arXiv subjects

Harmender Gahlawat

Publications and source records attributed to Harmender Gahlawat.

At least 19 recordsLinked to original sources

arXiv2026

The Closed Geodetic Game: algorithms and strategies

The geodetic closure of a set S of vertices of a graph is the set of all vertices in shortest paths between pairs of vertices of S. A set S of vertices in a graph is geodetic if its geodetic closure contains all the vertices of the graph. Several authors have studied variants of games around constructing geodetic sets. The most studied of those, the Geodetic Game, was introduced by Harary in 1984 and developed by Buckley and Harary in 1985. It is an achievement game: both players construct together a geodetic set by alternately adding vertices to the set, the winner being the one who plays last. However, this version of the game allows the players to select vertices that already are in the geodetic closure of the current set. We study the more natural variant, called the Closed Geodetic Game, where the players alternate adding to S vertices that are not in the geodetic closure of S. This variant was also introduced in another, less-noticed, paper by Buckley and Harary in 1985, and only studied since then in the context of trees by Araujo et al. in 2024. We provide a full characterization of the Sprague-Grundy values (equivalence values for games) of graph classes such as paths and cycles, of the outcomes of the Cartesian product of several graphs in function of their individual outcomes, and give polynomial-time algorithms to determine the Sprague-Grundy values in cactus and block graphs.

math.CO↗

arXiv2026

Algorithms and complexity for geodetic sets on interval and chordal graphs

We study the computational complexity of finding the geodetic number of a graph on chordal graphs and interval graphs. A set $S$ of vertices of a graph $G$ is a \textit{geodetic set} if every vertex of $G$ lies in a shortest path between some pair of vertices of $S$. The \textsc{Minimum Geodetic Set (MGS)} problem is to find a geodetic set with minimum cardinality of a given graph. We show that \textsc{Minimum Geodetic Set} is fixed parameter tractable for chordal graphs when parameterized by its \emph{tree-width} (which equals its clique number). This implies a polynomial-time algorithm for $k$-trees, for fixed $k$. Then, we show that \textsc{Minimum Geodetic Set} is NP-hard on interval graphs, thereby answering a question of Ekim et al. (LATIN, 2012), who showed that \textsc{Minimum Geodetic Set} is polynomial-time solvable on proper interval graphs. As interval graphs are very constrained, to prove the latter result, we design a rather sophisticated reduction technique to work around their inherent linear structure.

cs.DS↗

arXiv2026

Algorithms and complexity for geodetic sets on planar and chordal graphs

A set $S$ of vertices of a graph $G$ is a \emph{geodetic set} if every vertex of $G$ lies in a shortest path between some pair of vertices of $S$. The \textsc{Minimum Geodetic Set (MGS)} problem is to find a geodetic set with minimum cardinality of a given graph. A \emph{grid embedding} of a graph is a set of points in two dimensions with integer coordinates such that each point in the set represents a vertex of the graph and, for each edge, the points corresponding to its endpoints are at Euclidean distance~$1$. A graph is a \emph{partial grid} if it has a grid embedding. In this paper, we first prove that \textsc{Minimum Geodetic Set} remains NP-hard even for subcubic partial grids of arbitrary girth. This jointly strengthens three existing hardness results: for bipartite graphs (Dourado et al., Discrete. Math, 2010), subcubic graphs (Bueno et al., Inf. Process. Lett., 2018)~\cite{bueno2018}, and planar graphs (Chakraborty et al., CALDAM, 2020). The \emph{area} of an internal face is the number of integer points lying on the boundary or interior of the face. A graph is a \emph{solid grid} if it has a grid embedding such that all interior faces have area exactly four. To complement the above hardness result, we design a linear-time algorithm for \textsc{Minimum Geodetic Set} on solid grids, improving on a $3$-approximation algorithm by Chakraborty et al. (CALDAM, 2020). Our results hold for \textsc{Edge Geodetic Set} as well. A set $S$ of vertices of a graph $G$ is a \emph{geodetic set} if every edge of $G$ lies in a shortest path between some pair of vertices of $S$. The \textsc{Minimum Edge Geodetic Set (MEGS)} problem is to find an edge geodetic set with minimum cardinality of a given graph. As corollaries, we obtain that \textsc{MEGS} remains NP-hard on partial grids and is linear-time solvable on solid grids.

cs.DM↗

arXiv2026

Hardness and Approximation for Coloring Digraphs

The dichromatic number $\vecχ(D)$ of a digraph is the minimum number $k$ such that $V(D)$ can be partitioned into $k$ subsets, each inducing an acyclic digraph. The acyclic number $\vecα(D)$ is the cardinality of a largest induced acyclic subdigraph of $D$. We study these problems from an approximation point of view. We begin with establishing that even when restricted to tournaments, approximating $\vecχ$ and $\vecα$ remain as challenging as their undirected counterparts on general graphs. Specifically, we establish that for every $ε>0$, it is hard to approximate both $\vecα$ and $\vecχ$ up to a factor of $n^{1-ε}$ even when restricted to tournaments. We next consider approximate coloring of digraphs in special cases. We begin with establishing that we can color $\ell$-dicolorable digraphs using at most $\ell \cdot n^{1-\frac{1}{\ell}}$ colors in time $O(n^{2\ell})$; in particular, we can color $2$-dicolorable digraphs with $2\sqrt{n}$ colors in polynomial time. We then focus on bounding the dichromatic number of dense digraphs as a function of the independence number $α$ of the underlying graph. We consider two special cases in this regard: digraphs with $\vecχ(D)\leq 2$ and digraphs that do not contain any directed triangle. For these cases, we present algorithms which generalize and improve existing tools and results.

cs.DS↗

arXiv2026

Individual Rationality in Constrained Hedonic Games: Additively Separable and Fractional Preferences

Hedonic games are an archetypal problem in coalition formation, where a set of selfish agents want to partition themselves into stable coalitions. In this work, we focus on two natural constraints on the possible outcomes. First, we require that exactly k coalitions are created. Then, loosely following the model of Bilò et al. (AAAI 2022), we assume that each of the k coalitions is additionally associated with a lower and upper bound on its size. The notion of stability that we study is that of individual rationality (IR), which requires that no agent strictly prefers to be alone compared to being in his or her coalition. Although IR is trivially satisfiable even in the most general models of hedonic games, the complexity picture of deciding whether an IR allocation exists, considering the above constraints, is unexpectedly rich. We reveal that tractable fragments of this computational problem require surprisingly nontrivial arguments, even if we restrict ourselves to additively separable and fractional hedonic games. Our tractability results, achieved by exploiting the structure of the underlying preference graph, are also complemented by their intractability counterparts, painting a fairly complete picture of the tractability landscape of this problem.

cs.GT↗

arXiv2025

Pushing Cops and Robber on Graphs of Maximum Degree 4

\textsc{Cops and Robber} is a game played on graphs where a set of \textit{cops} aim to \textit{capture} the position of a single \textit{robber}. The main parameter of interest in this game is the \textit{cop number}, which is the minimum number of cops that are sufficient to guarantee the capture of the robber. In a directed graph $\overrightarrow{G}$, the \textit{push} operation on a vertex $v$ reverses the orientation of all arcs incident on $v$. We consider a variation of classical \textsc{Cops and Robber} on oriented graphs, where in its turn, each cop can either move to an out-neighbor of its current vertex or push some vertex of the graph, whereas, the robber can move to an adjacent vertex in its turn. [Das et al., CALDAM, 2023] introduced this variant and established that if $\overrightarrow{G}$ is an orientation of a subcubic graph, then one cop with push ability has a winning strategy. We extend these results to establish that if $\overrightarrow{G}$ is an orientation of a $3$-degenerate graph, or of a graph with maximum degree $4$, then one cop with push ability has a winning strategy.

math.CO↗

arXiv2025

Hunting a rabbit: complexity, approximability and some characterizations

In the Hunters and Rabbit game, $k$ hunters attempt to shoot an invisible rabbit on a given graph $G$. In each round, the hunters select $k$ vertices to shoot at, while the rabbit moves along an edge of $G$. The hunters win if, at any point, the rabbit is shot. The hunting number of $G$, denoted $h(G)$, is the minimum integer $k$ such that $k$ hunters have a winning strategy regardless of the rabbit's moves. The computational complexity of determining $h(G)$ has been one of the longest-standing open questions about the game. Our first main contribution resolves this by proving that computing $h(G)$ is NP-hard, even for bipartite simple graphs. We further show that the problem remains NP-hard even when $h(G) = O(n^ε)$ or when $n - h(G) = O(n^ε)$, where $n$ is the order of $G$. In addition, we prove that it is NP-hard to approximate $h(G)$ additively within $O(n^{1-ε})$. When a time limit $l$ is imposed on the hunting process, we show that computing $h(G)$ remains NP-hard for any $l \ge 2$ bounded by a polynomial in $n$. On the positive side, we present a polynomial-time $l$-factor approximation algorithm for computing the hunting number with time limit $l$, and we show that $h(G)$ can be computed in polynomial time for bipartite graphs when only two time slots are allowed ($l = 2$). Finally, we provide a forbidden-subgraph characterization of graphs with loops that satisfy $h(G) = 1$, extending a known characterization for simple graphs.

math.CO↗

arXiv2025

Coarse cops and robber in graphs and groups

(abstract shortened to meet arxiv's length requirements) We investigate two variants of the classical Cops and robber game in graphs, recently introduced by Lee, Martínez-Pedroza, and Rodríguez-Quinche. The two versions are played in infinite graphs and the goal of the cops is to prevent the robber to visit some ball of finite radius (chosen by the robber) infinitely many times. Moreover the cops and the robber move at a different speed, and the cops can choose a radius of capture before the game starts. Depending on the order in which the parameters are chosen, this naturally defines two games, a weak version and a strong version (in which the cops are more powerful), and thus two variants of the cop number of a graph $G$: the weak cop number and the strong cop number. It turns out that these two parameters are invariant under quasi-isometry and thus we can investigate these parameters in finitely generated groups by considering any of their Cayley graphs; the parameters do not depend on the chosen set of generators. We answer a number of questions raised by Lee, Martínez-Pedroza, and Rodríguez-Quinche, and more recently by Cornect and Martínez-Pedroza. This includes a proof that the weak and strong cop numbers are monotone under quasi-isometric embedding, characterizations of graphs of weak cop number 1 and graphs of strong cop number 1, and a proof that the grid has infinite strong cop number. Moreover we tie the weak cop number of a graph $G$ to the existence of asymptotic minors of large tree-width in $G$, and use the result to prove that any finitely presented group has weak cop number 1 or $\infty$. We have learned very recently that some of our results have been obtained independently by Appenzeller and Klinge, using fairly different arguments.

math.CO↗

arXiv2025

The Parameterized Complexity of Computing the VC-Dimension

The VC-dimension is a well-studied and fundamental complexity measure of a set system (or hypergraph) that is central to many areas of machine learning. We establish several new results on the complexity of computing the VC-dimension. In particular, given a hypergraph $\mathcal{H}=(\mathcal{V},\mathcal{E})$, we prove that the naive $2^{\mathcal{O}(|\mathcal{V}|)}$-time algorithm is asymptotically tight under the Exponential Time Hypothesis (ETH). We then prove that the problem admits a $1$-additive fixed-parameter approximation algorithm when parameterized by the maximum degree of $\mathcal{H}$ and a fixed-parameter algorithm when parameterized by its dimension, and that these are essentially the only such exploitable structural parameters. Lastly, we consider a generalization of the problem, formulated using graphs, which captures the VC-dimension of both set systems and graphs. We design a $2^{\mathcal{O}(\rm{tw}\cdot \log \rm{tw})}\cdot |V|$-time algorithm for any graph $G=(V,E)$ of treewidth $\rm{tw}$ (which, for a set system, applies to the treewidth of its incidence graph). This is in contrast with closely related problems that require a double-exponential dependency on the treewidth (assuming the ETH).

cs.CC↗

arXiv2025

Exact Algorithms and Lower Bounds for Forming Coalitions of Constrained Maximum Size

Imagine we want to split a group of agents into teams in the most \emph{efficient} way, considering that each agent has their own preferences about their teammates. This scenario is modeled by the extensively studied \textsc{Coalition Formation} problem. Here, we study a version of this problem where each team must additionally be of bounded size. We conduct a systematic algorithmic study, providing several intractability results as well as multiple exact algorithms that scale well as the input grows (FPT), which could prove useful in practice. Our main contribution is an algorithm that deals efficiently with tree-like structures (bounded \emph{treewidth}) for ``small'' teams. We complement this result by proving that our algorithm is asymptotically optimal. Particularly, there can be no algorithm that vastly outperforms the one we present, under reasonable theoretical assumptions, even when considering star-like structures (bounded \emph{vertex cover number}).

cs.DS↗

arXiv2025

Learning Small Decision Trees with Few Outliers: A Parameterized Perspective

Decision trees are a fundamental tool in machine learning for representing, classifying, and generalizing data. It is desirable to construct ``small'' decision trees, by minimizing either the \textit{size} ($s$) or the \textit{depth} $(d)$ of the \textit{decision tree} (\textsc{DT}). Recently, the parameterized complexity of \textsc{Decision Tree Learning} has attracted a lot of attention. We consider a generalization of \textsc{Decision Tree Learning} where given a \textit{classification instance} $E$ and an integer $t$, the task is to find a ``small'' \textsc{DT} that disagrees with $E$ in at most $t$ examples. We consider two problems: \textsc{DTSO} and \textsc{DTDO}, where the goal is to construct a \textsc{DT} minimizing $s$ and $d$, respectively. We first establish that both \textsc{DTSO} and \textsc{DTDO} are W[1]-hard when parameterized by $s+δ_{max}$ and $d+δ_{max}$, respectively, where $δ_{max}$ is the maximum number of features in which two differently labeled examples can differ. We complement this result by showing that these problems become \textsc{FPT} if we include the parameter $t$. We also consider the kernelization complexity of these problems and establish several positive and negative results for both \textsc{DTSO} and \textsc{DTDO}.

cs.LG↗

arXiv2025

(Almost-)Optimal FPT Algorithm and Kernel for $T$-Cycle on Planar Graphs

Research of cycles through specific vertices is a central topic in graph theory. In this context, we focus on a well-studied computational problem, \textsc{$T$-Cycle}: given an undirected $n$-vertex graph $G$ and a set of $k$ vertices $T\subseteq V(G)$ termed \textit{terminals}, the objective is to determine whether $G$ contains a simple cycle $C$ through all the terminals. Our contribution is twofold: (i) We provide a $2^{O(\sqrt{k}\log k)}\cdot n$-time fixed-parameter deterministic algorithm for \textsc{$T$-Cycle} on planar graphs; (ii) We provide a $k^{O(1)}\cdot n$-time deterministic kernelization algorithm for \textsc{$T$-Cycle} on planar graphs where the produced instance is of size $k\log^{O(1)}k$. Both of our algorithms are optimal in terms of both $k$ and $n$ up to (poly)logarithmic factors in $k$ under the ETH. In fact, our algorithms are the first subexponential-time fixed-parameter algorithm for \textsc{$T$-Cycle} on planar graphs, as well as the first polynomial kernel for \textsc{$T$-Cycle} on planar graphs. This substantially improves upon/expands the known literature on the parameterized complexity of the problem.

cs.DS↗

arXiv2025

On the Cop Number of String Graphs

Cops and Robber is a well-studied two-player pursuit-evasion game played on a graph, where a group of cops tries to capture the robber. The \emph{cop number} of a graph is the minimum number of cops required to capture the robber. Gavenčiak et al.~[Eur. J. of Comb. 72, 45--69 (2018)] studied the game on intersection graphs and established that the cop number for the class of string graphs is at most 15, and asked as an open question to improve this bound for string graphs and subclasses of string graphs. We address this question and establish that the cop number of a string graph is at most 13. To this end, we develop a novel \textit{guarding} technique. We further establish that this technique can be useful for other Cops and Robber games on graphs admitting a representation. In particular, we show that four cops have a winning strategy for a variant of Cops and Robber, named Fully Active Cops and Robber, on planar graphs, addressing an open question of Gromovikov et al.~[Austr. J. Comb. 76(2), 248--265 (2020)]. In passing, we also improve the known bounds on the cop number of boxicity 2 graphs. Finally, as a corollary of our result on the cop number of string graphs, we establish that the chromatic number of string graphs with girth at least $5$ is at most $14$.

cs.DM↗

arXiv2023

Parameterized Complexity of Incomplete Connected Fair Division

\textit{Fair division} of resources among competing agents is a fundamental problem in computational social choice and economic game theory. It has been intensively studied on various kinds of items (\textit{divisible} and \textit{indivisible}) and under various notions of \textit{fairness}. We focus on Connected Fair Division (\CFDO), the variant of fair division on graphs, where the \textit{resources} are modeled as an \textit{item graph}. Here, each agent has to be assigned a connected subgraph of the item graph, and each item has to be assigned to some agent. We introduce a generalization of \CFDO, termed Incomplete Connected Fair Division (\CFD), where exactly $p$ vertices of the item graph should be assigned to the agents. This might be useful, in particular when the allocations are intended to be ``economical'' as well as fair. We consider four well-known notions of fairness: \PROP, \EF, \EFO, \EFX. First, we prove that \EF-\CFD, \EFO-\CFD, and \EFX-\CFD are W[1]-hard parameterized by $p$ plus the number of agents, even for graphs having constant \textit{vertex cover number} ($\mathsf{vcn}$). In contrast, we present a randomized \FPT algorithm for \PROP-\CFD parameterized only by $p$. Additionally, we prove both positive and negative results concerning the kernelization complexity of \CFD under all four fairness notions, parameterized by $p$, $\mathsf{vcn}$, and the total number of different valuations in the item graph ($\mathsf{val}$).

cs.GT↗

arXiv2023

Further results on the Hunters and Rabbit game through monotonicity

Hunters and Rabbit game is played on a graph $G$ where the Hunter player shoots at $k$ vertices in every round while the Rabbit player occupies an unknown vertex and, if not shot, must move to a neighbouring vertex after each round. The Rabbit player wins if it can ensure that its position is never shot. The Hunter player wins otherwise. The hunter number $h(G)$ of a graph $G$ is the minimum integer $k$ such that the Hunter player has a winning strategy (i.e., allowing him to win whatever be the strategy of the Rabbit player). This game has been studied in several graph classes, in particular in bipartite graphs (grids, trees, hypercubes...), but the computational complexity of computing $h(G)$ remains open in general graphs and even in trees. To progress further, we propose a notion of monotonicity for the Hunters and Rabbit game imposing that, roughly, a vertex that has already been shot ``must not host the rabbit anymore''. This allows us to obtain new results in various graph classes. Let the monotone hunter number be denoted by $mh(G)$. We show that $pw(G) \leq mh(G) \leq pw(G)+1$ for any graph $G$ with pathwidth $pw(G)$, implying that computing $mh(G)$, or even approximating $mh(G)$ up to an additive constant, is NP-hard. Then, we show that $mh(G)$ can be computed in polynomial time in split graphs, interval graphs, cographs and trees. These results go through structural characterisations which allow us to relate the monotone hunter number with the pathwidth in some of these graph classes. In all cases, this allows us to specify the hunter number or to show that there may be an arbitrary gap between $h$ and $mh$, i.e., that monotonicity does not help. In particular, we show that, for every $k\geq 3$, there exists a tree $T$ with $h(T)=2$ and $mh(T)=k$. We conclude by proving that computing $h$ (resp., $mh$) is FPT parameterised by the minimum size of a vertex cover.

cs.DM↗

arXiv2023

Kernels for the Disjoint Paths Problem on Subclasses of Chordal Graphs

Given an undirected graph $G$ and a multiset of $k$ terminal pairs $\mathcal{X}$, the Vertex-Disjoint Paths (\VDP) and Edge-Disjoint Paths (\EDP) problems ask whether $G$ has $k$ pairwise internally vertex-disjoint paths and $k$ pairwise edge-disjoint paths, respectively, connecting every terminal pair in~$\mathcal{X}$. In this paper, we study the kernelization complexity of \VDP~and~\EDP~on subclasses of chordal graphs. For \VDP, we design a $4k$ vertex kernel on split graphs and an $\mathcal{O}(k^2)$ vertex kernel on well-partitioned chordal graphs. We also show that the problem becomes polynomial-time solvable on threshold graphs. For \textsc{EDP}, we first prove that the problem is $\mathsf{NP}$-complete on complete graphs. Then, we design an $\mathcal{O}(k^{2.75})$ vertex kernel for \EDP~on split graphs, and improve it to a $7k+1$ vertex kernel on threshold graphs. Lastly, we provide an $\mathcal{O}(k^2)$ vertex kernel for \EDP~on block graphs and a $2k+1$ vertex kernel for clique paths. Our contributions improve upon several results in the literature, as well as resolve an open question by Heggernes et al.~[Theory Comput. Syst., 2015].

cs.DS↗

arXiv2023

Parameterized Analysis of the Cops and Robber Problem

\textit{Pursuit-evasion games} have been intensively studied for several decades due to their numerous applications in artificial intelligence, robot motion planning, database theory, distributed computing, and algorithmic theory. \textsc{Cops and Robber} (\CR) is one of the most well-known pursuit-evasion games played on graphs, where multiple \textit{cops} pursue a single \textit{robber}. The aim is to compute the \textit{cop number} of a graph, $k$, which is the minimum number of cops that ensures the \textit{capture} of the robber. From the viewpoint of parameterized complexity, \CR is W[2]-hard parameterized by $k$~[Fomin et al., TCS, 2010]. Thus, we study structural parameters of the input graph. We begin with the \textit{vertex cover number} ($\mathsf{vcn}$). First, we establish that $k \leq \frac{\mathsf{vcn}}{3}+1$. Second, we prove that \CR parameterized by $\mathsf{vcn}$ is \FPT by designing an exponential kernel. We complement this result by showing that it is unlikely for \CR parameterized by $\mathsf{vcn}$ to admit a polynomial compression. We extend our exponential kernels to the parameters \textit{cluster vertex deletion number} and \textit{deletion to stars number}, and design a linear vertex kernel for \textit{neighborhood diversity}. Additionally, we extend all of our results to several well-studied variations of \CR.

cs.DM↗

arXiv2023

Cops and robber on variants of retracts and subdivisions of oriented graphs

\textsc{Cops and Robber} is one of the most studied two-player pursuit-evasion games played on graphs, where multiple \textit{cops}, controlled by one player, pursue a single \textit{robber}. The main parameter of interest is the \textit{cop number} of a graph, which is the minimum number of cops that can ensure the \textit{capture} of the robber. \textsc{Cops and Robber} is also well-studied on directed/oriented graphs. In directed graphs, two kinds of moves are defined for players: \textit{strong move}, where a player can move both along and against the orientation of an arc to an adjacent vertex; and \textit{weak move}, where a player can only move along the orientation of an arc to an \textit{out-neighbor}. We study three variants of \textsc{Cops and Robber} on oriented graphs: \textit{strong cop model}, where the cops can make strong moves while the robber can only make weak moves; \textit{normal cop model}, where both cops and the robber can only make weak moves; and \textit{weak cop model}, where the cops can make weak moves while the robber can make strong moves. We study the cop number of these models with respect to several variants of retracts on oriented graphs and establish that the strong and normal cop number of an oriented graph remains invariant in their strong and distributed retracts, respectively. Next, we go on to study all three variants with respect to the subdivisions of graphs and oriented graphs. Finally, we establish that all these variants remain computationally difficult even when restricted to the class of 2-degenerate bipartite graphs.

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