arXiv subjects

Kamal Santra

Publications and source records attributed to Kamal Santra.

18 recordsLinked to original sources

arXiv2026

Maximum Edge Open Packing in Permutation, Interval, and Well-Partitioned Chordal Graphs

Edge open packing is a relaxation of induced matching in which the selected edges may induce disjoint stars. We study the \textsc{Maximum Edge Open Packing} problem on permutation graphs, interval graphs, and well-partitioned chordal graphs. For the first two classes, we introduce an oriented star-conflict graph whose vertices are ordered edges. We prove that its compatibility graph admits a natural transitive orientation: a product-order orientation for permutation graphs and a left-to-right orientation for interval graphs. In each case, a maximum edge open packing is obtained from a maximum clique, equivalently a longest directed path, in the compatibility graph. Given the corresponding representation, both algorithms run in $O(n^2+m^2)\leq O(n^4)$ time, where $n=|V(G)|$ and $m=|E(G)|$. For well-partitioned chordal graphs, we give a dynamic program over a partition tree. Its states use the fact that the endpoint set of an edge open packing meets each clique bag in at most two vertices. Given a partition-tree representation, the edge open packing number is computed in $O(n^4)$ time, and an optimal packing can be reconstructed within the same time bound.

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arXiv2026

An Improved Upper Bound for the Strong Odd Chromatic Number of Planar Graphs

A proper coloring of a graph is called a strong odd coloring if, for every vertex $v$ and every color appearing in the open neighborhood of $v$, that color appears an odd number of times in $N(v)$. The corresponding minimum number of colors is the strong odd chromatic number, denoted by $χ_{\mathrm{so}}(G)$. Caro et al.~\cite{CaroPetrusevskiSkrekovskiTuzaStrongOdd} proved that every planar graph has strong odd chromatic number at most $388$. Manattu et al.~\cite{ManattuVinayLakshmanan2026} later constructed planar graphs with strong odd chromatic number $17$ and asked whether larger values are possible and whether the upper bound $388$ can be improved. We address these questions as follows. First, we improve the general upper bound by proving that every planar graph $G$ satisfies $χ_{\mathrm{so}}(G)\le 368$. This follows by improving the auxiliary proper facially odd coloring bound for loopless $2$-connected plane multigraphs from $97$ colors to $92$ colors and combining this with the reduction of Caro et al. and the Four Color Theorem. Second, we give a different explicit planar construction with $χ_{\mathrm{so}}(G)=20$, together with a self-contained proof of the exact value. We emphasize that Goetze et al.~\cite{GoetzeKluteKnauerParadaPenaUeckerdt2025} had already posted an arXiv preprint in May 2025 containing a planar example with strong odd chromatic number $20$. Thus our construction is not a priority claim for the value $20$, but rather an independent and fully verified construction whose value exceeds $17$, the value that motivated Problem~1 of Manattu et al.

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arXiv2026

Edge open packing on subclasses of chordal graphs

Packing problems in graphs are fundamental in combinatorial optimization and arise naturally in applications such as resource allocation, scheduling, and communication networks. A classical example is the \emph{induced matching} problem, where one seeks a set of edges whose endpoints induce a matching. In 2022, Chelladurai et al. introduced the notion of \emph{edge open packing}, which can be viewed as a relaxation of induced matching: instead of forcing the selected edges to induce only isolated edges, edge open packing allows them to induce disjoint stars. For a graph $G=(V,E)$, two edges $e_1,e_2\in E(G)$ are said to have a common edge if there exists an edge $e\in E(G)\setminus\{e_1,e_2\}$ joining an endpoint of $e_1$ to an endpoint of $e_2$. A set $D\subseteq E(G)$ is an \emph{edge open packing set} if no two edges in $D$ have a common edge, and the maximum cardinality of such a set is the \emph{edge open packing number} $ρ_e^o(G)$. The corresponding optimization problem is the \textsc{Maximum Edge Open Packing Problem}. In this paper, we study the computational complexity of the \textsc{Maximum Edge Open Packing Problem}. Motivated by an open question posed by Brešar and Samadi concerning chordal graphs, we investigate the problem on three subclasses of chordal graphs. We give an $O(n^2)$-time algorithm for proper interval graphs, an $O(n+m)$-time algorithm for block graphs, where $n=|V(G)|$ and $m=|E(G)|$, and an $O(n^3)$-time algorithm for split graphs. These results provide partial answers to the open question and contribute to the algorithmic understanding of edge packing parameters in chordal graph classes.

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arXiv2026

On the Complexity and Algorithms for the Upper Domatic Number of Graphs

Let $G$ be a graph. For two disjoint vertex sets $A,B\subseteq V(G)$, we say that $A$ dominates $B$ if every vertex of $B$ has a neighbour in $A$. An upper domatic partition of $G$ is a partition $π=\{V_1,V_2,\ldots,V_k\}$ of $V(G)$ such that, for every two distinct parts $V_i$ and $V_j$, either $V_i$ dominates $V_j$, or $V_j$ dominates $V_i$, or both. The maximum order of such a partition is the upper domatic number of $G$, denoted by $D(G)$. In this paper, we study the computational complexity of the upper domatic number from both hardness and algorithmic perspectives. Motivated by the complexity questions raised in Phillips's thesis, we first prove that deciding whether $D(G)\geq k$ is NP-complete when $k$ is part of the input. Our reduction is from \textsc{Clique}, and it also shows NP-completeness for connected graphs of diameter two. On the positive side, we give exact algorithms for several graph classes. We prove that, for cographs, $D(G)=\Tr(G)$, and we provide an $O(n^3)$-time cotree dynamic program. For unicyclic graphs, we use the known equality $D(G)=\Tr(G)$. By deleting an edge of the unique cycle, we reduce the computation of $\Tr(G)$ to the transitivity of a tree and one additional decision problem testing whether adding back the deleted edge increases the transitivity by one. This yields an $O(n^3)$-time algorithm for computing the transitivity, and hence the upper domatic number, of unicyclic graphs. We further prove that $D(G)=\Tr(G)$ for complements of bipartite graphs, which yields a linear-time algorithm for complements of bipartite chain graphs. Finally, we show that $D(G)=\Tr(G)$ for split graphs and obtain a linear-time algorithm for this class.

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arXiv2026

Roman Domination on Circular-Convex, Triad-Convex Bipartite Graphs and $P_4$-Tidy Graphs

The Roman Domination Problem (RDP) on a graph $G=(V,E)$ asks for a labeling function $f:V\rightarrow\{0,1,2\}$ such that every vertex assigned value $0$ is adjacent to a vertex assigned value $2$. The objective is to minimize the total weight $\sum_{v\in V} f(v)$; this minimum value is the Roman domination number of $G$, denoted by $γ_R(G)$. In this paper, we study RDP on graph classes motivated by convexity and induced-$P_4$ structure. First, we consider circular-convex bipartite graphs, a natural superclass of convex bipartite graphs, where RDP is already known to be polynomial-time solvable. Assuming that a circular-convex representation is given, we compute $γ_R(G)$ in $O(n^6)$ time by cutting the circular order, separating interval and wrap-around vertices, and branching over at most two wrap-around vertices assigned value $2$. Second, we study triad-convex bipartite graphs, a restricted subclass of tree-convex bipartite graphs whose convexity tree is a subdivision of $K_{1,3}$. Although RDP is hard on broader tree-convex subclasses such as star-convex and comb-convex bipartite graphs, we show that $γ_R(G)$ can be computed in $O(n^7)$ time on triad-convex bipartite graphs. Finally, we study $P_4$-tidy graphs, which properly extend cographs. Using the Giakoumakis et al. structural decomposition of $P_4$-tidy graphs, we give a direct, exact algorithm that computes $γ_R(G)$ in $O(n+m)$ time. These results extend the algorithmic boundary of Roman domination on convexity-based bipartite graphs and $P_4$-structured graph classes.

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arXiv2026

Roman-Type Domination on Convex and Chordal Bipartite Graphs: Algorithms and Hardness

Roman domination and its variants form an important family of domination-type graph parameters motivated by protection, fault tolerance, and resource allocation. A Roman dominating function of a graph $G$ is a function $f:V(G)\rightarrow\{0,1,2\}$ such that every vertex $v$ with $f(v)=0$ has a neighbour $u$ with $f(u)=2$. The weight of $f$ is $w(f)=\sum_{v\in V(G)}f(v)$, and the minimum weight of a Roman dominating function of $G$ is the Roman domination number, denoted by $γ_R(G)$. In this paper, we study four variants of Roman domination on two natural subclasses of bipartite graphs, namely convex bipartite graphs and chordal bipartite graphs. On the positive side, we develop a unified left-to-right dynamic programming framework for Roman-$\{2\}$ domination, double Roman domination, perfect Roman domination, and unique response Roman domination on convex bipartite graphs. The algorithms exploit the interval structure of one bipartition class and represent all unfinished requirements using a constant number of boundary indices. Consequently, each of the four parameters can be computed in $O(n^6)$ time, where $n=|V(G)|$. On the negative side, we prove that Roman-$\{2\}$ domination, perfect Roman domination, and unique response Roman domination remain NP-complete on chordal bipartite graphs. These results establish a clear algorithmic separation between convex bipartite graphs, where the interval ordering yields polynomial-time solvability, and the broader class of chordal bipartite graphs, where several Roman-type domination problems remain computationally intractable.

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arXiv2026

Roman domination on subclasses of bipartite graphs

The Roman Domination Problem (RDP) on a simple, finite, undirected graph $G=(V,E)$ asks for a labeling function $f:V\rightarrow\{0,1,2\}$ such that every vertex assigned value $0$ is adjacent to at least one vertex assigned value $2$. The objective is to minimize the total weight $\sum_{v\in V} f(v)$, and this minimum value is called the Roman domination number of $G$, denoted by $γ_R(G)$. Since the RDP is NP-complete for bipartite graphs, a natural direction is to study its complexity on restricted subclasses of bipartite graphs. The problem remains NP-complete even under strong structural restrictions, such as star-convex and comb-convex bipartite graphs. Therefore, identifying the borderline subclasses where the problem changes from NP-complete to polynomial-time solvable remains an important challenge. In this paper, we investigate the RDP on convex bipartite graphs and on their superclass, chordal bipartite graphs. First, we present a dynamic programming algorithm for convex bipartite graphs. The algorithm uses the interval ordering of one bipartition class and keeps a compact boundary state, which is sufficient to control the domination requirements of both processed and future vertices. This gives an $O(n^3)$-time algorithm for computing $γ_R(G)$ on an $n$ vertex convex bipartite graph. In contrast, we prove that the decision version of the RDP is NP-complete on chordal bipartite graphs by a polynomial reduction from \textsc{Dominating Set} on chordal bipartite graphs. Thus, our results show a clear separation between the tractability of convex bipartite graphs and the hardness of the larger chordal bipartite class.

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arXiv2026

Algorithms for the Maximum Edge Open Packing Problem

Packing problems form a central theme in graph theory, owing to their relevance in modeling conflict-free resource allocation, network design, and communication constraints. Motivated by applications in wireless networks where each device can participate in at most one communication at a time and simultaneous links must avoid interference we consider a generalization of induced matching known as \emph{edge open packing}. Two edges of a graph are said to conflict if a third edge connects one endpoint of each; an \emph{edge open packing set} is a set of edges containing no such conflicting pair. The largest cardinality of such a set is the \emph{edge open packing number} of a graph. In this work, we study the computational complexity of the Maximum Edge Open Packing Problem. We give a polynomial-time algorithm for the problem in \emph{distance-hereditary graphs}, exploiting their canonical decomposition via twin-set interactions. We further show that the problem remains polynomial-time solvable on \emph{biconvex bipartite graphs}, thereby identifying a tractable subclass within bipartite graphs, in contrast to the known NP-hardness of the problem on Eulerian bipartite graphs. Finally, we initiate the parameterized complexity study of the problem and present a fixed-parameter tractable algorithm for \emph{chordal graphs}, parameterized by the clique number $ω$, running in $O(2^ω\cdot\mathrm{poly}(n))$ time.

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arXiv2025

Vertex-edge domination on subclasses of bipartite graphs

Given a simple undirected graph $G = (V, E)$, the open neighbourhood of a vertex $v \in V$ is defined as $N_G(v) = \{u \in V \mid uv \in E\}$, and the closed neighbourhood as $N_G[v] = N_G(v) \cup \{v\}$. A subset $D \subseteq V$ is called a vertex-edge dominating set if, for every edge $uv \in E$, at least one vertex from $D$ appears in $N_G[u] \cup N_G[v]$; that is, $\vert (N_G[u] \cup N_G[v]) \cap D\vert \geq 1$. Intuitively, a vertex-edge dominating set ensures that every edge, as well as all edges incident to either of its endpoints, is dominated by at least one vertex from the set. The \textsc{Min-VEDS} problem asks for a vertex-edge dominating set of minimum size in a given graph. This problem is known to be NP-complete even for bipartite graphs. In this paper, we strengthen this hardness result by proving that the problem remains NP-complete for two specific subclasses of bipartite graphs: star-convex and comb-convex bipartite graphs. For a graph $G$ on $n$ vertices, it is known that the \textsc{Min-VEDS} problem cannot be approximated within a factor of $(1 - ε)\ln |V|$ for any $ε> 0$, unless $\text{NP} \subseteq \text{DTIME}(|V|^{O(\log \log |V|)})$. We also prove that this inapproximability result holds even for star-convex and comb-convex bipartite graphs. On the positive side, we present a polynomial-time algorithm for computing a minimum vertex-edge dominating set in convex bipartite graphs. A polynomial-time algorithm for this graph class was also proposed by B{ü}y{ü}k{ç}olak et al.~\cite{buyukccolak2025linear}, but we show that their algorithm has certain flaws by providing instances where it fails to produce an optimal solution. We address this issue by presenting a modified algorithm that correctly computes an optimal solution.

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arXiv2025

Improved upper bounds on color reversal by local inversions

We study the problem of color reversal in bicolored graphs under local inversions. A \emph{bicoloration} of a graph $G=(V,E)$ is a mapping $β: V \to \{-1,1\}$. A \emph{local inversion} at a vertex $v \in V$ consists of reversing the colors of all neighbors of $v$ and replacing the subgraph induced by these neighbors with its complement, while leaving $v$ and the rest of $G$ unchanged. Sabidussi (Discrete Mathematics, 1987) showed that any bicolored graph on $n$ vertices without isolated vertices can be color-reversed (that is, all vertex colors flipped while preserving the underlying graph) in at most $6n+3$ local inversions, and that any bicolored graph can be transformed into another bicolored graph on the same underlying graph in at most $9n$ local inversions. We improve both bounds: we prove that the first task can be accomplished in at most $4n-3$ local inversions, and the second in at most $ \left \lfloor \frac{11n-3}{2} \right \rfloor$ local inversions. Furthermore, we show that for stars and complete graphs, color reversal can be performed with at most $3n$ local inversions.

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arXiv2025

Edge open packing: further characterizations

Let $G=(V, E)$ be a graph where $V(G)$ and $E(G)$ are the vertex and edge sets, respectively. In a graph $G$, two edges $e_1, e_2\in E(G)$ are said to have \emph{common edge} $e\neq e_1, e_2$ if $e$ joins an endpoint of $e_1$ to an endpoint of $e_2$ in $G$. A subset $D\subseteq E(G)$ is called an \emph{edge open packing set} in $G$ if no two edges in $D$ share a common edge in $G$, and the largest size of such a set in $G$ is known as \emph{edge open packing number}, represented by $ρ_{e}^o(G)$. In the introductory paper (Chelladurai et al. (2022)), necessary and sufficient conditions for $ρ_{e}^o(G)=1, 2$ were provided, and the graphs $G$ with $ρ_{e}^o(G)\in \{m-2, m-1, m\}$ were characterized, where $m$ is the number of edges of $G$. In this paper, we further characterize the graphs $G$. First, we show necessary and sufficient conditions for $ρ_{e}^o(G)=t$, for any integer $t\geq 3$. Finally, we characterize the graphs with $ρ_{e}^o(G)=m-3$.

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arXiv2025

On total 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 that $A$ \emph{dominates} $B$ if every vertex of $B$ is adjacent to at least one vertex of $A$. A vertex partition $π= \{V_1, V_2, \ldots, V_k\}$ 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 study a variation of the transitive partition, namely the \emph{total transitive partition}. The total transitivity $Tr_t(G)$ is defined as the maximum order of a vertex partition $π= \{V_1, V_2, \ldots, V_k\}$ of $G$ obtained by repeatedly removing a total dominating set from $G$ until no vertices remain. Thus, $V_1$ is a total dominating set of $G$, $V_2$ is a total dominating set of the graph $G_1 = G - V_1$, and, for $2 \leq i \leq k - 1$, $V_{i+1}$ is a total dominating set in the graph $G_i = G - \bigcup_{j=1}^i V_j$. A vertex partition of order $Tr_t(G)$ is called a $Tr_t$-partition. The \textsc{Maximum Total Transitivity Problem} is to find a total transitive partition of a given graph with the maximum number of parts. First, we characterize split graphs with total transitivity equal to $1$ and $ω(G) - 1$. Moreover, for a split graph $G$ and $1 \leq p \leq ω(G) - 1$, we provide necessary conditions for $Tr_t(G) = p$. Furthermore, we show that the decision version of this problem is NP-complete for bipartite graphs. On the positive side, we prove that this problem can be solved in linear time for bipartite chain graphs. Finally, we design a polynomial-time algorithm to solve the \textsc{Maximum Total Transitivity Problem} in trees.

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arXiv2024

Tournament 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 $A$ \textit{dominates} $B$ if every vertex of $B$ is adjacent to at least one vertex of $A$ in $G$. A vertex partition $π= \{V_1, V_2, \ldots, V_k\}$ of $G$ is called a \emph{transitive partition} of size $k$ if $V_i$ dominates $V_j$ for all $1\leq i<j\leq k$. A vertex partition $π= \{V_1, V_2, \ldots, V_k\}$ of $G$ is called a \emph{tournament transitive partition} of size $k$ if $V_i$ dominates $V_j$ for all $1\leq i<j\leq k$ and $V_j$ does not dominate $V_i$ for $i<j$. The maximum integer $k$ for which the above partition exists is called \emph{tournament transitivity} of $G$, and it is denoted by $TTr(G)$. The \textsc{Maximum Tournament Transitivity Problem} is to find a tournament transitive partition of a given graph with the maximum number of parts. In this article, we study this variation of transitive partition from a structure and algorithmic point of view. We show that the decision version of this problem is NP-complete for chordal graphs (connected), perfect elimination bipartite graphs (disconnected) and doubly chordal graphs (disconnected). On the positive side, we prove that this problem can be solved in polynomial time for trees. Furthermore, we characterize \textup{Type-I BCG} with equal transitivity and tournament transitivity and find some sufficient conditions under which the above two parameters are equal for a \textup{Type-II BCG}. Finally, we show that for \textup{Type-III BCG}, these two parameters are never equal.

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arXiv2023

Algorithmic study on $2$-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 $π= \{V_1, V_2, \ldots, V_k\}$ 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 study a variation of transitive partition, namely \emph{$2$-transitive partition}. For two disjoint subsets $A$ and $B$ of $V$, we say $A$ \emph{$2$-dominates} $B$ if every vertex of $B$ is adjacent to at least two vertices of $A$. A vertex partition $π= \{V_1, V_2, \ldots, V_k\}$ of $G$ is called a \emph{$2$-transitive partition} of size $k$ if $V_i$ $2$-dominates $V_j$ for all $1\leq i<j\leq k$. The \textsc{Maximum $2$-Transitivity Problem} is to find a $2$-transitive partition of a given graph with the maximum number of parts. We show that the decision version of this problem is NP-complete for chordal and bipartite graphs. On the positive side, we design three linear-time algorithms for solving \textsc{Maximum $2$-Transitivity Problem} in trees, split and bipartite chain graphs.

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arXiv2023

Strong transitivity of a graph

A vertex partition $π= \{V_1, V_2, \ldots, V_k\}$ of $G$ is called a \emph{transitive partition} of size $k$ if $V_i$ dominates $V_j$ for all $1\leq i<j\leq k$. For two disjoint subsets $A$ and $B$ of $V$, we say $A$ \emph{strongly dominates} $B$ if for every vertex $y\in B$, there exists a vertex $x\in A$, such that $xy\in E$ and $deg_G(x)\geq deg_G(y)$. A vertex partition $π= \{V_1, V_2, \ldots, V_k\}$ of $G$ is called a \emph{strong transitive partition} of size $k$ if $V_i$ strongly dominates $V_j$ for all $1\leq i<j\leq k$. The \textsc{Maximum Strong Transitivity Problem} is to find a strong transitive partition of a given graph with the maximum number of parts. In this article, we initiate the study of this variation of transitive partition from algorithmic point of view. We show that the decision version of this problem is NP-complete for chordal graphs. On the positive side, we prove that this problem can be solved in linear time for trees and split graphs.

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arXiv2023

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 $π= \{V_1, V_2, \ldots, V_k\}$ 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 $π= \{V_1, V_2, \ldots, V_k\}$ 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.

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arXiv2022

Transitivity on subclasses of chordal 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 $A$ \textit{dominates} $B$ if every vertex of $B$ is adjacent to at least one vertex of $A$ in $G$. A vertex partition $π= \{V_1, V_2, \ldots, V_k\}$ of $G$ is called a \emph{transitive $k$-partition} if $V_i$ dominates $V_j$ for all $i,j$, where $1\leq i<j\leq k$. The maximum integer $k$ for which the above partition exists is called \emph{transitivity} of $G$ and it is denoted by $Tr(G)$. The \textsc{Maximum Transitivity Problem} is to find a transitive partition of a given graph with the maximum number of partitions. It was known that the decision version of \textsc{Maximum Transitivity Problem} is NP-complete for chordal graphs [Iterated colorings of graphs, \emph{Discrete Mathematics}, 278, 2004]. In this paper, we first prove that this problem can be solved in linear time for \emph{split graphs} and for the \emph{complement of bipartite chain graphs}, two subclasses of chordal graphs. We also discuss Nordhaus-Gaddum type relations for transitivity and provide counterexamples for an open problem posed by J. T. Hedetniemi and S. T. Hedetniemi [The transitivity of a graph, \emph{J. Combin. Math. Combin. Comput}, 104, 2018]. Finally, we characterize transitively critical graphs having fixed transitivity.

math.CO↗

arXiv2022

Transitivity on subclasses of bipartite graphs

Let $G=(V, E)$ be a graph where $V$ and $E$ are the vertex and edge set, respectively. For two disjoint subsets $A$ and $B$, we say $A$ dominates $B$ if every vertex of $B$ is adjacent to at least one vertex of $A$. A vertex partition $π= \{V_1, V_2, \ldots, V_k\}$ of $G$ is called a \emph{transitive $k$-partition} if $V_i$ dominates $V_j$ for all $i,j$ where $1\leq i<j\leq k$. The maximum integer $k$ for which the above partition exists is called \emph{transitivity} of $G$ and it is denoted by $Tr(G)$. The \textsc{Maximum Transitivity Problem} is to find a transitive partition of a given graph with the maximum number of partitions. It was known that the decision version of \textsc{Maximum Transitivity Problem} is NP-complete for general graphs, which was proved by Hedetniemi et al. [Iterated colorings of graphs, \emph{Discrete Mathematics}, 278, 2004]. This paper first strengthens the NP-completeness result by showing that this problem remains NP-complete for perfect elimination bipartite graphs. On the other hand, we propose a linear-time algorithm for finding the transitivity of a given bipartite chain graph. We then characterize graphs with transitivity at least $t$ for any integer $t$. This result answers two open questions posed by J. T. Hedetniemi and S. T. Hedetniemi [The transitivity of a graph, \emph{J. Combin. Math. Combin. Comput}, 104, 2018].

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