Algorithmic study of $d_2$-transitivity of graphs

Let $G=(V, E)$ be a graph where $V$ and $E$ are the vertex and edge sets, respectively. For two disjoint subsets $A$ and $B$ of $V$, we say $A$ \emph{dominates} $B$ if every vertex of $B$ is adjacent to at least one vertex of $A$. A vertex partition $\pi = \{V_1, V_2, \ldots, V_k\}$ of $G$ is called a \emph{transitive partition} of size $k$ if $V_i$ dominates $V_j$ for all $1\leq i<j\leq k$. In this article, we initiate the study of a generalization of transitive partition, namely \emph{$d_2$-transitive partition}. For two disjoint subsets $A$ and $B$ of $V$, we say $A$ \emph{$d_2$-dominates} $B$ if, for every vertex of $B$, there exists a vertex in $A$, such that the distance between them is at most two. A vertex partition $\pi = \{V_1, V_2, \ldots, V_k\}$ of $G$ is called a \emph{$d_2$-transitive partition} of size $k$ if $V_i$ $d_2$-dominates $V_j$ for all $1\leq i<j\leq k$. The maximum integer $k$ for which the above partition exists is called \emph{$d_2$-transitivity} of $G$, and it is denoted by $Tr_{d_2}(G)$. The \textsc{Maximum $d_2$-Transitivity Problem} is to find a $d_2$-transitive partition of a given graph with the maximum number of parts. We show that this problem can be solved in linear time for the complement of bipartite graphs and bipartite chain graphs. On the negative side, we prove that the decision version of the \textsc{Maximum $d_2$-Transitivity Problem} is NP-complete for split graphs, bipartite graphs, and star-convex bipartite graphs.


Introduction
Partitioning a graph is one of the fundamental problems in graph theory.In the partitioning problem, the objective is to partition the vertex set (or edge set) into some parts with desired properties, such as independence, minimal edges across partite sets, etc.In literature, partitioning the vertex set into certain parts so that the partite sets follow particular domination relations among themselves have been studied.Let G be a graph with V (G) as its vertex set and E(G) as its edge set.When the context is clear, V and E are used instead of V (G) and E(G).The neighbourhood of a vertex v ∈ V in a graph G = (V, E) is the set of all adjacent vertices of v and is denoted by N G (v).The degree of a vertex v in G, denoted as deg G (v), is the number of edges incident to v. A vertex v is said to dominate itself and all its neighbouring vertices.A dominating set of G = (V, E) is a subset of vertices D such that every vertex x ∈ V \ D has a neighbour y ∈ D, that is, x is dominated by some vertex y of D. For two disjoint subsets A and B of V , we say A dominates B if every vertex of B is adjacent to at least one vertex of A.
There has been a lot of research on graph partitioning problems that are based on a domination relationship between the different sets.Cockayne and Hedetniemi introduced the concept of domatic partition of a graph G = (V, E) in 1977, in which the vertex set is partitioned into k parts, say π = {V 1 , V 2 , . . ., V k }, such that each V i is a dominating set of G [CH77].The number that represents the highest possible order of a domatic partition is referred to as the domatic number of G, and it is denoted by d(G).Another similar type of partitioning problem is the Grundy partition.Christen and Selkow introduced a Grundy partition of a graph G = (V, E) in 1979 [CS79].In the Grundy partitioning problem, the vertex set is partitioned into k parts, say π = {V 1 , V 2 , . . ., V k }, such that each V i is an independent set and for all 1 ≤ i < j ≤ k, V i dominates V j .The maximum order of such a partition is called the Grundy number of G, and it is denoted by Γ(G).In 2018, J. T. Hedetniemi and S. T. Hedetniemi [HH18] introduced a transitive partition as a generalization of the Grundy partition.A transitive partition of size k is defined as a partition of the vertex set into k parts, say π = {V 1 , V 2 , . . ., V k }, such that for all 1 ≤ i < j ≤ k, V i dominates V j .The maximum order of such a transitive partition is called transitivity of G and is denoted by T r(G).Recently, in 2020, Haynes et al. generalized the idea of domatic partition as well as transitive partition and introduced the concept of upper domatic partition of a graph G, where the vertex set is partitioned into k parts, say π = {V 1 , V 2 , . . ., V k }, such that for each i, j, with 1 ≤ i < j ≤ k, either V i dominates V j or V j dominates V i or both [HHH + 20].The maximum order of such an upper domatic partition is called upper domatic number of G, and it is denoted by D(G).All these problems, domatic number [Cha94,Zel80,Zel83], Grundy number [Eff17, FGSS08, HHB82, Zak05, Zak06], transitivity [HH18, HHH + 19, PS23a, PS23b], upper domatic number [HHH + 20, SJ20] have been extensively studied both from an algorithmic and structural point of view.Clearly, a Grundy partition is a transitive partition with the additional restriction that each partite set must be independent.In a transitive partition π = {V 1 , V 2 , . . ., V k } of G, we have domination property in one direction, that is, V i dominates V j for 1 ≤ i < j ≤ k.However, in a upper domatic partition π = {V 1 , V 2 , . . ., V k } of G, for all 1 ≤ i < j ≤ k, either V i dominates V j or V j dominates V i or both.The definition of each vertex partitioning problem ensures the following inequalities for any graph G.For any graph G, In this article, we introduce a similar graph partitioning problem, namely d 2 -transitive partition, which is a generalization of transitive partition.For two disjoint subsets A and B, we say A d 2 -dominates B if, for every vertex of B, there exists a vertex in A, such that the distance between them is at most two.A d 2 -transitive partition of size k is defined as a partition of the vertex set into k parts, say Note that every transitive partition is also a d 2 -transitive partition.Therefore, for any graph G, 1 ≤ T r(G) ≤ T r d2 (G) ≤ n.Moreover, the difference between T r d2 (G) and T r(G) can be arbitrarily large.For complete bipartite graphs K t,t , T r d2 (G) = 2t whereas T r(G) = t + 1.From the complexity point of view, there are some graph classes where transitivity can be solved in linear time, but d 2transitivity is NP-complete.For example, in split graphs, the transitivity problem can be solved in linear time [PS23b], but later in this paper, we show that d 2 -transitivity is NP-complete in split graphs.There are some vertex partition parameters where the value of the parameter in a subgraph can be greater than the original graph.The upper domatic number is one such example.But in the case of a d 2 -transitive partition, T r d2 (H) ≤ T r d2 (G), for every subgraph H of G.As a consequence, for a disconnected graph, the d 2 -transitivity is equal to the maximum d 2 -transitivity among all of its components.Therefore, we focus only on connected graphs in this paper.
In this paper, we study the computational complexity of the d 2 -transitivity of graphs.The main contributions are summarized below: 1. We show that the d 2 -transitivity can be computed in linear time for the complement of bipartite graphs and bipartite chain graphs.
2. We show that the Md 2 TDP is NP-complete for split graphs, bipartite graphs, and star-convex bipartite graphs.
The rest of the paper is organized as follows.Section 2 contains basic definitions and notations that are followed throughout the article.We have also discussed the basic properties of d 2 -transitivity of graphs in this section.Section 3 describes linear-time algorithms for the complement of bipartite graphs and bipartite chain graphs.In Section 4, we show the Md 2 TDP is NP-complete in split graphs, bipartite graphs, and star-convex bipartite graphs.Finally, Section 5 concludes the article.

Definitions and notations
Let G = (V, E) be a graph with V and E as its vertex and edge sets, respectively.A graph H = (V ′ , E ′ ) is said to be a subgraph of a graph G = (V, E) if and only if V ′ ⊆ V and E ′ ⊆ E. For a subset S ⊆ V , the induced subgraph on S of G is defined as the subgraph of G whose vertex set is S and edge set consists of all of the edges in E that have both endpoints in S, and it is denoted by G The open neighbourhood of a vertex x ∈ V is the set of vertices y that are adjacent to x, and it is denoted by N G (x).The closed neighborhood of a vertex x ∈ V , denoted as N G [x], is defined by N G [x] = N G (x) ∪ {x}.For any x, y ∈ V , the distance between x and y is defined as the number of edges in the shortest path starting at x and ending at y in G, and it is denoted by d(x, y).The diameter of a graph G is defined as the greatest length of the shortest path between each pair of vertices, and it is denoted by diam(G).Let G be a graph; the square of G is a graph with the same set of vertices as G and for any x, y ∈ V , xy is an edge in the square graph if and only if d(x, y) ≤ 2. The square graph of a graph G is denoted by G 2 .A subset of S ⊆ V is said to be an independent set of G if every pair of vertices in S are non-adjacent.A subset of K ⊆ V is said to be a clique of G if every pair of vertices in K are adjacent.The cardinality of a clique of maximum size is called clique number of G, and it is denoted by ω(G).
A graph is called bipartite if its vertex set can be partitioned into two independent sets.A bipartite graph G = (X ∪Y, E) is called a bipartite chain graph if there exists an ordering of vertices of X and Y , say . Such an ordering of X and Y is called a chain ordering, and it can be computed in linear time [HK07].A bipartite graph G = (X ∪ Y, E) is said to be a starconvex bipartite graph if a star S can be defined on the vertex set X such that for every vertex y in Y , the neighbourhood of y in G induces a subtree of S. Star-convex bipartite graphs are recognizable in linear time, and associated star S can also be constructed in linear time [SBZ12].A graph G = (V, E) is said to be a split graph if V can be partitioned into an independent set S and a clique K.

Basic properties of d 2 -transitivity
In this subsection, we present some basic properties of d 2 -transitivity.First, we show the following bounds for d 2 -transitivity.
First, we show that then we are done.Otherwise, let us assume that l < k −1.Hence, there are some sets in π that do not contain any vertex from N G (x).Let V i be such a set in π and y ∈ V i d 2 -dominates x.This implies that y is adjacent to some vertex of N G (x), and hence one vertex from V i contributes one to the sum l i=1 deg(x i ).Also, if V j is a set in π that contains a vertex from N G (x), then the vertex x contributes one to the sum l i=1 deg(x i ).In either case, we have a contribution of one to the sum corresponding to each set in π, except the last set, V k .Hence, The d 2 -transitivity of paths and cycles in the following propositions is immediately due to the above bound.
Proposition 2. Let P n be a path of n vertices, and then the d 2 -transitivity of P n is given as follows: Proposition 3. Let C n be a cycle of n vertices, and then the d 2 -transitivity of C n is given as follows: Next, we characterize graphs with small d 2 -transitivity.
Lemma 4. Let G be a connected graph.Then we have the following results: Proof.Proofs for (a) and (b) are trivial, so we omit them. (c Therefore, G is either a path or a cycle.From Proposition 2, we know that P 3 and P 4 are the only two paths for which d 2 -transitivity is 3. On the other hand, from Proposition 3, we know that C 3 (equivalently K 3 ) is the only cycle for which d 2 -transitivity is 3.
We also characterize the graphs as having d 2 -transitivity equal to n, where n is the number of vertices of the graph.
Lemma 5. Let G be a graph with n vertices.Then T r d2 (G) = n if and only if diam(G) ≤ 2.
Proof.If T r d2 (G) = n, then every vertex in G d 2 -dominates every other vertex in G. Therefore, the distance between every pair of vertices is at most two.Therefore, diam(G) ≤ 2.
On the other hand, if diam(G) ≤ 2, then the distance between every pair of vertices is at most two.Hence, by putting every vertex in different sets, we get a d 2 -transitive partition of size n.Therefore, T r d2 (G) = n.
Remark 1.Many important graph classes like threshold graphs, (2K 2 , P 4 )-free graphs, connected strongly regular graphs, etc. have a diameter of at most two.The lemma 5 implies that for these graph classes, we can solve Md 2 TP trivially.

Algorithms for Md 2 TP
In this section, we find the d 2 -transitivity for the complement of bipartite graphs and bipartite chain graphs.

The complement of bipartite graphs
In this subsection, we find the d 2 -transitivity of the complement of bipartite graphs.Let G be the complement of a bipartite graph G = (X ∪ Y, E).
Proof.Let M = {e 1 , e 2 , . . ., e t } be a maximum matching in G[X ′ ∪ Y ′ ] of size t and e i = x i y i for all 1 ≤ i ≤ t.Let X t = {x 1 , x 2 , . . ., x t } and Y t = {y 1 , y 2 , . . ., y t }.Note that since G[X ′ ∪ Y ′ ] forms a complete bipartite graph, M saturates either X ′ or Y ′ .So, without loss of generality, let us assume that Y t = Y ′ .Consider a vertex partition, say π = {V 1 , V 2 , . . ., V n+m−t }, of G of size n + m − t as follows: V i = {x i , y i } for all 1 ≤ i ≤ t, and every V j contains exactly one vertex from (X \ X t ) ∪ (Y \ Y t ) for all t + 1 ≤ j ≤ n + m − t.We show that π is a d 2 -transitive partition of G.Note that every vertex of Y \ Y t is adjacent to at least one vertex of X \ X ′ in G. Therefore, every pair of vertices of (X \ X t ) ∪ (Y \ Y t ) are within distance two from each other.Therefore, V p d 2 -dominates V q for all t + 1 ≤ p < q ≤ n + m − t.Also, since every set in {V 1 , V 2 , . . ., V t } contains vertices from X and Y both, V i d 2 -dominates every set in π for all 1 ≤ i ≤ t.Therefore, π is a d 2 -transitive partition of G. Hence, T r d2 (G) ≥ n + m − t.
On the other hand, let us assume that G has a d 2 -transitive partition, say π, of size more than n + m − t.Since there are n + m − 2t vertices in (X \ X t ) ∪ (Y \ Y t ), at most n + m − 2t many sets of π contains vertices from (X \ X t ) ∪ (Y \ Y t ).Therefore, there are at least t + 1 many remaining sets in π that contain vertices from X t ∪ Y t .By the pigeonhole principle, there are at least two sets in π, say V i and V j , such that V i contains vertices only from X t and V j contains vertices only from Y t .But, in that case, neither set d 2 -dominates the other because the distance between any vertex of X t and any vertex of Y t is more than two.This contradicts the fact that π is a d 2 -transitive partition of G. So, we have T r d2 (G) ≤ n + m − t.
Therefore, the d 2 -transitivity of G is n + m − t.
From the above lemma, we can compute the d 2 -transitivity as follows: given the complement of a bipartite graph G, first, we compute the sizes of X ′ and Y ′ in linear time.The size t of the maximum matching M is clearly the minimum of these sizes.Then T r d2 (G) = n + m − t.Hence, we have the following theorem.
Theorem 7. The Maximum d 2 -Transitivity Problem can be solved in linear time for the complement of bipartite graphs.

Bipartite chain graphs
In this subsection, we find the d 2 -transitivity of a given bipartite chain graph G.To find the d 2transitivity of a given bipartite chain graph G, first we show that the d 2 -transitive partition of a graph G is the same as the transitivity of its square graph, namely G 2 .Then we show that if G is a connected bipartite chain graph, then G 2 is the complement of another bipartite chain graph H, that is, Then, by the definition of d 2 -transitive partition, V i d 2 -dominates V j for all 1 ≤ i < j ≤ k.So, for all y ∈ V j , there exists a x ∈ V i , such that d(x, y) ≤ 2 in G. From the definition of G 2 , it follows that there is an edge xy in G 2 .This implies that V i dominates V j in G 2 and hence π is a transitive partition of G 2 of size k.Therefore, T r d2 (G) ≤ T r(G 2 ).
On the other hand, let T r(G 2 ) = k and π = {V 1 , V 2 , . . ., V k } be a transitive partition of G 2 .By the definition of transitive partition, V i dominates V j for all 1 ≤ i < j ≤ k.So, for all y ∈ V j , there exists a x ∈ V i , such that xy is an edge in G 2 .From the definition of G 2 , it follows that d(x, y) ≤ 2 in G.This implies that V i d 2 -dominates V j in G, and hence π is a d 2 -transitive partition of G of size k.Therefore, T r d2 (G) ≥ T r(G 2 ).So, we have T r d2 (G) = T r(G 2 ).
Next, we show that if G is a connected bipartite chain graph, then G 2 = H, where H is another bipartite chain graph.
Lemma 9.If G is a connected bipartite chain graph, then G 2 is the complement of another bipartite chain graph, H.
Proof.Let G = (X ∪Y, E) be a connected bipartite chain graph with chain ordering σ X = (x 1 , x 2 , . . ., x m ) and σ Y = (y 1 , y 2 , . . ., y n ) such that N (x m ) ⊆ N (x m−1 ) ⊆ . . .⊆ N (x 1 ) and N (y n ) ⊆ N (y n−1 ) ⊆ . . .⊆ N (y 1 ).Note that every pair of vertices of X are at a distance of two as N (y 1 ) = X.Hence, X forms a clique in G 2 .Similarly, Y forms another clique in G 2 .Moreover, for any two vertices x ∈ X and y ∈ Y , the distance between x and y is either 1 or greater or equal to 3. Hence, all the edges of G are present in G 2 , and there is no other edge in G 2 .Now, the complement of G 2 is a bipartite graph.Let H be the bipartite graph such that H = G 2 .Next, we prove that H is a bipartite chain graph.Since the edges across X and Y are the same in G and G 2 , N (x m ) ⊆ N (x m−1 ) ⊆ . . .⊆ N (x 1 ) and N (y n ) ⊆ N (y n−1 ) ⊆ . . .⊆ N (y 1 ) in G 2 as well.Therefore, in H, we have N (x 1 ) ⊆ N (x 2 ) ⊆ . . .⊆ N (x m ) and N (y 1 ) ⊆ N (y 2 ) ⊆ . . .⊆ N (y n ).Therefore, vertices of H have chain ordering, and hence H is a bipartite chain graph.
From the Lemma 8 and Lemma 9, we know that finding the d 2 -transitivity of a bipartite chain graph G is the same as finding the transitivity of the complement of another bipartite chain graph H. From the proof of Lemma 9, it follows that we can obtain H by taking the complement of G and then deleting edges inside X and Y .Note that H contains some isolated vertices as N (x m ) = N (y n ) = ∅.Hence, we can compute H in linear time.Moreover, the transitivity of the complement of a bipartite chain graph can be computed in linear time [PS23b].Therefore, we have the following theorem: Theorem 10.The Maximum d 2 -Transitivity Problem can be solved in linear time for bipartite chain graphs.

NP-completeness of Md 2 TDP
In this section, we present three NP-completeness results for Maximum d 2 -Transitivity Decision Problem, namely in split graphs, bipartite graphs, and star-convex bipartite graphs.

Split graphs
In this subsection, we show that the Maximum d 2 -Transitivity Decision Problem (Md 2 TDP) is NP-complete for split graphs, an important subclass of chordal graphs.
Theorem 11.The Maximum d 2 -Transitivity Decision Problem is NP-complete for split graphs.
Proof.Given a vertex partition π = {V 1 , V 2 , . . ., V k } of a given split graph, we can verify in polynomial time whether π is a d 2 -transitive partition of that graph or not.Hence, the Maximum d 2 -Transitivity Decision Problem (Md 2 TDP) is in NP.To prove that this problem is NP-hard, we show a polynomial time reduction from the Maximum Transitivity Decision Problem in general graphs, which is known to be NP-complete [HHM + 04].The reduction is as follows: given an instance of the Maximum Transitivity Decision Problem, that is, a graph G = (V, E) and an integer k, we first subdivide each edge of G exactly once.Let u i be the subdivision vertex corresponding to the edge e i ∈ E for every 1 ≤ i ≤ m.Then we put edges between every pair of subdivision vertices.Let the new graph be G ′ = (V ′ , E ′ ).Clearly, G ′ is a split graph having n + m vertices and m 2 +3m 2 edges.The construction of G ′ is illustrated in Figure 1.Next, we prove the following claim.
Finally, every vertex of G ′ , other than the subdivision vertices, is adjacent to at least one subdivision vertex in G ′ .Hence, . ., V pt be the sets in π that do not contain any subdivision vertex, where p 1 < p 2 < . . .< p t .Since there are m subdivision vertices in G ′ , there exist at least k such sets in π.Therefore, t ≥ k.Let us consider the vertex partition, say and V ′ 1 contains the rest of the vertices of G. Since V pi d 2 -dominates V pj in G ′ , every vertex of V pj must be adjacent to some vertex of V pi in G. Therefore, π ′ is a transitive partition of G of size k.
From the above claim, it follows that the Maximum d 2 -Transitivity Decision Problem is NPcomplete for split graphs.

Bipartite graphs
In this subsection, we show that the Maximum d 2 -Transitivity Decision Problem is NP-complete for bipartite graphs as well.
Theorem 13.The Maximum d 2 -Transitivity Decision Problem is NP-complete for bipartite graphs.
Proof.Given a vertex partition π = {V 1 , V 2 , . . ., V k } of a bipartite graph, we can verify in polynomial time whether π is a d 2 -transitive partition of that graph or not.Hence, the Maximum d 2 -Transitivity Decision Problem is in NP.To prove that this problem is NP-hard, we show a polynomial time reduction from the Maximum Transitivity Decision Problem in general graphs, which is known to be NPcomplete [HHM + 04].The reduction is as follows: given an instance of the Maximum Transitivity Decision Problem, that is, a graph G = (V, E) and an integer k, we construct another graph G ′ = (V ′ , E ′ ).Let V = {v 1 , v 2 , . . ., v n } and E = {e 1 , e 2 , . . ., e m } vertices and edges of G, respectively.For each vertex v i ∈ V , we take two vertices v 1 i and v 2 i in V ′ , and for each edge e j ∈ E, we take two vertices u 1 j and u 2 j in V ′ .Hence, Next, we add edges between v 1 i and u 1 j if e j is incident on v i in G. Similarly, we add edges between v 2 i and u 2 j if e j is incident on v i in G. Finally, we add edges between every vertex of U 1 and every vertex of U 2 ; that is, U 1 and U 2 induce a complete bipartite graph in G ′ .Clearly, G ′ is a bipartite graph with V 1 ∪ U 2 and V 2 ∪ U 1 forming the bipartition, and G ′ has 2(m + n) vertices and m 2 + 4m edges.The construction is illustrated in Figure 2. Next, we prove the following claim.
Note that every vertex of U 1 is at a distance of two from every vertex of V 2 , and every vertex of U 2 is at a distance of two from every vertex of V 1 .Therefore, every W . ., W pt be the sets in π that do not contain any vertex from U 1 ∪U 2 , where p 1 < p 2 < . . .< p t .Since there are 2m vertices in U 1 ∪ U 2 , there exist at least k such sets in π.Therefore, t ≥ k.Note that since the distance between any vertex of V 1 and any vertex of V 2 is more than two, if W pt contains a vertex from V 1 (or V 2 ), then W pi for all i < t contains at least one vertex from V 1 (respectively from V 2 ).Let us assume that W pt contains vertices from V 1 .Consider a vertex partition, say π and W ′ 1 contains every other vertex of G.Note that if v 1 r ∈ W pj for some 2 ≤ j ≤ k, then every W pi , with i < j, contains at least one vertex, say v 1 s , such that d(v 1 s , v 1 r ) = 2 in G ′ .By the construction of G ′ , it follows that v s and v r are adjacent in G. Hence, W ′ i dominates W ′ j for all 1 ≤ i < j ≤ k.Therefore, π ′ is a transitive partition of G of size k.For the case where W pt does not contain any vertex from V 1 , that is, it contains vertices from V 2 only, we can argue in a similar way by considering vertices from V 2 and show that π ′ is a transitive partition of G of size k.
The above claim shows that the Maximum d 2 -Transitivity Decision Problem is NP-complete for bipartite graphs.

Star-convex bipartite graphs
In this subsection, we strengthen the NP-completeness result by showing that the Maximum d 2 -Transitivity Decision Problem remains NP-complete for star-convex bipartite graphs.
Theorem 15.The Maximum d 2 -Transitivity Decision Problem is NP-complete for star-convex bipartite graphs.
Proof.Given a vertex partition π = {V 1 , V 2 , . . ., V k } of a star-convex bipartite graph, we can check in polynomial time whether π is a d 2 -transitive partition of that graph or not.Hence, the Maximum d 2 -Transitivity Decision Problem is in NP.To prove that this problem is NP-hard, we show a polynomial time reduction from the Maximum d 2 -Transitivity Decision Problem in bipartite graphs.The reduction is as follows: given a bipartite graph G = (X ∪ Y, E) and an integer k, we add a vertex x to X ∪ Y and join x to every vertex of Y to construct the new graph G ′ = (X ′ ∪ Y ′ , E ′ ), where X ′ = X ∪ {x} and Y ′ = Y and E ′ = E ∪ {xy|y ∈ Y }.We can define a star S on the set X ′ with x as the centre vertex.Now, if we take any vertex y ∈ Y ′ , N G ′ (y) induces a substar of S. Hence, the constructed graph G ′ is a star-convex bipartite graph.The construction is illustrated in Figure 3. Next, we prove the following claim.Proof.Let {V 1 , V 2 , . . ., V k } be a d 2 -transitive partition of G of size k.Then, clearly, {V 1 , V 2 , . . ., V k , {x}} is a d 2 -transitive partition of G ′ of size k + 1 as {x} is d 2 -dominated by every V i in G ′ .Conversely, let π ′ = {V ′ 1 , V ′ 2 , . . ., V ′ k+1 } be a d 2 -transitive partition of G ′ of size k + 1.If x ∈ V ′ 1 , then consider the partition π = {V 1 , V 2 , . . ., V k } of G, where V 1 = V ′ 2 ∪ (V ′ 1 \ {x}) and V i = V ′ i+1 for all 2 ≤ i ≤ k.Since V ′ 2 d 2 -dominates V j for all j ≥ 3 in G ′ , V 1 also d 2 -dominates every set of π in G. Hence, π is a d 2 -transitive partition of G of size k.On the other hand, if x ∈ V ′ i where i ̸ = 1, then consider the partition π = {V 1 , V 2 , . . ., V k } of G, where V 1 = V ′ 1 ∪ (V ′ i \ {x}), V p = V ′ p , 2 ≤ p ≤ i − 1 and V q = V ′ q+1 , i ≤ q ≤ k.Since V ′ 1 d 2 -dominates every set of π ′ in G ′ , V 1 also d 2 -dominates every set of π in G. Hence, π is a d 2 -transitive partition of G of size k.

G G
The above claim shows that Maximum d 2 -Transitivity Decision Problem is NP-complete for star-convex bipartite graphs.

Conclusion
In this paper, we have introduced the notion of d 2 -transitivity in graphs, which is a generalization of transitivity.First, we have shown some basic properties for d 2 -transitivity.We have also proved the d 2 -transitivity can be computed in linear time for the complement of bipartite graphs and bipartite chain graphs.On the negative side, we have shown that the Maximum d 2 -Transitivity Decision Problem is NP-complete for split graphs, bipartite graphs, and star-convex bipartite graphs.It would be interesting to investigate the complexity status of this problem in other graph classes.Designing an approximation algorithm for this problem would be another challenging open problem.
The maximum order of such a d 2 -transitive partition is called d 2 -transitivity of G and is denoted by T r d2 (G).The Maximum d 2 -Transitivity Problem and its corresponding decision version are defined as follows: Maximum d 2 -Transitivity Problem(Md 2 TP) Instance: A graph G = (V, E) Solution: An d 2 -transitive partition of G Measure: Order of the d 2 transitive partition of G Maximum d 2 -Transitivity Decision Problem(Md 2 TDP) Instance: A graph G = (V, E), integer k Question: Does G have a d 2 -transitive partition of order at least k?