About this Journal Submit a Manuscript Table of Contents
Journal of Discrete Mathematics
Volume 2013 (2013), Article ID 892839, 9 pages
http://dx.doi.org/10.1155/2013/892839
Research Article

Induced Graphoidal Decompositions in Product Graphs

1Department of Mathematics, Christ University, Bangalore, Karnataka 560029, India
2Department of Mathematics, The Madura College, Madurai, Tamil Nadu 625011, India

Received 25 July 2012; Revised 20 October 2012; Accepted 5 November 2012

Academic Editor: Annalisa De Bonis

Copyright © 2013 Mayamma Joseph and I. Sahul Hamid. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let be a nontrivial, simple, finite, connected, and undirected graph. A graphoidal decomposition (GD) of is a collection of nontrivial paths and cycles in that are internally disjoint such that every edge of lies in exactly one member of . By restricting the members of a GD to be induced, the concept of induced graphoidal decomposition (IGD) of a graph has been defined. The minimum cardinality of an IGD of a graph is called the induced graphoidal decomposition number and is denoted by (). An IGD of without any cycles is called an induced acyclic graphoidal decomposition (IAGD) of , and the minimum cardinality of an IAGD of is called the induced acyclic graphoidal decomposition number of , denoted by (). In this paper we determine the value of () and () when is a product graph, the factors being paths/cycles.

1. Introduction

By a graph we mean a nontrivial, finite, connected, and undirected graph having no loops and multiple edges. The order and size of graph are denoted by and respectively. For terms not defined here we refer the reader to [1].

A decomposition of is a collection of edge-disjoint subgraphs of such that every edge of belongs to exactly one . Graph decomposition problems constitute a major area of research because of their theoretical and practical implications. Designing interconnection networks and drug designing are examples for the application of graph decomposition problems. Among the variants of decompositions of graphs that abound in the literature, path decomposition problems assume a prominent position. Harary [2] introduced the concept of path decomposition of graphs in 1970 which was further studied by Harary and Schwenk [3], Péroche [4], and Stanton et al. [5]. As a special case of path decomposition, Acharya and Sampathkumar [6] introduced the notion of graphoidal decomposition which is a decomposition of a graph into internally disjoint paths/cycles. By imposing the condition that the members of a graphoidal decomposition are induced paths/cycles, Arumugam [7] introduced the concept of induced graphoidal decomposition as well as that of induced acyclic graphoidal decomposition. Studies on these decompositions were initiated by Ratan Singh and Das [8, 9] and were further extended by Sahul Hamid and Joseph [10, 11] by obtaining certain bounds of the related parameters and and solving some characterization problems. In this paper we determine the value of and for a class of product graphs, namely, products of paths and cycles.

2. Induced Graphoidal Decomposition

The concept of graphoidal cover (we say graphoidal decomposition) introduced by Acharya and Sampathkumar [6] is defined as follows.

Definition 1 (see [6]). A graphoidal decomposition (GD) of a graph is a collection of non-trivial paths and cycles of such that(i)every vertex of is an internal vertex of at most one member of , (ii)every edge of is in exactly one member of .

The minimum cardinality of a GD of a graph is called the graphoidal decomposition number of and is denoted by .

Note that if is a path, not necessarily open, in a graph , then and are called terminal vertices and are called internal vertices of . For cycles (considered as closed paths), there is an inherent “ordering” of vertices as in paths. So, when we say that a cycle of a graph is a member of a GD of , we should mention the vertex at which the cycle begins, and this particular vertex is considered as the terminal vertex of and all other vertices on are called internal vertices of . Given a GD of , a vertex is said to be interior to if is an internal vertex of an element of and is called exterior to otherwise.

As a variation of GD, Arumugam [7] formulated the concept of induced graphoidal decomposition by restricting the members of a GD to be induced paths/cycles.

Definition 2 (see [7]). An induced graphoidal decomposition (IGD) of a graph is a GD of in which every member of is an induced path/cycle. The minimum cardinality of an IGD of is called the induced graphoidal decomposition number of and is denoted by or .

Arumugam [7] further defined an induced acyclic graphoidal decomposition (IAGD) of a graph as a GD of in which every member is an induced path. The minimum cardinality of an IAGD of is called the induced acyclic graphoidal decomposition number of and is denoted by or .

The study of the parameters and was initiated by Ratan Singh and Das [8, 9] by determining the values of these path-decomposition parameters for families of graphs such as complete graphs, complete bipartite graphs, unicyclic graphs, and bicyclic graphs. Sahul Hamid and Joseph [10, 11] extended this study further and obtained bounds for as well as in terms of the diameter, girth, and maximum degree of a vertex of and also characterized graphs obtaining these bounds.

The following theorem which gives the value of in terms of the size of a graph and the maximum number of interior vertices of an IGD of is useful in determining the value of for a given graph.

Theorem 3 (see [10]). For every IGD of a graph , let denote the number of vertices interior to and let , where the maximum is taken over all the IGD of . Then .

Remark 4. If there exists an edge-disjoint collection of internally disjoint induced paths/cycles of such that each vertex of is an internal vertex of some element of , then and vice versa. This is because together with the edges of not belonging to members of yields an IGD of with .

We require the following theorem giving the value of for complete graphs.

Theorem 5 (see [8]). For the complete graph one has

The following result is analogous to Theorem 3 and is useful in determining for a given graph .

Theorem 6 (see [11]). For any graph , , where denotes the maximum number of vertices of that can be made interior to an IAGD of .

Remark 7. if and only if there exists an edge-disjoint collection of internally disjoint induced paths of such that each vertex of is an internal vertex of some path in .

3. Induced Graphoidal Decomposition Number of Products of Paths and Cycles

Given two graphs and , their product is a graph whose vertex set is while the edge set varies according to the nature of the product. Various kinds of graph products are defined in [12]. We consider only the Cartesian product, strong product, and lexicographic product of two paths/cycles. The Cartesian product of graphs and denoted by has the edge set , and the strong product of graphs and is denoted by with edge set . The lexicographic product or the composition of graphs and is denoted by where .

All through this section and the next, we use the following notations. For positive integers and ,

For all the product graphs we have considered, the vertex set , where .

First we determine the value of for the Cartesian products of two paths/cycles.

Theorem 8. For the graph , where , one has

Proof. We have ; .
If , then and obviously . When and , let , and . Then and along with the set form an IGD of with cardinality so that . Also for any IGD of , at least one vertex of each of the cycles and will be exterior and therefore . Hence by Theorem 3 we get proving that .
Now, suppose . Then consider the following paths and cycles: Then is an edge-disjoint collection of internally disjoint induced paths and cycles of having all the vertices of as internal vertices of members of . Therefore by Remark 4, we have , thus proving the theorem.

Theorem 9. If , where , then .

Proof. The edge set . Now, let
Then is a collection of induced paths and cycles that are edge-disjoint as well as internally disjoint where each vertex of is internal to a path/cycle of . Hence it follows from Remark 4 that .

Theorem 10. For the graph , where , .

Proof. We have . Let
Then by taking when and for we get the collection of internally disjoint induced paths satisfying the property mentioned in Remark 4 and therefore .

Next we proceed to obtain the value of for the strong product of two paths/cycles.

Theorem 11. Let where . Then one has

Proof. Let .
The cases when , , and can easily be verified.
Suppose and . To prove that , by Remark 4, it is sufficient to obtain an edge-disjoint and internally-disjoint collection of induced paths and cycles such that every vertex of is an internal vertex of an element of . Let us consider the following cycles and paths of :
If , let ; if , let , and when , let thus obtaining the required collection , proving that .
When , we consider the collection where And for and , we consider the following paths and cycles
to obtain the collection mentioned in Remark 4 so that thus completing the proof of the theorem.

Theorem 12. For the graph where

Proof. We have .
When clearly , and therefore by Theorem 5, . If and , consider
Then forms an internally-disjoint and edge-disjoint collection of induced paths and cycles having all the vertices of as internal vertices. Hence by Remark 4 we have the desired result.

Theorem 13. If , where and , then .

Proof. The edge set of is given by .
In this case also, we will show the existence of a collection of internally-disjoint and edge-disjoint induced paths and cycles in such that each vertex of is an internal vertex of an element of so that by Remark 4, we get .
When , consider to obtain the desired collection of induced cycles. Next we consider
so that when ,   and when , , form the required collection of edge-disjoint and internally-disjoint induced paths/cycles.
If , we consider to obtain the collection so that by Remark 4   , thus completing the proof.

Theorem 14. Let , where such that . Then

Proof. We know that .
When , , and therefore by Theorem 5, . If and , consider
Then the collection of internally-disjoint induced paths and cycles satisfies the property mentioned in Remark 4 and therefore .
When , we observe that all the edges of the strong product are included in . Therefore using the same argument as that of Theorem 11 we get thus completing the proof of the theorem.

Theorem 15. For the graph , where with one has

Proof. For the given graph the edge set .
When , , and therefore by Theorem 5, . If and , then the same set we have considered in Theorem 12 is sufficient to prove that .

Remark 16. The lexicographic product of two graphs is not commutative. Therefore the case when is to be considered separately. However, using the same argument followed in the previous theorems, it is not difficult to show that when and whereas in all other cases , whether or .

Theorem 17. Let , where and . Then .

Proof. The edge set of is given by .
This result follows immediately from Theorem 13 as the collection of induced paths and cycles considered in the set in various cases there is applicable to the graph as well.

Remark 18. It can also be proved that by showing the existence of a collection mentioned in Remark 4.

4. Induced Acyclic Graphoidal Decomposition Number of Products of Paths and Cycles

Many of the path decomposition problems in the literature deal with splitting of a graph into paths (and not cycles) with a specified property. Therefore in this section we consider the concept of induced acyclic graphoidal decomposition for product graphs. As seen before, an induced acyclic graphoidal decomposition (IAGD) of a graph is an induced graphoidal decomposition where each member of is an induced path. The minimum cardinality of an IAGD of is called the induced acyclic graphoidal decomposition number of , and it is denoted by .

As in the previous section, we determine the value of for nine types of product graphs. We begin with the Cartesian product of two paths.

Theorem 19. For the graph where ,

Proof. If , , and so, . For and , let , let , and let . Then and along with the set form an IAGD of with cardinality so that . Also for any IAGD of , one of the vertices and will be exterior and the same is in the case with the vertices and . Therefore, —the maximum number of vertices interior to any IAGD of —is , and hence by Theorem 6   proving the desired result.
The case when also is similar to the previous case where at least two vertices of will be exterior to any IAGD of , and, therefore, by Theorem 6. Further, the paths along with the edges and form an IAGD of cardinality , and hence it follows that .
Next assume that and . Consider the induced paths , , and , where
Then the collection together with the edges , , and forms an IAGD of such that proving that . Also it can be verified that at least one vertex of is exterior to any IAGD of , and therefore by Theorem 6 thus proving the required result.
If , then as in the previous case any IAGD of will have at least one vertex exterior to it so that . Also there exists an IAGD of cardinality consisting of the induced paths , and where and the edges , , , , and so that and hence the result.
Finally, assume that and . Consider the paths
Then is an edge-disjoint collection of internally-disjoint induced paths of such that every vertex of is an internal vertex of some path in . Therefore by Remark 7, .

Theorem 20. If , where , then .

Proof. For the given graph we will show that there exists a collection of internally-disjoint and edge-disjoint induced paths such that each vertex of is internal to a member of . If , let where
If , let
Then forms the desired collection of internally-disjoint and edge-disjoint induced paths such that all the vertices of are internal to some path in , and by Remark 7, .

Theorem 21. If , then .

Proof. The result follows from Theorem 10 and Remark 7 as the collection considered there does not contain any cycle.

Theorem 22. Let where . Then one has

Proof. When , the result is immediate as which has its edge set as a minimum IAGD as does not contain any induced paths of length greater than one.
In all other cases it can easily be observed that the vertices , , , and cannot be made interior to any IAGD of . Therefore , and hence by Theorem 6, . Now, let Then the collection of induced paths together with the edges of not covered by the paths in forms an IAGD such that , proving that , and this completes the proof of the theorem.

Theorem 23. If is the graph , then

Proof. When , we have , and therefore as the edge set of is a minimum IAGD of .
If and , we consider the collection of induced paths , , , , , , , , , , , which satisfy the property mentioned in Remark 7, and therefore in this case.
If and , we consider the following induced paths: Then the collection of all the paths given previous constitutes an edge-disjoint collection of internally-disjoint induced paths in such that each vertex of is internal to some path in . Therefore it follows from Remark 7 that , thus completing the proof.

Theorem 24. Let where . Then

Proof. Suppose . Then it can be observed that the vertices and cannot be made internal to any induced path of . Therefore, . Further, the three paths ; , and together with the edges of not belonging to the induced paths , , and form an IAGD of cardinality so that thus proving that .
Next assume that . If , consider the induced paths
so that is a collection of edge-disjoint and internally-disjoint induced paths such that each vertex of is an internal vertex of a path in . Therefore by Remark 7, .
If and , let
Then is a collection of internally-disjoint and edge-disjoint induced paths mentioned in Remark 7 so that .

Theorem 25. For the graph where and one has

Proof. When , , and therefore .
Next assume that and . Then at least two of the vertices , , , will be exterior to any IAGD of , and hence by Theorem 6. Now, let
Then the collection along with the edges of not covered by the paths in forms an IAGD of cardinality , and hence it follows that .
If and , then we get a collection of induced paths given by which satisfies the condition mentioned in Remark 7 by considering the following paths: Thus we have . Next assume that and . In this case let us consider the paths Then is an internally-disjoint and edge-disjoint collection of induced paths in such that each vertex of is internal to some path in and so by Remark 7, . Finally, when , consider the induced paths following: Then is the collection satisfying the condition mentioned in Remark 7 so that , and this completes the proof of the theorem.

Theorem 26. For the graph where and

Proof. When , is a complete graph on nine vertices, and therefore . Suppose and . Consider the induced paths
Then is a collection of induced paths that are edge-disjoint and internally-disjoint such that each vertex of is an internal vertex of some path in . Therefore by Remark 7, .

Remark 27. Since the lexicographic product of two graphs is not commutative, the case when is to be considered separately. However, in this case using the same argument followed in Theorems 25 and 26 we get when and otherwise. Also, when .

Theorem 28. Let where and . Then

Proof. The result follows from Theorem 24 as , and all the induced paths considered in Theorem 24 to obtain the collection mentioned in Remark 7 are induced paths in as well.

Remark 29. For the graph , it can easily be shown that if and and in all other cases.

5. Scope for Further Research

Given any graph , the lower bound for as well as is . This is because the maximum value that and can take is . It has been found in [10, 11] that the families of graphs for which this lower bound is attained are very large. In this study on product graphs, we observe that and attain the bound for several product graphs. Therefore, the present study leads us to problems like the following that need further investigation.(i)Characterize the product graphs for which .(ii)Characterize the product graphs for which .(iii)Characterize the product graphs for which