Induced Graphoidal Decompositions in Product Graphs

Abstract

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

1. Introduction

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

A decomposition of 𝐺 is a collection 𝜓=(𝐻₁,𝐻₂,𝐻₃,…,𝐻_𝑘) of edge-disjoint subgraphs of 𝐺 such that every edge of 𝐺 belongs to exactly one 𝐻_𝑖. 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 𝜂_𝑖 and 𝜂_𝑖𝑎 and solving some characterization problems. In this paper we determine the value of 𝜂_𝑖 and 𝜂_𝑖𝑎 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 𝐺 is a collection 𝜓 of non-trivial paths and cycles of 𝐺 such that(i)every vertex of 𝐺 is an internal vertex of at most one member of 𝜓, (ii)every edge of 𝐺 is in exactly one member of 𝜓.

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

Note that if 𝑃=(𝑣₀,𝑣₁,…,𝑣_𝑘) is a path, not necessarily open, in a graph 𝐺, then 𝑣₀ and 𝑣_𝑘 are called terminal vertices and 𝑣₁,𝑣₂,…,𝑣_𝑘−1 are called internal vertices of 𝑃. For cycles (considered as closed paths), there is an inherent “ordering” of vertices as in paths. So, when we say that a cycle 𝐶 of a graph 𝐺 is a member of a GD 𝜓 of 𝐺, we should mention the vertex at which the cycle 𝐶 begins, and this particular vertex is considered as the terminal vertex of 𝐶 and all other vertices on 𝐶 are called internal vertices of 𝐶. Given a GD 𝜓 of 𝐺, a vertex 𝑣 is said to be interior to 𝜓 if 𝑣 is an internal vertex of an element of 𝜓 and is called exterior to 𝜓 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 𝐺 is a GD 𝜓 of 𝐺 in which every member of 𝜓 is an induced path/cycle. The minimum cardinality of an IGD of 𝐺 is called the induced graphoidal decomposition number of 𝐺 and is denoted by 𝜂_𝑖(𝐺) or 𝜂_𝑖.

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

The study of the parameters 𝜂_𝑖 and 𝜂_𝑖𝑎 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 𝜂_𝑖(𝐺) as well as 𝜂_𝑖𝑎(𝐺) in terms of the diameter, girth, and maximum degree of a vertex of 𝐺 and also characterized graphs obtaining these bounds.

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

Theorem 3 (see [10]). For every IGD 𝜓 of a graph 𝐺, let 𝑡_𝜓 denote the number of vertices interior to 𝜓 and let 𝑡_𝑖(𝐺)=max𝑡_𝜓, where the maximum is taken over all the IGD 𝜓 of 𝐺. Then 𝜂_𝑖(𝐺)=𝑚−𝑡_𝑖(𝐺).

Remark 4. If there exists an edge-disjoint collection 𝒮 of internally disjoint induced paths/cycles of 𝐺 such that each vertex of 𝐺 is an internal vertex of some element of 𝒮, then 𝜂_𝑖(𝐺)=𝑚−𝑛 and vice versa. This is because 𝒮 together with the edges of 𝐺 not belonging to members of 𝒮 yields an IGD 𝜓 of 𝐺 with 𝑡_𝑖(𝐺)=𝑛.

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

Theorem 5 (see [8]). For the complete graph 𝐾_𝑛 one has 𝜂_𝑖𝐾_𝑛=1,if𝑛=3,4,if𝑛=4,𝑚−𝑛,if𝑛≥6,𝑛iseven,𝑚−𝑛+1,if𝑛≥5,𝑛isodd. (1)

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

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

Remark 7. 𝜂_𝑖𝑎(𝐺)=𝑚−𝑛 if and only if there exists an edge-disjoint collection 𝒮 of internally disjoint induced paths of 𝐺 such that each vertex of 𝐺 is an internal vertex of some path in 𝒮.

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

Given two graphs 𝐺 and 𝐻, their product is a graph whose vertex set is 𝑉(𝐺)×𝑉(𝐻)={(𝑔,ℎ)∶𝑔∈𝑉(𝐺)𝑎𝑛𝑑ℎ∈𝑉(𝐻)} 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 𝐺 and 𝐻 denoted by 𝐺□𝐻 has the edge set 𝐸(𝐺□𝐻)={((𝑔,ℎ),(𝑔^′,ℎ^′))∶𝑔=𝑔^′,(ℎ,ℎ^′)∈𝐸(𝐻)𝑜𝑟(𝑔,𝑔^′)∈𝐸(𝐺),ℎ=ℎ^′}, and the strong product of graphs 𝐺 and 𝐻 is denoted by 𝐺⊠𝐻 with edge set 𝐸(𝐺⊠𝐻)={((𝑔,ℎ),(𝑔^′,ℎ^′))∶𝑔=𝑔^′,(ℎ,ℎ^′)∈𝐸(𝐻);(𝑔,𝑔^′)∈𝐸(𝐺),ℎ=ℎ^′𝑜𝑟(𝑔,𝑔^′)∈𝐸(𝐺)and(ℎ,ℎ^′)∈𝐸(𝐻)}. The lexicographic product or the composition of graphs 𝐺 and 𝐻 is denoted by 𝐺∘𝐻 where 𝐸(𝐺∘𝐻)={((𝑔,ℎ),(𝑔^′,ℎ^′))∶(𝑔,𝑔^′)∈𝐸(𝐺)𝑜𝑟𝑔=𝑔^′,(ℎ,ℎ^′)∈𝐸(𝐻)}.

All through this section and the next, we use the following notations. For positive integers 𝑛₁ and 𝑛₂, 𝑃_𝑛1=𝑢₁,𝑢₂,𝑢₃,…,𝑢_𝑛1,𝑃_𝑛2=𝑣₁,𝑣₂,𝑣₃,…,𝑣_𝑛2,𝐶_𝑛1=𝑢₁,𝑢₂,𝑢₃,…,𝑢_𝑛1,𝑢₁,𝐶_𝑛2=𝑣₁,𝑣₂,𝑣₃,…,𝑣_𝑛2,𝑣₁. (2)

For all the product graphs 𝐺 we have considered, the vertex set 𝑉(𝐺)={𝑤_𝑖𝑗∶1≤𝑖≤𝑛₁;1≤𝑗≤𝑛₂}, where 𝑤_𝑖𝑗=(𝑢_𝑖,𝑣_𝑗).

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

Theorem 8. For the graph 𝐺=𝑃_𝑛1□𝑃_𝑛2, where 𝑛₁,𝑛₂≥2, one has 𝜂_𝑖(𝐺)=1,if𝑛₁=𝑛₂=2,𝑚−𝑛+2,if𝑛₁=2,𝑛₂≥3,𝑚−𝑛,otherwise. (3)

Proof. We have 𝐸(𝐺)={𝑤_𝑖𝑗𝑤_𝑖(𝑗+1)∶1≤𝑖≤𝑛₁;1≤𝑗≤(𝑛₂−1)}∪{𝑤_𝑖𝑗𝑤_(𝑖+1)𝑗∶1≤𝑖≤(𝑛₁−1); 1≤𝑗≤𝑛₂}.
If 𝑛₁=𝑛₂=2, then 𝐺=𝐶₄ and obviously 𝜂_𝑖(𝐺)=1. When 𝑛₁=2 and 𝑛₂≥3, let 𝑄₁=(𝑤₂₁,𝑤₁₁,𝑤₁₂,𝑤₁₃,…,𝑤_1𝑛2,𝑤_2𝑛2), 𝑄₂=(𝑤₂₁,𝑤₂₂,𝑤₂₃,…,𝑤_2𝑛2) and 𝑆^′={𝑤_1𝑗𝑤_2𝑗∶2≤𝑗≤(𝑛₂−1)}. Then 𝑄₁ and 𝑄₂ along with the set 𝑆^′ form an IGD 𝜓 of 𝐺 with cardinality 𝑛₂ so that 𝜂_𝑖(𝐺)≤|𝜓|=𝑛₂=𝑚−𝑛+2. Also for any IGD 𝜓 of 𝐺, at least one vertex of each of the cycles (𝑤₁₁,𝑤₂₁,𝑤₂₂,𝑤₁₂,𝑤₁₁) and (𝑤_1𝑛2,𝑤_1(𝑛2−1),𝑤_2(𝑛2−1),𝑤_2𝑛2,𝑤_1𝑛2) will be exterior and therefore 𝑡_𝑖(𝐺)≤𝑛−2. Hence by Theorem 3 we get 𝜂_𝑖(𝐺)≥𝑚−𝑛+2 proving that 𝜂_𝑖(𝐺)=𝑚−𝑛+2.
Now, suppose 𝑛₁,𝑛₂≥3. Then consider the following paths and cycles: 𝑄^′=𝑤₂₂,𝑤₂₁,𝑤₁₁,𝑤₁₂,𝑤₂₂,𝑄₁=𝑤₁₂,𝑤₁₃,…,𝑤_1𝑛2,𝑤_2𝑛2,…,𝑤_𝑛1𝑛2,𝑤_{𝑛1(𝑛2−1)}𝑤,…,_𝑛11,𝑤_(𝑛1−1)1…,𝑤₂₁,𝑄₂=𝑤_𝑛12,𝑤_(𝑛1−1)2,…,𝑤₃₂,𝑤₂₂,𝑤₂₃,…,𝑤_2𝑛2,𝑄_𝑖=𝑤_𝑖2,𝑤_𝑖3,…,𝑤_𝑖𝑛2foreach𝑖=3,4,5,…,𝑛₁−1. (4) Then 𝒮={𝑄^′,𝑄₁,𝑄₂,…,𝑄_(𝑛1−1)} is an edge-disjoint collection of internally disjoint induced paths and cycles of 𝐺 having all the vertices of 𝐺 as internal vertices of members of 𝒮. Therefore by Remark 4, we have 𝜂_𝑖(𝐺)=𝑚−𝑛, thus proving the theorem.

Theorem 9. If 𝐺=𝐶_𝑛1□𝐶_𝑛2, where 𝑛₁,𝑛₂≥3, then 𝜂_𝑖(𝐺)=𝑚−𝑛.

Proof. The edge set 𝐸(𝐺)={𝑤_𝑖𝑗𝑤_𝑖(𝑗+1)∶1≤𝑖≤𝑛₁;1≤𝑗≤(𝑛₂−1)}∪{𝑤_𝑖𝑛2𝑤_𝑖1∶1≤𝑖≤𝑛₁}∪{𝑤_𝑖𝑗𝑤_(𝑖+1)𝑗∶1≤𝑖≤(𝑛₁−1);1≤𝑗≤𝑛₂}∪{𝑤_𝑛1𝑗𝑤_1𝑗∶1≤𝑗≤𝑛₂}. Now, let 𝑄₁=𝑤₁₁,𝑤₁₂,𝑤₁₃,…,𝑤_1𝑛2,𝑤₁₁,𝑄₂=𝑤_1𝑛2,𝑤_2𝑛2,𝑤_3𝑛2,…,𝑤_𝑛1𝑛2,𝑤_1𝑛2,𝑄₃=𝑤_𝑛1𝑛2,𝑤_{𝑛1(𝑛2−1)},𝑤_{𝑛1(𝑛2−2)},…,𝑤_𝑛11,𝑤_𝑛1𝑛2,𝑄₄=𝑤_𝑛11,𝑤_(𝑛1−1)1,𝑤_(𝑛1−2)1,…,𝑤₁₁,𝑤_𝑛11,𝑄₅=𝑤_𝑛12,𝑤_(𝑛1−1)2,…,𝑤₃₂,𝑤₂₂,𝑤₂₃,…,𝑤_2𝑛2,𝑄^′_(𝑖−2)=𝑤_𝑖2,𝑤_𝑖3,…,𝑤_𝑖𝑛2foreach𝑖=3,4,…,𝑛₁−1. (5)
Then 𝒮={𝑄₁,𝑄₂,…,𝑄₅,𝑄^′₁,𝑄^′₂,…,𝑄^′_(𝑛1−3)} is a collection of induced paths and cycles that are edge-disjoint as well as internally disjoint where each vertex of 𝐺 is internal to a path/cycle of 𝒮. Hence it follows from Remark 4 that 𝜂_𝑖(𝐺)=𝑚−𝑛.

Theorem 10. For the graph 𝐺=𝑃_𝑛1□𝐶_𝑛2, where 𝑛₁≥2,𝑛₂≥3, 𝜂_𝑖(𝐺)=𝑚−𝑛.

Proof. We have 𝐸(𝐺)={𝑤_𝑖𝑗𝑤_𝑖(𝑗+1)∶1≤𝑖≤𝑛₁;1≤𝑗≤(𝑛₂−1)}∪{𝑤_𝑖𝑛2𝑤_𝑖1∶1≤𝑖≤𝑛₁}∪{𝑤_𝑖𝑗𝑤_(𝑖+1)𝑗∶1≤𝑖≤(𝑛₁−1);1≤𝑗≤𝑛₂}. Let 𝑄₁=𝑤₁₂,𝑤₁₁,𝑤₂₁,𝑤_2𝑛2,𝑄₂=𝑤₂₁,𝑤₂₂,𝑤₁₂,𝑤₁₃,…,𝑤_1𝑛2,𝑄₃=𝑤₂₂,𝑤₂₃,𝑤₂₄,…,𝑤_2𝑛2,𝑤_1𝑛2,𝑤₁₁,𝑄^′=𝑤_2𝑛2,𝑤_3𝑛2,…,𝑤_𝑛1𝑛2,𝑤_𝑛11,𝑄_𝑗^′′=𝑤_2𝑗,𝑤_3𝑗,𝑤_4𝑗,…,𝑤_𝑛1𝑗,𝑤_{𝑛1(𝑗+1)},𝑗=1,2,…𝑛₂−1. (6)
Then by taking 𝒮={𝑄₁,𝑄₂,𝑄₃} when 𝑛₁=2 and 𝒮={𝑄₁,𝑄₂,𝑄₃,𝑄^′,𝑄₁^′′,𝑄₂^′′,…,𝑄^′′_(𝑛2−1)} for 𝑛₁≥3 we get the collection of internally disjoint induced paths satisfying the property mentioned in Remark 4 and therefore 𝜂_𝑖(𝐺)=𝑚−𝑛.

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

Theorem 11. Let 𝐺=𝑃_𝑛1⊠𝑃_𝑛2 where 𝑛₁,𝑛₂≥2. Then one has 𝜂_𝑖(𝐺)=4,if𝑛₁=𝑛₂=2,𝑚−𝑛+1,if𝑛₁=2,𝑛₂=3,𝑚−𝑛,otherwise. (7)

Proof. Let 𝐸(𝐺)={𝑤_𝑖𝑗𝑤_𝑖(𝑗+1)∶1≤𝑖≤𝑛₁;1≤𝑗≤𝑛₂−1}∪{𝑤_𝑖𝑗𝑤_(𝑖+1)𝑗∶1≤𝑖≤𝑛₁−1;1≤𝑗≤𝑛₂}∪{𝑤_𝑖𝑗𝑤_{(𝑖+1)(𝑗+1)}∶1≤𝑖≤(𝑛₁−1);1≤𝑗≤(𝑛₂−1)}∪{𝑤_𝑖𝑗𝑤_{(𝑖+1)(𝑗−1)}∶1≤𝑖≤(𝑛₁−1);2≤𝑗≤𝑛₂}.
The cases when 𝑛₁=𝑛₂=2, 𝑛₁=2, and 𝑛₂=3 can easily be verified.
Suppose 𝑛₁=2 and 𝑛₂≥4. To prove that 𝜂_𝑖(𝐺)=𝑚−𝑛, by Remark 4, it is sufficient to obtain an edge-disjoint and internally-disjoint collection 𝒮 of induced paths and cycles such that every vertex of 𝐺 is an internal vertex of an element of 𝒮. Let us consider the following cycles and paths of 𝐺: 𝑄₁=𝑤₂₂,𝑤₂₁,𝑤₁₁,𝑤₂₂,𝑄₂=𝑤_2(𝑛2−1),𝑤_2𝑛2,𝑤_1𝑛2,𝑤_2(𝑛2−1),𝑄₃=𝑤₁₂,𝑤₂₂,𝑤₂₃,𝑤₁₂,𝑄₄=𝑤₂₃,𝑤₂₄,𝑤₁₄,𝑤₂₃,𝑄₅=𝑤_1(𝑛2−1),𝑤_2(𝑛2−1),𝑤_2(𝑛2−2),𝑤_1(𝑛2−1),𝑄₆=𝑤₁₁,𝑤₁₂,𝑤₁₃,…,𝑤_1(𝑛2−1),𝑤_1𝑛2,𝑄₇=𝑤₁₁,𝑤₁₂,𝑤₁₃,𝑤₁₄,𝑄₈=𝑤₂₃,𝑤₂₄,𝑤₂₅,…,𝑤_2(𝑛2−2). (8)
If 𝑛₂=4, let 𝒮={𝑄₁,𝑄₂,𝑄₃,𝑄₆}; if 𝑛₂=5, let 𝒮={𝑄₁,𝑄₂,𝑄₃,𝑄₄,𝑄₇}, and when 𝑛₂≥6, let 𝒮={𝑄₁,𝑄₂,𝑄₃,𝑄₅,𝑄₆,𝑄₈} thus obtaining the required collection 𝒮, proving that 𝜂_𝑖(𝐺)=𝑚−𝑛.
When 𝑛₁=𝑛₂=3, we consider the collection 𝒮={𝑄₁,𝑄₂,𝑄₃,𝑄₄,𝑄₅} where 𝑄₁=𝑤₂₂,𝑤₂₁,𝑤₁₁,𝑤₂₂,𝑄₂=𝑤₂₂,𝑤₂₃,𝑤₁₃,𝑤₂₂,𝑄₃=𝑤₂₁,𝑤₃₁,𝑤₃₂,𝑤₂₁,𝑄₄=𝑤₃₂,𝑤₃₃,𝑤₂₂,𝑤₃₂,𝑄₅=𝑤₁₁,𝑤₁₂,𝑤₁₃. (9) And for 𝑛₁≥3 and 𝑛₂≥4, we consider the following paths and cycles 𝑄₁=𝑤₂₂,𝑤₁₁,𝑤₁₂,𝑤₂₂,𝑄₂=𝑤_2(𝑛2−1),𝑤_1𝑛2,𝑤_1(𝑛2−1),𝑤_2(𝑛2−1),𝑄₃=𝑤_(𝑛1−1)2,𝑤_𝑛11,𝑤_𝑛12,𝑤_(𝑛1−1)2,𝑄₄=𝑤_{(𝑛1−1)(𝑛2−1)},𝑤_𝑛1𝑛2,𝑤_{𝑛1(𝑛2−1)},𝑤_{(𝑛1−1)(𝑛2−1)},𝑄₅=𝑤₁₁,𝑤₂₁,𝑤₃₁,…,𝑤_𝑛11,𝑄₆=𝑤_1𝑛2,𝑤_2𝑛2,𝑤_3𝑛2,…,𝑤_𝑛1𝑛2,𝑄₇=𝑤₁₂,𝑤₁₃,𝑤₁₄,…,𝑤_1(𝑛2−1),𝑄₈=𝑤_𝑛12,𝑤_𝑛13,𝑤_𝑛14,…,𝑤_{𝑛1(𝑛2−1)},𝑄^′_𝑖−1=𝑤_𝑖1,𝑤_𝑖2,𝑤_𝑖3,…,𝑤_𝑖𝑛2foreach𝑖=2,3,4,…,𝑛₁−1 (10)
to obtain the collection 𝒮={𝑄₁,𝑄₂,…,𝑄₈,𝑄^′₁,𝑄^′₂,…,𝑄^′_(𝑛1−2)} mentioned in Remark 4 so that 𝜂_𝑖(𝐺)=𝑚−𝑛 thus completing the proof of the theorem.

Theorem 12. For the graph 𝐺=𝐶_𝑛1⊠𝐶_𝑛2 where 𝑛₁,𝑛₂≥3𝜂_𝑖(𝐺)=𝑚−𝑛+1,if𝑛₁=𝑛₂=3,𝑚−𝑛,otherwise. (11)

Proof. We have 𝐸(𝐺)={𝑤_𝑖𝑗𝑤_𝑖(𝑗+1)∶1≤𝑖≤𝑛₁;1≤𝑗≤𝑛₂−1}∪{𝑤_𝑖𝑛2𝑤_𝑖1∶1≤𝑖≤𝑛₁}∪{𝑤_𝑖𝑛2𝑤_(𝑖+1)1∶1≤𝑖≤(𝑛₁−1)}∪{𝑤_𝑖1𝑤_{(𝑖+1)𝑛2}∶1≤𝑖≤(𝑛₁−1)}∪{𝑤_𝑖𝑗𝑤_(𝑖+1)𝑗∶1≤𝑖≤𝑛₁−1;1≤𝑗≤𝑛₂}∪{𝑤_𝑛1𝑗𝑤_1𝑗∶1≤𝑗≤𝑛₂}∪{𝑤_1𝑗𝑤_{𝑛1(𝑗+1)}∶1≤𝑖≤(𝑛₂−1)}∪{𝑤_𝑛1𝑗𝑤_1(𝑗+1)∶1≤𝑖≤(𝑛₂−1)}∪{𝑤_𝑖𝑗𝑤_{(𝑖+1)(𝑗+1)}∶1≤𝑖≤(𝑛₁−1);1≤𝑗≤(𝑛₂−1)}∪{𝑤_𝑖𝑗𝑤_{(𝑖+1)(𝑗−1)}∶1≤𝑖≤(𝑛₁−1);2≤𝑗≤𝑛₂}∪{𝑤₁₁𝑤_𝑛1𝑛2,𝑤_1𝑛2𝑤_𝑛11}.
When 𝑛₁=𝑛₂=3 clearly 𝐺=𝐾₉, and therefore by Theorem 5, 𝜂_𝑖(𝐺)=𝑚−𝑛+1. If 𝑛₁≥3 and 𝑛₂≥4, consider 𝑄₁=𝑤₁₁,𝑤₁₂,𝑤₁₃,…,𝑤_1𝑛2,𝑤₁₁,𝑄₂=𝑤_1𝑛2,𝑤_2𝑛2,𝑤_3𝑛2,…,𝑤_𝑛1𝑛2,𝑤_1𝑛2,𝑄₃=𝑤_𝑛1𝑛2,𝑤_{𝑛1(𝑛2−1)},𝑤_{𝑛1(𝑛2−2)},…,𝑤_𝑛11,𝑤_𝑛1𝑛2,𝑄₄=𝑤_𝑛11,𝑤_(𝑛1−1)1,𝑤_(𝑛1−2)1,…,𝑤₁₁,𝑤_𝑛11,𝑄₅=𝑤₁₁,𝑤₂₂,𝑤₂₃,𝑤₁₁,𝑄₆=𝑤₂₂,𝑤₃₂,𝑤₄₂,…,𝑤_𝑛12,𝑄₇=𝑤₂₃,𝑤₃₃,𝑤₄₃,…,𝑤_𝑛13,𝑄^′_𝑖−1=𝑤_𝑖3,𝑤_𝑖4,𝑤_𝑖5,…,𝑤_{𝑖(𝑛2−1)},𝑤_𝑖𝑛2;2≤𝑖≤𝑛₁−1. (12)
Then 𝒮={𝑄₁,𝑄₂,…,𝑄₇,𝑄^′₁,𝑄^′₂,…,𝑄^′_(𝑛1−2)} forms an internally-disjoint and edge-disjoint collection of induced paths and cycles having all the vertices of 𝐺 as internal vertices. Hence by Remark 4 we have the desired result.

Theorem 13. If 𝐺=𝑃_𝑛1⊠𝐶_𝑛2, where 𝑛₁≥2 and 𝑛₂≥3, then 𝜂_𝑖(𝐺)=𝑚−𝑛.

Proof. The edge set of 𝐺 is given by 𝐸(𝐺)={𝑤_𝑖𝑗𝑤_𝑖(𝑗+1)∶1≤𝑖≤𝑛₁;1≤𝑗≤𝑛₂−1}∪{𝑤_𝑖𝑛2𝑤_𝑖1∶1≤𝑖≤𝑛₁}∪{𝑤_𝑖𝑛2𝑤_(𝑖+1)1∶1≤𝑖≤(𝑛₁−1)}∪{𝑤_𝑖1𝑤_{(𝑖+1)𝑛2}∶1≤𝑖≤(𝑛₁−1)}∪{𝑤_𝑖𝑗𝑤_(𝑖+1)𝑗∶1≤𝑖≤𝑛₁−1;1≤𝑗≤𝑛₂}∪{𝑤_𝑖𝑗𝑤_{(𝑖+1)(𝑗+1)}∶1≤𝑖≤(𝑛₁−1);1≤𝑗≤(𝑛₂−1)}∪{𝑤_𝑖𝑗𝑤_{(𝑖+1)(𝑗−1)}∶1≤𝑖≤(𝑛₁−1);2≤𝑗≤𝑛₂}.
In this case also, we will show the existence of a collection 𝒮 of internally-disjoint and edge-disjoint induced paths and cycles in 𝐺 such that each vertex of 𝐺 is an internal vertex of an element of 𝒮 so that by Remark 4, we get 𝜂_𝑖(𝐺)=𝑚−𝑛.
When 𝑛₁=2, consider 𝑄^′=𝑤_1𝑛2,𝑤_2𝑛2,𝑤₁₁,𝑤_1𝑛2,𝑄_𝑗=𝑤_1𝑗,𝑤_2𝑗,𝑤_1(𝑗+1),𝑤_1𝑗foreach𝑗=1,2,…,𝑛₂−1 (13) to obtain the desired collection 𝒮={𝑄^′,𝑄₁,𝑄₂,…,𝑄_(𝑛2−1)} of induced cycles. Next we consider 𝑄₁=𝑤₂₁,𝑤₁₁,𝑤_1𝑛2,𝑤₂₁,𝑄₂=𝑤₂₁,𝑤₃₁,𝑤_3𝑛2,𝑤₂₁,𝑄₃=𝑤₁₂,𝑤₂₁,𝑤₃₂,𝑄₄=𝑤₁₁,𝑤₁₂,𝑤₂₂,𝑤₁₁,𝑄₅=𝑤₂₂,𝑤₂₃,𝑤₃₂,𝑤₂₂,𝑄^′₁=𝑤₁₂,𝑤₁₃,…,𝑤_1𝑛2,𝑄^′₂=𝑤₃₂,𝑤₃₃,…,𝑤_3𝑛2,𝑄^′_(𝑗−1)=𝑤_1𝑗,𝑤_2𝑗,𝑤_3𝑗,foreach𝑗=4,5,…,𝑛₂ (14)
so that when 𝑛₁=𝑛₂=3,  𝒮={𝑄₁,𝑄₂,𝑄₃,𝑄₄,𝑄₅} and when 𝑛₁=3, 𝑛₂≥4, 𝒮={𝑄₁,𝑄₂,…,𝑄₅,𝑄^′₁,𝑄^′₂,…,𝑄^′_(𝑛2−1)} form the required collection of edge-disjoint and internally-disjoint induced paths/cycles.
If 𝑛₁≥4, we consider 𝑄₁=𝑤₂₁,𝑤₁₁,𝑤_1𝑛2,𝑤₂₁,𝑄₂=𝑤_(𝑛1−1)1,𝑤_𝑛11,𝑤_𝑛1𝑛2,𝑤_(𝑛1−1)1,𝑄₃=𝑤₁₂,𝑤₂₁,𝑤₃₁,…,𝑤_(𝑛1−1)1,𝑤_𝑛12,𝑄₄=𝑤₁₁,𝑤₁₂,𝑤₂₂,𝑤₁₁,𝑄₅=𝑤_𝑛11,𝑤_𝑛12,𝑤_(𝑛1−1)2,𝑤_𝑛11,𝑄^′₁=𝑤₁₂,𝑤₁₃,…,𝑤_1𝑛2,𝑄^′₂=𝑤_𝑛12,𝑤_𝑛13,…,𝑤_𝑛1𝑛2,𝑄^′_𝑗=𝑤_1𝑗,𝑤_2𝑗,…,𝑤_𝑛1𝑗,foreach𝑗=3,4,…𝑛₂ (15) to obtain the collection 𝒮={𝑄₁,𝑄₂,…,𝑄₅,𝑄^′₁,𝑄^′₂,…,𝑄^′_𝑛2} so that by Remark 4   𝜂_𝑖(𝐺)=𝑚−𝑛, thus completing the proof.

Theorem 14. Let 𝐺=𝑃_𝑛1∘𝑃_𝑛2, where 𝑛₁,𝑛₂≥2 such that 𝑛₂≥𝑛₁. Then 𝜂_𝑖(𝐺)=4,if𝑛₁=𝑛₂=2,𝑚−𝑛,otherwise. (16)

Proof. We know that 𝐸(𝐺)={𝑤_𝑖𝑗𝑤_(𝑖+1)𝑘∶1≤𝑖≤𝑛₁−1;1≤𝑗≤𝑛₂;1≤𝑘≤𝑛₂}∪{𝑤_𝑖𝑗𝑤_𝑖(𝑗+1)∶1≤𝑖≤𝑛₁;1≤𝑗≤(𝑛₂−1)}.
When 𝑛₁=𝑛₂=2, 𝐺=𝐾₄, and therefore by Theorem 5, 𝜂_𝑖(𝐺)=4. If 𝑛₁=2 and 𝑛₂≥3, consider 𝑄_𝑗=𝑤_1𝑗,𝑤_2𝑗,𝑤_1(𝑗+1),𝑤_1𝑗where1≤𝑗≤𝑛₂,𝑄−1_′=𝑤_2(𝑛2−1),𝑤_2𝑛2,𝑤₁₁,𝑤_2(𝑛2−1). (17)
Then the collection 𝒮={𝑄₁,𝑄₂,…,𝑄_(𝑛2−1),𝑄^′} of internally-disjoint induced paths and cycles satisfies the property mentioned in Remark 4 and therefore 𝜂_𝑖(𝐺)=𝑚−𝑛.
When 𝑛₁,𝑛₂≥3, we observe that all the edges of the strong product 𝑃_𝑛1⊠𝑃_𝑛2 are included in 𝐸(𝑃_𝑛1∘𝑃_𝑛2). Therefore using the same argument as that of Theorem 11 we get 𝜂_𝑖(𝐺)=𝑚−𝑛 thus completing the proof of the theorem.

Theorem 15. For the graph 𝐺=𝐶_𝑛1∘𝐶_𝑛2, where 𝑛₁,𝑛₂≥3 with 𝑛₂≥𝑛₁ one has 𝜂_𝑖(𝐺)=𝑚−𝑛+1,if𝑛₁=𝑛₂=3,𝑚−𝑛,otherwise. (18)

Proof. For the given graph 𝐺 the edge set 𝐸(𝐺)={𝑤_𝑖𝑗𝑤_(𝑖+1)𝑘∶1≤𝑖≤(𝑛₁−1);1≤𝑗≤𝑛₂;1≤𝑘≤𝑛₂}∪{𝑤_𝑖𝑗𝑤_𝑖(𝑗+1)∶1≤𝑖≤𝑛₁;1≤𝑗≤(𝑛₂−1)}∪{𝑤_𝑖1𝑤_𝑖𝑛2∶1≤𝑖≤𝑛₁}∪{𝑤_1𝑗𝑤_𝑛1𝑘∶1≤𝑗≤𝑛₂;1≤𝑘≤𝑛₂}.
When 𝑛₁=𝑛₂=3, 𝐺=𝐾₉, and therefore by Theorem 5, 𝜂_𝑖(𝐺)=𝑚−𝑛+1. If 𝑛₁≥3 and 𝑛₂≥4, then the same set 𝒮={𝑄₁,𝑄₂,…,𝑄₇,𝑄^′₁,𝑄^′₂,…,𝑄^′_(𝑛1−2)} we have considered in Theorem 12 is sufficient to prove that 𝜂_𝑖(𝐺)=𝑚−𝑛.

Remark 16. The lexicographic product of two graphs is not commutative. Therefore the case when 𝑛₁>𝑛₂ is to be considered separately. However, using the same argument followed in the previous theorems, it is not difficult to show that 𝜂_𝑖(𝑃_𝑛1∘𝑃_𝑛2)=𝑚−𝑛+1 when 𝑛₁=3 and 𝑛₂=2 whereas in all other cases 𝜂_𝑖(𝐺)=𝑚−𝑛, whether 𝐺=𝑃_𝑛1∘𝑃_𝑛2 or 𝐺=𝐶_𝑛1∘𝐶_𝑛2.

Theorem 17. Let 𝐺=𝑃_𝑛1∘𝐶_𝑛2, where 𝑛₁≥2 and 𝑛₂≥3. Then 𝜂_𝑖(𝐺)=𝑚−𝑛.

Proof. The edge set of 𝐺 is given by 𝐸(𝐺)={𝑤_𝑖𝑗𝑤_(𝑖+1)𝑘∶1≤𝑖≤(𝑛₁−1);1≤𝑗≤𝑛₂;1≤𝑘≤𝑛₂}∪{𝑤_𝑖𝑗𝑤_𝑖(𝑗+1)∶1≤𝑖≤𝑛₁;1≤𝑗≤(𝑛₂−1)}∪{𝑤_𝑖1𝑤_𝑖𝑛2∶1≤𝑖≤𝑛₁}.
This result follows immediately from Theorem 13 as the collection of induced paths and cycles considered in the set 𝒮 in various cases there is applicable to the graph 𝐺=𝑃_𝑛1∘𝐶_𝑛2 as well.

Remark 18. It can also be proved that 𝜂_𝑖(𝐶_𝑛2∘𝑃_𝑛1)=𝑚−𝑛 by showing the existence of a collection 𝒮 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 𝐺 is an induced graphoidal decomposition 𝜓 where each member of 𝜓 is an induced path. The minimum cardinality of an IAGD of 𝐺 is called the induced acyclic graphoidal decomposition number of 𝐺, and it is denoted by 𝜂_𝑖𝑎(𝐺).

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

Theorem 19. For the graph 𝐺=𝑃_𝑛1□𝑃_𝑛2 where 𝑛₁,𝑛₂≥2, 𝜂_𝑖𝑎(𝐺)=𝑚−𝑛+2,if𝑛₁=2,𝑛₂≥2or𝑛₁=𝑛₂=3𝑚−𝑛+1,if𝑛₁=3or4,𝑛₂=4𝑚−𝑛,otherwise. (19)

Proof. If 𝑛₁=𝑛₂=2, 𝐺=𝐶₄, and so, 𝜂_𝑖𝑎(𝐺)=2=𝑚−𝑛+2. For 𝑛₁=2 and 𝑛₂≥3, let 𝑄₁=(𝑤₁₁,𝑤₂₁,𝑤₂₂,𝑤₂₃,…,𝑤_2𝑛2,𝑤_1𝑛2), let 𝑄₂=(𝑤₁₁,𝑤₁₂,𝑤₁₃,…,𝑤_1𝑛2), and let 𝑆^′={𝑤_1𝑗𝑤_2𝑗∶2≤𝑗≤(𝑛₂−1)}. Then 𝑄₁ and 𝑄₂ along with the set 𝑆^′ form an IAGD 𝜓 of 𝐺 with cardinality 𝑛₂=𝑚−𝑛+2 so that 𝜂_𝑖𝑎(𝐺)≤|𝜓|=𝑚−𝑛+2. Also for any IAGD 𝜓 of 𝐺, one of the vertices 𝑤₁₁ and 𝑤₂₁ will be exterior and the same is in the case with the vertices 𝑤_1𝑛2 and 𝑤_2𝑛2. Therefore, 𝑡_𝑖𝑎(𝐺)—the maximum number of vertices interior to any IAGD of 𝐺—is ≤(𝑛−2), and hence by Theorem 6  𝜂_𝑖𝑎(𝐺)≥𝑚−𝑛+2 proving the desired result.
The case when 𝑛₁=𝑛₂=3 also is similar to the previous case where at least two vertices of 𝐺 will be exterior to any IAGD of 𝐺, and, therefore, 𝜂_𝑖𝑎(𝐺)≥𝑚−𝑛+2 by Theorem 6. Further, the paths 𝑄₁=𝑤₁₁,𝑤₂₁,𝑤₃₁,𝑤₃₂,𝑤₃₃,𝑤₂₃,𝑤₁₃,𝑄₂=𝑤₁₁,𝑤₁₂,𝑤₁₃,𝑄₃=𝑤₂₁,𝑤₂₂,𝑤₂₃ (20) along with the edges 𝑤₁₂𝑤₂₂ and 𝑤₂₂𝑤₃₂ form an IAGD of cardinality 5=𝑚−𝑛+2, and hence it follows that 𝜂_𝑖𝑎(𝐺)=𝑚−𝑛+2.
Next assume that 𝑛₁=3 and 𝑛₂=4. Consider the induced paths 𝑄₁, 𝑄₂, and 𝑄₃, where 𝑄₁=𝑤₁₁,𝑤₂₁,𝑤₃₁,𝑤₃₂,𝑤₃₃,𝑤₃₄,𝑤₂₄,𝑤₁₄,𝑤₁₃,𝑄₂=𝑤₁₁,𝑤₁₂,𝑤₁₃,𝑤₂₃,𝑄₃=𝑤₂₁,𝑤₂₂,𝑤₂₃,𝑤₂₄. (21)
Then the collection 𝜓₁={𝑄₁,𝑄₂,𝑄₃} together with the edges 𝑒₁=𝑤₁₂𝑤₂₂, 𝑒₂=𝑤₂₂𝑤₃₂, and 𝑒₃=𝑤₂₃𝑤₃₃ forms an IAGD 𝜓 of 𝐺 such that |𝜓|=6=𝑚−𝑛+1 proving that 𝜂_𝑖𝑎≤𝑚−𝑛+1. Also it can be verified that at least one vertex of 𝐺 is exterior to any IAGD of 𝐺, and therefore 𝜂_𝑖𝑎≥𝑚−𝑛+1 by Theorem 6 thus proving the required result.
If 𝑛₁=4=𝑛₂, then as in the previous case any IAGD of 𝐺 will have at least one vertex exterior to it so that 𝜂_𝑖𝑎(𝐺)≥𝑚−𝑛+1. Also there exists an IAGD 𝜓 of cardinality 9=𝑚−𝑛+1 consisting of the induced paths 𝑄₁,𝑄₂,𝑄₃, and 𝑄₄ where 𝑄₁=𝑤₁₁,𝑤₂₁,𝑤₃₁,𝑤₄₁,𝑤₄₂,𝑤₄₃,𝑤₄₄,𝑤₃₄,𝑤₂₄,𝑤₁₄,𝑤₁₃,𝑄₂=𝑤₁₁,𝑤₁₂,𝑤₁₃,𝑤₂₃,𝑄₃=𝑤₂₁,𝑤₂₂,𝑤₂₃,𝑤₂₄,𝑄₄=𝑤₃₁,𝑤₃₂,𝑤₃₃,𝑤₃₄ (22) and the edges 𝑒₁=𝑤₁₂𝑤₂₂, 𝑒₂=𝑤₂₂𝑤₃₂, 𝑒₃=𝑤₃₂𝑤₄₂, 𝑒₄=𝑤₂₃𝑤₃₃, and 𝑒₅=𝑤₃₃𝑤₄₃ so that 𝜂_𝑖𝑎≤𝑚−𝑛+1 and hence the result.
Finally, assume that 𝑛₁≥3 and 𝑛₂≥5. Consider the paths 𝑄₁=𝑤₁₂,𝑤₁₁,𝑤₂₁,…,𝑤_𝑛11,𝑤_𝑛12𝑤,…,_𝑛1𝑛2,𝑤_{(𝑛1−1)𝑛2},𝑤_1𝑛2,𝑤_1(𝑛2−1),𝑄^′=𝑤₂₂,𝑤₁₂,𝑤₁₃,…,𝑤_1(𝑛2−1),𝑤_2(𝑛2−1),𝑄_𝑖=𝑤_𝑖1,𝑤_𝑖2,…,𝑤_𝑖𝑛2foreach𝑖=2,3,…𝑛₁−1. (23)
Then 𝒮={𝑄₁,𝑄₂,…,𝑄_(𝑛1−1),𝑄^′} is an edge-disjoint collection of internally-disjoint induced paths of 𝐺 such that every vertex of 𝐺 is an internal vertex of some path in 𝒮. Therefore by Remark 7, 𝜂_𝑖𝑎(𝐺)=𝑚−𝑛.

Theorem 20. If 𝐺=𝐶_𝑛1□𝐶_𝑛2, where 𝑛₁,𝑛₂≥3, then 𝜂_𝑖𝑎(𝐺)=𝑚−𝑛.

Proof. For the given graph we will show that there exists a collection 𝒮 of internally-disjoint and edge-disjoint induced paths such that each vertex of 𝐺 is internal to a member of 𝒮. If 𝑛₁=3, let 𝒮={𝑄₁,𝑄₂,𝑄₃,𝑄₄} where 𝑄₁=𝑤₂₁,𝑤₁₁,𝑤_1𝑛2,𝑤_3𝑛2,𝑤_3(𝑛2−1),…,𝑤₃₂,𝑄₂=𝑤₁₁,𝑤₁₂,…,𝑤_1(𝑛2−1),𝑤_2(𝑛2−1),𝑤_2𝑛2,𝑤_3𝑛2,𝑄₃=𝑤_2(𝑛2−1),𝑤_2(𝑛2−2),…,𝑤₂₁,𝑤₃₁,𝑤_3𝑛2,𝑄₄=𝑤₃₁,𝑤₃₂,𝑤₂₃. (24)
If 𝑛₁≥4, let 𝑄₁=𝑤₂₁,𝑤₁₁,𝑤_1𝑛2,𝑤_𝑛1𝑛2,𝑤_{𝑛1(𝑛2−1)},…,𝑤_𝑛12,𝑄₂=𝑤₁₁,𝑤₁₂,…,𝑤_1(𝑛2−1),𝑤_2(𝑛2−1),𝑤_2𝑛2,𝑤_3𝑛2,…,𝑤_𝑛1𝑛2,𝑄₃=𝑤_2(𝑛2−1),𝑤_2(𝑛2−2),…,𝑤₂₁,𝑤₃₁,…,𝑤_𝑛11,𝑤_𝑛1𝑛2,𝑄₄=𝑤_𝑛11,𝑤_𝑛12,𝑤_(𝑛1−1)2,𝑤_(𝑛1−1)3,…,𝑤_{(𝑛1−1)𝑛2},𝑄^′_𝑖−1=𝑤_2𝑗,𝑤_3𝑗,…,𝑤_{(𝑛1−1)𝑗}foreach𝑗=2,3,…,𝑛₂−1. (25)
Then 𝒮={𝑄₁,𝑄₂,𝑄₃,𝑄₄,𝑄^′₁,𝑄^′₂,…,𝑄^′_(𝑛2−2)} forms the desired collection of internally-disjoint and edge-disjoint induced paths such that all the vertices of 𝐺 are internal to some path in 𝒮, and by Remark 7, 𝜂_𝑖𝑎(𝐺)=𝑚−𝑛.

Theorem 21. If 𝐺=𝑃_𝑛1□𝐶_𝑛2, then 𝜂_𝑖𝑎(𝐺)=𝑚−𝑛.

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

Theorem 22. Let 𝐺=𝑃_𝑛1⊠𝑃_𝑛2 where 𝑛₁,𝑛₂≥2. Then one has 𝜂_𝑖𝑎(𝐺)=6,if𝑛₁=𝑛₂=2,𝑚−𝑛+4,otherwise. (26)

Proof. When 𝑛₁=𝑛₂=2, the result is immediate as 𝐺=𝐾₄ which has its edge set as a minimum IAGD as 𝐺 does not contain any induced paths of length greater than one.
In all other cases it can easily be observed that the vertices 𝑤₁₁, 𝑤_1𝑛1, 𝑤_𝑛11, and 𝑤_𝑛1𝑛2 cannot be made interior to any IAGD of 𝐺. Therefore 𝑡_𝑖𝑎(𝐺)≤𝑛−4, and hence by Theorem 6, 𝜂_𝑖𝑎≥𝑚−𝑛+4. Now, let 𝑄_𝑖=𝑤_𝑖1,𝑤_𝑖2,…,𝑤_𝑖𝑛2foreach𝑖=1,2,…,𝑛₁,𝑄^′=𝑤₁₁,𝑤₂₁,…,𝑤_𝑛11,𝑄^′′=𝑤_1𝑛2,𝑤_2𝑛2,…,𝑤_𝑛1𝑛2. (27) Then the collection of induced paths 𝑆₁={𝑄₁,𝑄₂,…,𝑄_𝑛1,𝑄^′,𝑄^′′} together with the edges of 𝐺 not covered by the paths in 𝑆₁ forms an IAGD 𝜓 such that |𝜓|=𝑛₁+2(𝑛₂−2)(𝑛₁−1)+2(𝑛₁−1)(𝑛₂−1)=𝑚−𝑛+4, proving that 𝜂_𝑖𝑎(𝐺)≤𝑚−𝑛+4, and this completes the proof of the theorem.

Theorem 23. If 𝐺 is the graph 𝐶_𝑛1⊠𝐶_𝑛2𝑤ℎ𝑒𝑟𝑒𝑛₁,𝑛₂≥3, then 𝜂_𝑖𝑎(𝐺)=𝑚,if𝑛₁=𝑛₂=3,𝑚−𝑛,otherwise. (28)

Proof. When 𝑛₁=𝑛₂=3, we have 𝐺=𝐾₉, and therefore 𝜂_𝑖(𝐺)=𝑚 as the edge set of 𝐺 is a minimum IAGD of 𝐺.
If 𝑛₁=3 and 𝑛₂=4, we consider the collection of induced paths 𝒮={(𝑤₁₂,𝑤₁₁,𝑤₁₄), (𝑤₂₂,𝑤₂₁,𝑤₂₄), (𝑤₃₂,𝑤₃₁,𝑤₃₄), (𝑤₂₁,𝑤₁₂,𝑤₂₃), (𝑤₃₁,𝑤₂₂,𝑤₃₃), (𝑤₂₁,𝑤₃₂,𝑤₂₃), (𝑤₁₂,𝑤₁₃,𝑤₂₄), (𝑤₂₂,𝑤₂₃,𝑤₃₄), (𝑤₃₂,𝑤₃₃,𝑤₂₄), (𝑤₁₃,𝑤₁₄,𝑤₃₁), (𝑤₂₃,𝑤₂₄,𝑤₃₁), (𝑤₃₃,𝑤₃₄,𝑤₁₁)} which satisfy the property mentioned in Remark 7, and therefore 𝜂_𝑖𝑎(𝐺)=𝑚−𝑛 in this case.
If 𝑛₁≥3 and 𝑛₂≥5, we consider the following induced paths: 𝑄_𝑖=𝑤_𝑖2,𝑤_𝑖1,𝑤_𝑖𝑛2,𝑤_{𝑖(𝑛2−1)}foreach𝑖=1,2,…,𝑛₁,𝑄^′_𝑖=𝑤_(𝑖+1)1,𝑤_𝑖2,𝑤_(𝑖+1)3foreach𝑖=1,2,…,𝑛₁,𝑄−1_∗=𝑤_(𝑛1−1)1,𝑤_𝑛12,𝑤_(𝑛1−1)3,𝑄_𝑖^′′=𝑤_{(𝑖+1)(𝑛2−2)},𝑤_{𝑖(𝑛2−1)},𝑤_{(𝑖+1)𝑛2}foreach𝑖=1,2,…,𝑛₁,𝑄−1_∗∗=𝑤_{(𝑛1−1)(𝑛2−2)},𝑤_{𝑛1(𝑛2−1)},𝑤_{(𝑛1−1)𝑛2},𝑄_𝑖^′′′=𝑤_𝑖2,𝑤_𝑖3,…,𝑤_{𝑖(𝑛2−1)}foreach𝑖=1,2,…,𝑛₁. (29) Then the collection 𝒮 of all the paths given previous constitutes an edge-disjoint collection of internally-disjoint induced paths in 𝐺 such that each vertex of 𝐺 is internal to some path in 𝒮. Therefore it follows from Remark 7 that 𝜂_𝑖𝑎=𝑚−𝑛, thus completing the proof.

Theorem 24. Let 𝐺=𝑃_𝑛1⊠𝐶_𝑛2 where 𝑛₁≥2;𝑛₂≥3. Then 𝜂_𝑖𝑎(𝐺)=𝑚−𝑛+6,if𝑛₂=3,𝑚−𝑛,otherwise. (30)

Proof. Suppose 𝑛₂=3. Then it can be observed that the vertices 𝑤₁₁,𝑤₁₂,𝑤₁₃,𝑤_𝑛11,𝑤_𝑛12 and 𝑤_𝑛13 cannot be made internal to any induced path of 𝐺. Therefore, 𝑡_𝑖𝑎(𝐺)≤(𝑛−6). Further, the three paths 𝑄₁={𝑤₁₁,𝑤₂₁,…,𝑤_𝑛11}; 𝑄₂={𝑤₁₂,𝑤₂₂,…,𝑤_𝑛12}, and 𝑄₃={𝑤₁₃,𝑤₂₃,…,𝑤_𝑛13} together with the edges of 𝐺 not belonging to the induced paths 𝑄₁, 𝑄₂, and 𝑄₃ form an IAGD of cardinality 𝑚−3(𝑛₁−1)+3=𝑚−𝑛+6 so that 𝜂_𝑖𝑎(𝐺)≤𝑚−𝑛+6 thus proving that 𝜂_𝑖𝑎(𝐺)=𝑚−𝑛+6.
Next assume that 𝑛₂≥4. If 𝑛₁=2, consider the induced paths 𝑄₁=𝑤₁₂,𝑤₁₁,𝑤_1𝑛2,𝑄₂=𝑤₂₁,𝑤₁₂,𝑤₂₃,𝑄₃=𝑤₂₂,𝑤₂₁,𝑤_2𝑛2,𝑄₄=𝑤₁₁,𝑤₂₂,𝑤₁₃,𝑄₅=𝑤₂₁,𝑤_1𝑛2,𝑤_2(𝑛2−1),𝑄₆=𝑤₁₁,𝑤_2𝑛2,𝑤_1(𝑛2−1),𝑄₇=𝑤₁₂,𝑤₁₃,…,𝑤_1𝑛2,𝑄₈=𝑤₂₂,𝑤₂₃,…,𝑤_2𝑛2 (31)
so that 𝒮={𝑄₁,𝑄₂,…,𝑄₈} is a collection of edge-disjoint and internally-disjoint induced paths such that each vertex of 𝐺 is an internal vertex of a path in 𝒮. Therefore by Remark 7, 𝜂_𝑖𝑎(𝐺)=𝑚−𝑛.
If 𝑛₁≥3 and 𝑛₂≥4, let 𝑄₁=𝑤₁₂,𝑤₁₁,𝑤_1𝑛2,𝑄₂=𝑤₂₁,𝑤₁₂,𝑤₂₃,𝑄₃=𝑤₂₁,𝑤_1𝑛2,𝑤_2(𝑛2−1),𝑄₄=𝑤_𝑛12,𝑤_𝑛11,𝑤_𝑛1𝑛2,𝑄₅=𝑤_(𝑛1−1)1,𝑤_𝑛12,𝑤_(𝑛1−1)3,𝑄₆=𝑤_(𝑛1−1)1,𝑤_𝑛1𝑛2,𝑤_{(𝑛1−1)(𝑛2−1)},𝑄₇=𝑤₁₂,𝑤₁₃,…,𝑤_1𝑛2,𝑄₈=𝑤_𝑛12,𝑤_𝑛13,…,𝑤_𝑛1𝑛2,𝑄^′_(𝑗−1)=𝑤_1𝑗,𝑤_2𝑗,…,𝑤_𝑛1𝑗foreach𝑗=2,3,…,𝑛₂. (32)
Then 𝒮={𝑄₁,𝑄₂,…,𝑄₈,𝑄^′₁,𝑄^′₂,…,𝑄^′_(𝑛2−1)} is a collection of internally-disjoint and edge-disjoint induced paths mentioned in Remark 7 so that 𝜂_𝑖𝑎(𝐺)=𝑚−𝑛.

Theorem 25. For the graph 𝐺=𝑃_𝑛1∘𝑃_𝑛2 where 𝑛₁,𝑛₂≥2 and 𝑛₂≥𝑛₁ one has 𝜂_𝑖𝑎(𝐺)=𝑚,if𝑛₁=𝑛₂=2,𝑚−𝑛+2,if𝑛₁=2,𝑛₂=3,𝑚−𝑛,otherwise. (33)

Proof. When 𝑛₁=𝑛₂=2, 𝐺=𝐾₄, and therefore 𝜂_𝑖𝑎(𝐺)=𝑚.
Next assume that 𝑛₁=2 and 𝑛₂=3. Then at least two of the vertices 𝑤₁₁, 𝑤_1𝑛2, 𝑤₂₁, 𝑤_2𝑛2 will be exterior to any IAGD of 𝐺, and hence 𝜂_𝑖𝑎≥𝑚−𝑛+2 by Theorem 6. Now, let 𝑄₁=𝑤₁₁,𝑤₂₁,𝑤₁₃,𝑄₂=𝑤₁₁,𝑤₂₂,𝑤₁₃,𝑄₃=𝑤₁₁,𝑤₂₃,𝑤₁₃,𝑄₄=𝑤₂₁,𝑤₁₂,𝑤₂₃. (34)
Then the collection 𝜓^′={𝑄₁,𝑄₂,𝑄₃,𝑄₄} along with the edges of 𝐺 not covered by the paths in 𝜓^′ forms an IAGD of cardinality 𝑚−𝑛+2, and hence it follows that 𝜂_𝑖𝑎=𝑚−𝑛+2.
If 𝑛₁=2 and 𝑛₂≥4, then we get a collection 𝒮 of induced paths given by 𝒮={𝑄₁,𝑄₂,…,𝑄₆} which satisfies the condition mentioned in Remark 7 by considering the following paths: 𝑄₁=𝑤₂₁,𝑤₁₁,𝑤₂₃,𝑄₂=𝑤₂₂,𝑤_1𝑛2,𝑤_2𝑛2,𝑄₃=𝑤₁₂,𝑤₂₁,𝑤₁₄,𝑄₄=𝑤₁₁,𝑤_2𝑛2,𝑤₁₃,𝑄₅=𝑤₁₁,𝑤₁₂,…,𝑤_1𝑛2,𝑄₆=𝑤₂₁,𝑤₂₂,…,𝑤_2𝑛2. (35) Thus we have 𝜂_𝑖𝑎(𝐺)=𝑚−𝑛. Next assume that 𝑛₁=3 and 𝑛₂≥3. In this case let us consider the paths 𝑄₁=𝑤₂₁,𝑤₁₁,𝑤_2𝑛2,𝑄₂=𝑤₂₁,𝑤_1𝑛2,𝑤_2𝑛2,𝑄₃=𝑤₂₁,𝑤₃₁,𝑤_2𝑛2,𝑄₄=𝑤₂₁,𝑤_3𝑛2,𝑤_2𝑛2,𝑄₅=𝑤₁₂,𝑤₂₁,𝑤₃₂,𝑄₆=𝑤₁₂,𝑤_2𝑛2,𝑤₃₂,𝑄^′_𝑖=𝑤_𝑖1,𝑤_𝑖2,…,𝑤_𝑖𝑛2foreach𝑖=1,2,3. (36) Then 𝒮={𝑄₁,𝑄₂,…,𝑄₆,𝑄^′₁,𝑄^′₂,𝑄^′₃} is an internally-disjoint and edge-disjoint collection of induced paths in 𝐺 such that each vertex of 𝐺 is internal to some path in 𝒮 and so by Remark 7, 𝜂_𝑖𝑎=𝑚−𝑛. Finally, when 𝑛₁,𝑛₂≥4, consider the induced paths following: 𝑄₁=𝑤₂₁,𝑤₁₁,𝑤_2𝑛2,𝑄₂=𝑤₂₁,𝑤_1𝑛2,𝑤_2𝑛2,𝑄₃=𝑤_(𝑛1−1)1,𝑤_𝑛11,𝑤_{(𝑛1−1)𝑛2},𝑄₄=𝑤_(𝑛1−1)1,𝑤_𝑛1𝑛2,𝑤_{(𝑛1−1)𝑛2},𝑄₅=𝑤₁₂,𝑤₂₁,𝑤₃₁,…,𝑤_(𝑛1−1)1,𝑤_𝑛12,𝑄₆=𝑤_1(𝑛2−1),𝑤_2𝑛2,𝑤_3𝑛2…,𝑤_{(𝑛1−1)𝑛2},𝑤_{𝑛1(𝑛2−1)},𝑄^′_𝑖=𝑤_𝑖1,𝑤_𝑖2,…,𝑤_𝑖𝑛2foreach𝑖=1,2…,𝑛₁. (37) Then 𝒮={𝑄₁,𝑄₂,…,𝑄₆,𝑄^′₁,𝑄^′₂…,𝑄^′_𝑛1} is the collection satisfying the condition mentioned in Remark 7 so that 𝜂_𝑖𝑎(𝐺)=𝑚−𝑛, and this completes the proof of the theorem.

Theorem 26. For the graph 𝐺=𝐶_𝑛1∘𝐶_𝑛2 where 𝑛₁,𝑛₂≥3 and 𝑛₂≥𝑛₁𝜂_𝑖𝑎(𝐺)=𝑚,if𝑛₁=𝑛₂=3,𝑚−𝑛,otherwise. (38)

Proof. When 𝑛₁=𝑛₂=3, 𝐺 is a complete graph on nine vertices, and therefore 𝜂_𝑖𝑎(𝐺)=𝑚. Suppose 𝑛₁≥3 and 𝑛₂≥4. Consider the induced paths 𝑄_𝑖=𝑤_𝑖2,𝑤_𝑖1,𝑤_𝑖𝑛2foreach𝑖=1,2,…,𝑛₁,𝑄^′_𝑖=𝑤_𝑖2,𝑤_𝑖3,𝑤_𝑖𝑛2foreach𝑖=1,2,…,𝑛₁,𝑄_𝑖^′′=𝑤_(𝑖+1)1,𝑤_𝑖2,𝑤_(𝑖+1)3foreach𝑖=1,2,…,𝑛₁,𝑄−1_{𝑖′′′}=𝑤_{(𝑖+1)(𝑛2−2)},𝑤_𝑖𝑛2,𝑤_{(𝑖+1)𝑛2}foreach𝑖=1,2,…,𝑛₁,𝑄−1_∗=𝑤_(𝑛1−1)1,𝑤_𝑛12,𝑤_(𝑛1−1)3,𝑄^∗∗=𝑤_(𝑛1−1)1,𝑤_𝑛1𝑛2,𝑤_(𝑛1−1)3. (39)
Then 𝒮={𝑄₁,𝑄₂,…,𝑄_𝑛1,𝑄^′₁,𝑄^′₂,…,𝑄^′_𝑛1,𝑄₁^′′,𝑄₂^′′,…,𝑄^′′_(𝑛1−1),𝑄₁^′′′,𝑄₂^′′′,…,𝑄^′′′_(𝑛1−1),𝑄^∗,𝑄^∗∗} is a collection of induced paths that are edge-disjoint and internally-disjoint such that each vertex of 𝐺 is an internal vertex of some path in 𝒮. Therefore by Remark 7, 𝜂_𝑖𝑎(𝐺)=𝑚−𝑛.

Remark 27. Since the lexicographic product of two graphs is not commutative, the case when 𝑛₁>𝑛₂ is to be considered separately. However, in this case using the same argument followed in Theorems 25 and 26 we get 𝜂_𝑖𝑎(𝑃_𝑛1∘𝑃_𝑛2)=𝑚−𝑛+4 when 𝑛₁=2 and 𝜂_𝑖𝑎(𝑃_𝑛1∘𝑃_𝑛2)=𝑚−𝑛 otherwise. Also, 𝜂_𝑖𝑎(𝐶_𝑛1∘𝐶_𝑛2)=𝑚−𝑛 when 𝑛₁>𝑛₂.

Theorem 28. Let 𝐺=𝑃_𝑛1∘𝐶_𝑛2 where 𝑛₁≥2 and 𝑛₂≥3. Then 𝜂_𝑖𝑎(𝐺)=𝑚−𝑛+6,if𝑛₂=3,𝑚−𝑛,otherwise. (40)

Proof. The result follows from Theorem 24 as 𝑃_𝑛1∘𝐶₃=𝑃_𝑛1⊠𝐶₃, and all the induced paths considered in Theorem 24 to obtain the collection 𝒮 mentioned in Remark 7 are induced paths in 𝑃_𝑛1∘𝐶_𝑛2 as well.

Remark 29. For the graph 𝐺=𝐶_𝑛1∘𝑃_𝑛2, it can easily be shown that 𝜂_𝑖𝑎(𝐺)=𝑚 if 𝑛₁=3 and 𝑛₂=2 and 𝜂_𝑖𝑎(𝐺)=𝑚−𝑛 in all other cases.

5. Scope for Further Research

Given any graph 𝐺, the lower bound for 𝜂_𝑖 as well as 𝜂_𝑖𝑎 is 𝑚−𝑛. This is because the maximum value that 𝑡_𝑖(𝐺) and 𝑡_𝑖𝑎(𝐺) can take is 𝑛. 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 𝜂_𝑖 and 𝜂_𝑖𝑎 attain the bound 𝑚−𝑛 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 𝜂_𝑖=𝑚−𝑛.(ii)Characterize the product graphs for which 𝜂_𝑖𝑎=𝑚−𝑛.(iii)Characterize the product graphs for which 𝜂_𝑖=𝜂_𝑖𝑎=𝑚−𝑛.

Acknowledgment

The authors would like to thank the two anonymous referees for their constructive and valuable suggestions that played a major role in enriching the content with more results and improving the presentation of the paper.