arXiv subjects

I. Vinod Reddy

Publications and source records attributed to I. Vinod Reddy.

15 recordsLinked to original sources

arXiv2026

On the complexity of global Roman domination problem in graphs

A Roman dominating function of a graph $G=(V,E)$ is a labeling $f: V \rightarrow{} \{0 ,1, 2\}$ such that for each vertex $u \in V$ with $f(u) = 0$, there exists a vertex $v \in N(u)$ with $f(v) =2$. A Roman dominating function $f$ is a global Roman dominating function if it is a Roman dominating function for both $G$ and its complement $\overline{G}$. The weight of $f$ is the sum of $f(u)$ over all the vertices $u \in V$. The objective of Global Roman Domination problem is to find a global Roman dominating function with minimum weight. The objective of Global Roman Domination is to compute a global Roman dominating function of minimum weight. In this paper, we study the algorithmic aspects of Global Roman Domination problem on various graph classes and obtain the following results. 1. We prove that Roman domination and Global Roman Domination problems are not computationally equivalent by identifying graph classes on which one is linear-time solvable, while the other is NP-complete. 2. We show that Global Roman Domination problem is NP-complete on split graphs, thereby resolving an open question posed by Panda and Goyal [Discrete Applied Mathematics, 2023]. 3. We prove that Global Roman Domination problem is NP-complete on chordal bipartite graphs, planar bipartite graphs with maximum degree five and circle graphs. 4. On the positive side, we present a linear-time algorithm for Global Roman domination problem on cographs.

math.CO↗

arXiv2025

On the Parameterized Complexity of Odd Coloring

A proper vertex coloring of a connected graph $G$ is called an odd coloring if, for every vertex $v$ in $G$, there exists a color that appears odd number of times in the open neighborhood of $v$. The minimum number of colors required to obtain an odd coloring of $G$ is called the \emph{odd chromatic number} of $G$, denoted by $χ_{o}(G)$. Determining $χ_o(G)$ known to be ${\sf NP}$-hard. Given a graph $G$ and an integer $k$, the \odc{} problem is to decide whether $χ_o(G)$ is at most $k$. In this paper, we study the parameterized complexity of the problem, particularly with respect to structural graph parameters. We obtain the following results: \begin{itemize} \item We prove that the problem admits a polynomial kernel when parameterized by the distance to clique. \item We show that the problem cannot have a polynomial kernel when parameterized by the vertex cover number unless ${\sf NP} \subseteq {\sf Co {\text -} NP/poly}$. \item We show that the problem is fixed-parameter tractable when parameterized by distance to cluster, distance to co-cluster, or neighborhood diversity. \item We show that the problem is ${\sf W[1]}$-hard parameterized by clique-width. \end{itemize} Finally, we study the complexity of the problem on restricted graph classes. We show that it can be solved in polynomial time on cographs and split graphs but remains NP-complete on certain subclasses of bipartite graphs.

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arXiv2024

Conflict-Free Coloring: Graphs of Bounded Clique Width and Intersection Graphs

A conflict-free coloring of a graph $G$ is a (partial) coloring of its vertices such that every vertex $u$ has a neighbor whose assigned color is unique in the neighborhood of $u$. There are two variants of this coloring, one defined using the open neighborhood and one using the closed neighborhood. For both variants, we study the problem of deciding whether the conflict-free coloring of a given graph $G$ is at most a given number $k$. In this work, we investigate the relation of clique-width and minimum number of colors needed (for both variants) and show that these parameters do not bound one another. Moreover, we consider specific graph classes, particularly graphs of bounded clique-width and types of intersection graphs, such as distance hereditary graphs, interval graphs and unit square and disk graphs. We also consider Kneser graphs and split graphs. We give (often tight) upper and lower bounds and determine the complexity of the decision problem on these graph classes, which improve some of the results from the literature. Particularly, we settle the number of colors needed for an interval graph to be conflict-free colored under the open neighborhood model, which was posed as an open problem.

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arXiv2023

On Locally Identifying Coloring of Cartesian Product and Tensor Product of Graphs

For a positive integer $k$, a proper $k$-coloring of a graph $G$ is a mapping $f: V(G) \rightarrow \{1,2, \ldots, k\}$ such that $f(u) \neq f(v)$ for each edge $uv$ of $G$. The smallest integer $k$ for which there is a proper $k$-coloring of $G$ is called the chromatic number of $G$, denoted by $χ(G)$. A locally identifying coloring (for short, lid-coloring) of a graph $G$ is a proper $k$-coloring of $G$ such that every pair of adjacent vertices with distinct closed neighborhoods has distinct set of colors in their closed neighborhoods. The smallest integer $k$ such that $G$ has a lid-coloring with $k$ colors is called locally identifying chromatic number (for short, lid-chromatic number) of $G$, denoted by $χ_{lid}(G)$. This paper studies the lid-coloring of the Cartesian product and tensor product of two graphs. We prove that if $G$ and $H$ are two connected graphs having at least two vertices then (a) $χ_{lid}(G \square H) \leq χ(G) χ(H)-1$ and (b) $χ_{lid}(G \times H) \leq χ(G) χ(H)$. Here $G \square H$ and $G \times H$ denote the Cartesian and tensor products of $G$ and $H$ respectively. We determine the lid-chromatic number of $C_m \square P_n$, $C_m \square C_n$, $P_m \times P_n$, $C_m \times P_n$ and $C_m \times C_n$, where $C_m$ and $P_n$ denote a cycle and a path on $m$ and $n$ vertices respectively.

math.CO↗

arXiv2023

Star Coloring of Tensor Product of Two Graphs

A star coloring of a graph $G$ is a proper vertex coloring such that no path on four vertices is bicolored. The smallest integer $k$ for which $G$ admits a star coloring with $k$ colors is called the star chromatic number of $G$, denoted as $χ_s(G)$. In this paper, we study the star coloring of tensor product of two graphs and obtain the following results. 1. We give an upper bound on the star chromatic number of the tensor product of two arbitrary graphs. 2. We determine the exact value of the star chromatic number of tensor product two paths. 3. We show that the star chromatic number of tensor product of two cycles is five, except for $C_3 \times C_3$ and $C_3 \times C_5$. 4. We give tight bounds for the star chromatic number of tensor product of a cycle and a path.

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arXiv2023

On Color Critical Graphs of Star Coloring

A \emph{star coloring} of a graph $G$ is a proper vertex-coloring such that no path on four vertices is $2$-colored. The minimum number of colors required to obtain a star coloring of a graph $G$ is called star chromatic number and it is denoted by $χ_s(G)$. A graph $G$ is called $k$-critical if $χ_s(G)=k$ and $χ_s(G -e) < χ_s(G)$ for every edge $e \in E(G)$. In this paper, we give a characterization of 3-critical, $(n-1)$-critical and $(n-2)$-critical graphs with respect to star coloring, where $n$ denotes the number of vertices of $G$. We also give upper and lower bounds on the minimum number of edges in $(n-1)$-critical and $(n-2)$-critical graphs.

math.CO↗

arXiv2022

On Structural Parameterizations of Star Coloring

A Star Coloring of a graph G is a proper vertex coloring such that every path on four vertices uses at least three distinct colors. The minimum number of colors required for such a star coloring of G is called star chromatic number, denoted by χ_s(G). Given a graph G and a positive integer k, the STAR COLORING PROBLEM asks whether $G$ has a star coloring using at most k colors. This problem is NP-complete even on restricted graph classes such as bipartite graphs. In this paper, we initiate a study of STAR COLORING from the parameterized complexity perspective. We show that STAR COLORING is fixed-parameter tractable when parameterized by (a) neighborhood diversity, (b) twin-cover, and (c) the combined parameters clique-width and the number of colors.

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arXiv2020

On Structural Parameterizations of Load Coloring

Given a graph $G$ and a positive integer $k$, the 2-Load coloring problem is to check whether there is a $2$-coloring $f:V(G) \rightarrow \{r,b\}$ of $G$ such that for every $i \in \{r,b\}$, there are at least $k$ edges with both end vertices colored $i$. It is known that the problem is NP-complete even on special classes of graphs like regular graphs. Gutin and Jones (Inf Process Lett 114:446-449, 2014) showed that the problem is fixed-parameter tractable by giving a kernel with at most $7k$ vertices. Barbero et al. (Algorithmica 79:211-229, 2017) obtained a kernel with less than $4k$ vertices and $O(k)$ edges, improving the earlier result. In this paper, we study the parameterized complexity of the problem with respect to structural graph parameters. We show that \lcp{} cannot be solved in time $f(w)n^{o(w)}$, unless ETH fails and it can be solved in time $n^{O(w)}$, where $n$ is the size of the input graph, $w$ is the clique-width of the graph and $f$ is an arbitrary function of $w$. Next, we consider the parameters distance to cluster graphs, distance to co-cluster graphs and distance to threshold graphs, which are weaker than the parameter clique-width and show that the problem is fixed-parameter tractable (FPT) with respect to these parameters. Finally, we show that \lcp{} is NP-complete even on bipartite graphs and split graphs.

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arXiv2020

Parameterized Coloring Problems on Threshold Graphs

In this paper, we study several coloring problems on graphs from the viewpoint of parameterized complexity. We show that Precoloring Extension is fixed-parameter tractable (FPT) parameterized by distance to clique and Equitable Coloring is FPT parameterized by the distance to threshold graphs. We also study the List k-Coloring and show that the problem is NP-complete on split graphs and it is FPT parameterized by solution size on split graphs.

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arXiv2017

On the Parallel Parameterized Complexity of the Graph Isomorphism Problem

In this paper, we study the parallel and the space complexity of the graph isomorphism problem (\GI{}) for several parameterizations. Let $\mathcal{H}=\{H_1,H_2,\cdots,H_l\}$ be a finite set of graphs where $|V(H_i)|\leq d$ for all $i$ and for some constant $d$. Let $\mathcal{G}$ be an $\mathcal{H}$-free graph class i.e., none of the graphs $G\in \mathcal{G}$ contain any $H \in \mathcal{H}$ as an induced subgraph. We show that \GI{} parameterized by vertex deletion distance to $\mathcal{G}$ is in a parameterized version of $\AC^1$, denoted $\PL$-$\AC^1$, provided the colored graph isomorphism problem for graphs in $\mathcal{G}$ is in $\AC^1$. From this, we deduce that \GI{} parameterized by the vertex deletion distance to cographs is in $\PL$-$\AC^1$. The parallel parameterized complexity of \GI{} parameterized by the size of a feedback vertex set remains an open problem. Towards this direction we show that the graph isomorphism problem is in $\PL$-$\TC^0$ when parameterized by vertex cover or by twin-cover. Let $\mathcal{G}'$ be a graph class such that recognizing graphs from $\mathcal{G}'$ and the colored version of \GI{} for $\mathcal{G}'$ is in logspace ($Ł$). We show that \GI{} for bounded vertex deletion distance to $\mathcal{G}'$ is in $Ł$. From this, we obtain logspace algorithms for \GI{} for graphs with bounded vertex deletion distance to interval graphs and graphs with bounded vertex deletion distance to cographs.

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arXiv2017

On Structural Parameterizations of Firefighting

The Firefighting problem is defined as follows. At time $t=0$, a fire breaks out at a vertex of a graph. At each time step $t \geq 0$, a firefighter permanently defends (protects) an unburned vertex, and the fire then spread to all undefended neighbors from the vertices on fire. This process stops when the fire cannot spread anymore. The goal is to find a sequence of vertices for the firefighter that maximizes the number of saved (non burned) vertices. The Firefighting problem turns out to be NP-hard even when restricted to bipartite graphs or trees of maximum degree three. We study the parameterized complexity of the Firefighting problem for various structural parameterizations. All our parameters measure the distance to a graph class (in terms of vertex deletion) on which the firefighting problem admits a polynomial time algorithm. Specifically, for a graph class $\mathcal{F}$ and a graph $G$, a vertex subset $S$ is called a modulator to $\mathcal{F}$ if $G \setminus S$ belongs to $\mathcal{F}$. The parameters we consider are the sizes of modulators to graph classes such as threshold graphs, bounded diameter graphs, disjoint unions of stars, and split graphs. To begin with, we show that the problem is W[1]-hard when parameterized by the size of a modulator to diameter at most two graphs and split graphs. In contrast to the above intractability results, we show that Firefighting is fixed parameter tractable (FPT) when parameterized by the size of a modulator to threshold graphs and disjoint unions of stars, which are subclasses of diameter at most two graphs. We further investigate the kernelization complexity of these problems to find that firefighting admits a polynomial kernel when parameterized by the size of a modulator to a clique, while it is unlikely to admit a polynomial kernel when parameterized by the size of a modulator to a disjoint union of stars.

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arXiv2017

Parameterized Algorithms for Conflict-free Colorings of Graphs

In this paper, we study the conflict-free coloring of graphs induced by neighborhoods. A coloring of a graph is conflict-free if every vertex has a uniquely colored vertex in its neighborhood. The conflict-free coloring problem is to color the vertices of a graph using the minimum number of colors such that the coloring is conflict-free. We consider both closed neighborhoods, where the neighborhood of a vertex includes itself, and open neighborhoods, where a vertex does not included in its neighborhood. We study the parameterized complexity of conflict-free closed neighborhood coloring and conflict-free open neighborhood coloring problems. We show that both problems are fixed-parameter tractable (FPT) when parameterized by the cluster vertex deletion number of the input graph. This generalizes the result of Gargano et al.(2015) that conflict-free coloring is fixed-parameter tractable parameterized by the vertex cover number. Also, we show that both problems admit an additive constant approximation algorithm when parameterized by the distance to threshold graphs. We also study the complexity of the problem on special graph classes. We show that both problems can be solved in polynomial time on cographs. For split graphs, we give a polynomial time algorithm for closed neighborhood conflict-free coloring problem, whereas we show that open neighborhood conflict-free coloring is NP-complete. We show that interval graphs can be conflict-free colored using at most four colors.

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arXiv2017

The Parameterized Complexity of Happy Colorings

Consider a graph $G = (V,E)$ and a coloring $c$ of vertices with colors from $[\ell]$. A vertex $v$ is said to be happy with respect to $c$ if $c(v) = c(u)$ for all neighbors $u$ of $v$. Further, an edge $(u,v)$ is happy if $c(u) = c(v)$. Given a partial coloring $c$ of $V$, the Maximum Happy Vertex (Edge) problem asks for a total coloring of $V$ extending $c$ to all vertices of $V$ that maximises the number of happy vertices (edges). Both problems are known to be NP-hard in general even when $\ell = 3$, and is polynomially solvable when $\ell = 2$. In [IWOCA 2016] it was shown that both problems are polynomially solvable on trees, and for arbitrary $k$, it was shown that MHE is \NPH{} on planar graphs and is \FPT{} parameterized by the number of precolored vertices and branchwidth. We continue the study of this problem from a parameterized prespective. Our focus is on both structural and standard parameterizations. To begin with, we establish that the problems are \FPT{} when parameterized by the treewidth and the number of colors used in the precoloring, which is a potential improvement over the total number of precolored vertices. Further, we show that both the vertex and edge variants of the problem is \FPT{} when parameterized by vertex cover and distance-to-clique parameters. We also show that the problem of maximizing the number of happy edges is \FPT{} when parameterized by the standard parameter, the number of happy edges. We show that the maximum happy vertex (edge) problem is \NPH{} on split graphs and bipartite graphs and polynomially solvable on cographs.

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arXiv2016

Polynomial-time Algorithm for Isomorphism of Graphs with Clique-width at most Three

The clique-width is a measure of complexity of decomposing graphs into certain tree-like structures. The class of graphs with bounded clique-width contains bounded tree-width graphs. We give a polynomial time graph isomorphism algorithm for graphs with clique-width at most three. Our work is independent of the work by Grohe et al. \cite{grohe2015isomorphism} showing that the isomorphism problem for graphs of bounded clique-width is polynomial time.

cs.CC↗

arXiv2015

Does diversity of papers affect their citations? Evidence from American Physical Society Journals

In this work, we study the correlation between interdisciplinarity of papers within physical sciences and their citations by using meta data of articles published in American Physical Society's Physical Review journals between 1985 to 2012. We use the Weitzman diversity index to measure the diversity of papers and authors, exploiting the hierarchical structure of PACS (Physics and Astronomy Classification Scheme) codes. We find that the fraction of authors with high diversity is increasing with time, where as the fraction of least diversity are decreasing, and moderate diversity authors have higher tendency to switch over to other diversity groups. The diversity index of papers is correlated with the citations they received in a given time period from their publication year. Papers with lower and higher end of diversity index receive lesser citations than the moderate diversity papers.

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