kn, n by G, see [57] and for SODCs of Kn by G, see, [810]. The images-GD with images is equivalent to the orthogonal double covers (ODCs) of Cayley graphs which have been studied in [11]. Also, the images-GD with images is equivalent to ODCs of Kn, n by G that have been investigated by many authors (see, e.g., [1215]). Studying the case when images, images, images and images is equivalent to studying the mutually orthogonal graph squares which have been studied by many authors (see, e.g., [7,1619]) and for more details see the survey [20]. Since SODCs and ODCs can be considered as graph designs, its construction tools can be used to construct new graph designs as will be done in this work.

Here, all graphs are assumed to be finite, simple and with non-empty edge set. We use the usual notations: images for the group of all residual classes modulo n, images for the empty set, Kn, n for the complete bipartite graphs, Kn for the complete graph, Pn+1 for the path graph with n edges, Sn for the star of size n, En the empty graph of order n, the circulant graph images is defined by images and images, see [21].

In our current study, we concentrate on the case when images or images and images or images. Note that, if images, then images > n.

From now on, all addition and subtraction shall be done modulo n.

The vertices of Kn, n shall be labeled by the elements of images. Namely, for images we shall write vi for the corresponding vertex and define images if and only if images, for all images and images. To avoid ambiguity, the edge images shall be written as (u, v).

All designs can be represented by a corresponding incidence matrix [22]. Following the method produced in [23], the incidence matrices can be used in coding error detection and corrections. Here, the suggested codes are not linear codes.

The arrangement of our paper is as follows: In Section 2, a new representation of graphs is introduced. In Section 3, a technique of constructing the group generated graph designs of Kn, n is studied. In Section 4, detection of error and its correction is suggested as an application of the codes generated by the constructed graph designs. In Section 5, we construct new group generated graph designs of Kn, n. In Section 6, a technique of constructing the group generated graph designs of the circulants is given with group generated graph designs of certain circulants. The conclusion shall be in Section 7.

2  New Representation of Graphs

In this section, we introduce a new representation of graphs following the method that has been introduced in [13]. In [13], the graphs have been represented by a vector whose entries are the labels of the vertices related to the lengths of edges linked to it. This method of graph representation has simplified the method of constructing the graph designs. Here, a general method is proposed and applied to construct new graph designs.

Let G be a spanning subgraph of H and let images. Then the graph G with

images

is called the images-translate of G. The length of an edge images is defined by l(e) = vu.

For any subgraph G of Kn, n, let images

be the multiset containing the length of every edge in G. For any two subgraphs G1 and G2 of H, let

images

be the multiset containing the distance of every pair of equal length edges in G1 and G2. Note that the distance set D(G, G) means the set of distances between the different edges in G which have the same lengths. For any collection of graph images, we define rd-matrix as a images matrix whose entries are images for images.

Let G be a graph of order n and its vertices are the elements of images, G can be represented by a map images from images to its power set (i.e., images) where for all images, images such that for all images, the edge images. images can be written in the form of n-tuple where

images

for all images (a vector whose ith entry is a set of vertices, from images, incident to the edges with length equal i). Then the following are clear.

images

and for all images, images and images.

Let images where images. Then

images

Let G and H be two spanning subgraphs of Kn, n, G and H are said to be orthogonal if they share at most one edge (i.e., images), see [8,10] or [24]. Then the collection images is mutually orthogonal if and only if all cells of images matrix are sets.

For H = r-regular images, the existence of (images)-GD immediately implies the following two necessary conditions that is recorded as

Lemma 2.1 Let images be a images-GD and e is the size of any element of B. Then

images

Proof. From Definition 1.1 of the (images)-GD, we have

images

Since all elements of images are isomorphic to one element of B and all elements of B have the same size e, this implies that images. Also, we have images by Definition 1.1, which imply that images.

3  Group Generated Graph Designs of images

Definition 3.1 Let images be a collection of spanning subgraphs of Kn, n. We call images a images-GD generator if it satisfies the following conditions:

1.    Every element of images appears exactly images times in the sum of the multisets

images

2.    For all pairs i, j with images, the cells of the images matrix are sets, that is D(Gi, Gj) are all sets.

The elements of the generator images are called images-GD pre-starters graphs.

Theorem 3.2 Let images be a images-GD generator. Then for all images, the collection of all the translates of images for all images, forms a images-GD by B.

Proof. It is clear that the collection of all translates covers every edge of Kn, n exactly images times. Now, It is to show that the collection of all translates are mutually orthogonal, that is any two graphs of the collection of all translates share at most one edge. Consider two translates images and images where images and assume that they share two edges e1 = (x, y) with length l1 = yx and e2 = (u, v) with length l2 = vu. Then the two edges images, images with lengths l1, l2 respectively and images, images with lengths l1, l2 respectively. Then the distance between the two edges with length l1 in Gi and Gj is images, and also the distance between the two edges with length l2 in Gi and Gj is images and then D(Gi, Gj) is not a set. This is a contradiction of the second condition in the Definition 3.1 of the images-GD generator. Consequently, all subgraphs in the collection of all translates of GD-generator are mutually orthogonal, that is a images-GD.

Lemma 3.3 Let images be a images-GD generator, then

  i)  the number of pre-starters in images images.

 ii)  For all images, if images then images,

iii)  For all images, if n is even then images.

Proof. (i) images-GD. Since s = kn then images and hence images.

(ii) Let D(Gi, Gi) contains images. then Gi contains four edges each pair of them has the same length l1 and l2, that is (x, x + l1), (x + d, x + d + l1), images.

Then Gi + d contains (x + d, x + d + l1), (x + 2d, x + 2d + l1), (u + d, u + d + l2), (u, u + l2) which imply that images which is a contradiction. Hence, for all images, if images then images.

(iii) For any images, let images.

So there exist two edges e1 = (x, x + l), e2 = (x + n/2, x + n/2+l) belong to E(Gi) with the same length l and D(e1, e2) = n/2.

Then Gi + n/2 contains also e1 = (x, x + l), e2 = (x + n/2, x + n/2+l) that means images which is a contradiction. Hence, for all images, if n is even then images.

Therefore, images is a necessary condition of the existence of the images-GD generator.

Lemma 3.4 Let images is a pre-starter of images-GD and

images

Then

images

Proof. Case 1. For n is even; for images, then images (the number of differences of the edges of length i), then D(G, G) is a multiset set. This is a contradiction, then images.

Case 2. For n is odd; for images, then images (the number of differences of the edges of length i), then D(G, G) is a multiset set. This is a contradiction, then images.

Proposition 3.5 Let images and images be any integers, B is a set of graphs of size e. If there exists a images-GD generator of Km, m by B, then there exists a images-GD generator of Kmn, mn by B.

Proof. Here, the element images is written as st. Let images be a images-GD generator of Km, m by B with respect to images that is every edge in Km, m appears images times in images and D(Gp, Gq) for all images, are sets, and images.

For all images and images, let images is a pre-starter graph images, that is images and images.

Let the set D(Gp, Gp) = D1 and the set D(Gp, Gq) = D2 and images.

For all images, and for all images, define images by

images

Then images. Then every edge in Kmn, mn (its vertices are images) appears images times in images. For any two graphs images which is a set and images which is a set then images is a images-GD generator of images by images with respect to images.

4  Coding Error Detection and Correction Application

The rows or columns of the incedence matrix of the GDs can be used as binary codes because all of its entries are 0 or 1. Let us define the GD’s Incedence matrix images as follows.

For the images-GD, since Kn, n has n2 edges and we have s blocks (GD subgraphs), define images as images integer matrix where its elements are 0 or 1 and displays the relation between the edges and the blocks where every row corresponds to a block (GD subgraph Gi) and every column corresponds to an edge (ej) in the graph Kn, n.

images

GD Incidence Matrix has the following properties:

As the incidence matrix images of a images-GD has the following properties.

1.    Every row has n number of 1s,

2.    Every column has images number of 1s,

3.    Two distinct columns both have 1s in at most 1 rows.

For illustration, the following example is produced.

The blocks of (K3, 3, S3, 2; K2)-GD is constructed as:

images

where ab is an edge between vertex a0 and vertex b1, see Fig. 1. The incedence matrix of this GD is

images

images

Figure 1: (K3, 3, S3, 2; K2)-GD

When a GD is transformed into an incidence matrix, the rows and the columns can be both viewed as a binary nonlinear code. The binary codes formed from the row denoted as images and binary codes from the column will be referred as images. As mentioned previously, by conversion of GD to incidence matrix, the incidence matrix of a GD retains certain properties that are inherited from GD. Using these properties, results can be obtained to evaluate the minimum Hamming distance (number of different bits in two codes) between codes from images or images. Where

images

and

images

The minimum Hamming distance images and images.

Distance in binary codes detects the number of errors a code can detect or correct [25]. As proved in [26], we have

•    a binary code images can be detected up to q errors iff the minimum distance images is greater or equivalent to q + 1.

•    a binary code images can be corrected up to q errors iff the minimum distance images is greater or equivalent to 2q + 1.

Then for our example images can detect upto 3 errors and correct upto one error.

Efficiency factor E is the the quality estimation of the design efficiency. The efficiency factor E is a numerical value lies between 0 and 1. The quality of a design is “good” if E is greater than 0.75 The efficiency of the images-BIBD design codes [27] is calculated as images which can be simplified for our graph design as images (put v = n2, the size of Kn, n and k = n, the size of G) which will be always greater than 0.75 where n is the size of the GD blocks. Then the efficiency of the codes from the GDs are very good and can be safely used in coding processes. For more details about the design efficiency, see [27]. For more applications of networks, see [2830].

To clear the proposed application, we use the above images for coding the following words shown in Tab. 1 and assuming that there is a possibility of occurring an error in at most two positions. From the structure of the corresponding GD, the number of ones must be 3 in any code.

Table 1: Words’ codes

images

If the code 111100001 is received. Since number of ones must be 3, the error is detected. To correct the error, the code with the minimum Hamming distance from the received one can be chosen that is 111000000. Then the message is “go,” and so on.

5  Graph Designs images-GD’s

Here, we use the above representation of graphs to construct images-GD for images by certain graph classes B.

5.1 Graph Designs images-GD’s

Lemma 5.1 Let images1 be a positive integer. There exists images-GD.

Proof. For n = 6, define images by

images

Then all graphs in images are isomorphic to C6 and

images

Since every cell of the images-matrix is a set satisfying Lemma 3.3, then images is a images-GD generator. Applying Proposition 3.5 completes the proof. images

Lemma 5.2 Let images be a positive integer. There exists images-GD.

Proof. For n = 10, define images by

images

Then all graphs in images are isomorphic to C10 and

images

Since every cell of the images-matrix is a set satisfying Lemma 3.3, then images is a images-GD generator. Applying Proposition 3.5 completes the proof. images

5.2 Graph Designs images-GD’s

The existence of images-GD still open for images. Nevertheless, we can record the following result as:

Lemma 5.3 For images. There is no images-GD generator.

Proof. Let P5 is a spanning subgraph of K4, 4. Then the following vectors and all of its translates are the all possible pre-starter vectors of P5 shown in Tab. 2. By careful inspection, we find that there are no images mutually orthogonal pre-starter vectors inside this collection, then the proof is complete.

Table 2: All possible pre-starter vectors of P5

images

images

Proposition 5.4 Let images be a positive integer. There exists a images-GD.

Proof. Define images as follows.

For all images.

images

Then all graphs in images are isomorphic to P4 and images, and

images

Since every cell of the images-matrix is a set satisfying Lemma 3.3, then images is a images-GD generator. images

Lemma 5.5 Let images be a positive integer. There exists a images-GD.

Proof. For n = 8, define images as.

images

Then all graphs in images are isomorphic to C4 and

images

Since every cell of the images-matrix is a set satisfying Lemma 3.3, then images is a images-GD generator. Applying Proposition 3.5 completes the proof. images

Lemma 5.6 Let images be a positive integer. There exists images-GD.

Proof. For n = 6, define images as.

images

Then all graphs in images are isomorphic to P7 and

images

Since every cell of the images-matrix is a set satisfying Lemma 3.3, then images is a images-GD generator, Applying Proposition 3.5 completes the proof. images

Lemma 5.7 Let images be a positive integer. There exists a images-GD.

Proof. For n = 6, define images by

images

Then all graphs in images are isomorphic to images and

images

Since every cell of the images is a set satisfying Lemma 3.3, then images is a images-GD generator. Applying Proposition 3.5 completes the proof. images

Lemma 5.8 Let images be a positive integer. There exists a images-GD.

Proof. For n = 6, define images by

images

Then images are isomorphic to C6, G2 is isomorphic to images and

images

Since every cell of the images-matrix is a set satisfying Lemma 3.3, then images is a images-GD generator. Applying Proposition 3.5 completes the proof.

Lemma 5.9 Let images be a positive integer. There exists a images-GD.

Proof. For n = 6, define images by

images

Then images are isomorphic to C6, G2 is isomorphic to images and

images

Since every cell of the images-matrix is a set satisfying Lemma 3.3, then images is a images-GD generator. Applying Proposition 3.5 completes the proof. images

Lemma 5.10 Let images be a positive integer and G is the class of the spanning sub-graphs isomorphic to the graph with vertices images and the 6 edges images. There exists a images-GD.

Proof. For n = 6, define images by

images

Then images are isomorphic to C6, G2 is isomorphic to G and

images

Since every cell of the imagesmatrix is a set satisfying Lemma 3.3, then images is a images-GD generator. Applying Proposition 3.5 completes the proof. images

Lemma 5.11 Let images be a positive integer. There exists a images-GD.

Proof. For n = 8, define images as.

images

Then all graphs in images are isomorphic to C6 and

images

Since every cell of the images is a set satisfying Lemma 3.3, then images is a images-GD generator. Applying Proposition 3.5 completes the proof. images

Lemma 5.12 Let images be a positive integer and G is a graph containing a cycle C4 in addition to an edge K2such that they share a vertex. There exists images-GD.

Proof. For n = 5, define images as:

images

Then images are isomorphic to C4, G2 is isomorphic to G and

images

Since every cell of the images is a set satisfying Lemma 3.3, then images is a images-GD generator. Applying Proposition 3.5 completes the proof. images

Lemma 5.13 Let images be a positive integer. There exists a images-GD.

Proof. For n = 5, define images as:

images

Then all graphs in images are isomorphic to P6 and

images

Since every cell of the images is a set satisfying Lemma 3.3, then images is a images-GD generator. Applying Proposition 3.5 completes the proof.

Lemma 5.14 Let images be a positive integer. There exists a images-GD.

Proof. For n = 5, define images as.

images

Then images are isomorphic to P6, G2 is isomorphic to images and

images

Since every cell of the images-matrix is a set satisfying Lemma 3.3, then images is a images-GD generator. Applying Proposition 3.5 completes the proof. images

5.3 Graph Designs images-GD’s

Lemma 5.15 Let images be a positive integer. There exists a images-GD.

Proof. For n = 5, define images as:

images

Then images are isomorphic to images, G3 is isomorphic to images and

images

Since every cell of the images-matrix is a set satisfying Lemma 3.3, then images is a images-GD generator. Applying Proposition 3.5 completes the proof. images

Lemma 5.16 Let images be a positive integer. There exists a images-GD.

Proof. For n = 5, define images as:

images

Then all graphs in images are isomorphic to images and

images

Since every cell of the images is a set satisfying Lemma 3.3, then images is a images-GD generator. Applying Proposition 3.5 completes the proof. images

6  Graph Designs images-GD’s

Definition 6.1 Let images be a collection of spanning subgraphs of H = r-regular images where images. We call images a images-GD generator if it satisfies the following conditions:

1.    Every element of A+ appears exactly images times in the sum of the multisets L(Gi), images.

2.    For all pairs i, j with images, the cells of the images matrix are sets, that is D(Gi, Gj) are all sets.

The elements of the generator images are called images-GD pre-starters graphs.

Theorem 6.2 Let images be a images-GD generator. Then for all images, the collection of all the translates of images for all images, forms a images-GD by B.

Proof. It is clear that the collection of all translates covers every edge of H exactly images times. Now, It is to show that the collection of all translates are mutually orthogonal, that is any two graphs of the collection of all translates share at most one edge. Consider two translates images and images where images and assume that they share two edges e1 = (x, y) with length l1 = yx and e2 = (u, v) with length l2 = vu. Then the two edges images, images with lengths l1, l2 respectively and images, images with lengths l1, l2 respectively. Then the distance between the two edges with length l1 in Gi and Gj is images, and also the distance between the two edges with length l1 in Gi and Gj is images and then D(Gi, Gj) is not a set. This is a contradiction of the second condition in the Definition 6.1 of the images-GD generator. Consequently, all subgraphs in the collection of all translates of GD-generator are mutually orthogonal, that is a images-GD. images

Lemma 6.3 Let images be a images-GD generator, then

  i)  the number of pre-starters in images images,

 ii)  For all images, if images images,

iii)  For all images, if n is even then images.

Proof. (i) images-GD. Since the s = gn then images and hence images.

(ii) Let D(Gi, Gi) contains images then Gi contains four edges each pair of them has the same length l1 and l2, that is (x, x + l1), (x + d, x + d + l1), (u, u + l2), images.

Then Gi + d contains (x + d, x + d + l1), (x + 2d, x + 2d + l1), (u + d, u + d + l2), (u, u + l2) which imply that images which is a contradiction. Hence, for all images, if images then images

(iii) For any images, let images.

So there exist two edges e1 = (x, x + l), e2 = (x + n/2, x + n/2+l) belong to E(Gi) with the same length l and D(e1, e2) = n/2.

Then Gi + n/2 contains also e1 = (x, x + l), e2 = (x + n/2, x + n/2+l) that means images which is a contradiction. Hence, for all images, if n is even then images.

Therefore, images is a necessary condition of the existence of the images-GD generator. images

Proposition 6.4 Let images and images be integers and let H = 2m-regular images where

images

where images. Then there exists images-GD.

Proof. Define images as:

For all images and for all images

images

Then all graphs in images are isomorphic to P4 and

images

Since every cell of the images-matrix is a set satisfying Lemma 6.3, then images is a images-GD generator. images

Proposition 6.5 Let images be an integer and let H = 8 -regular images where images where images. Then there exists images-GD.

Proof. Define images as:

images

Then all graphs in images are isomorphic to P5 and

images

Since every cell of the images is a set satisfying Lemma 6.3, then images is a images-GD generator. images

Proposition 6.6 Let images be a positive integer and H be 4-regular images where images where images such that images, images, images, images and images are all sets (i.e., all have different elements).Then there exists images-GD.

Proof. Define images as

images

Since images and images are sets then all graphs in images are isomorphic to P5 and

images

Since every cell of the images-matrix is a set satisfying Lemma 6.3, then images is a images-GD generator. For illustration, at n = 7 take l1 = 1 and l2 = 3.

Table 3: New graph designs

images

7  Conclusion

In this paper, we have studied the group generated graph designs. A new representation of graphs has been proposed that help in constructing new graph designs images-GD that can be summerized in Tab. 3. Where H is certain circulant graph. In addition, an efficient coding method has been proposed using the constructed graph designs which may open a new door to produce more research in this area. Finally, we can state that the constructed GD’s can be efficiently used to generate a code set.

Acknowledgement: The authors are thankful of the Taif University. Taif University researchers supporting project number (TURSP-2020/031), Taif University, Taif, Saudi Arabia.

Funding Statement: The authors received financial support from Taif University Researchers Supporting Project Number (TURSP-2020/031), Taif University, Taif, Saudi Arabia.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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