arXiv subjects

R. B. Sandeep

Publications and source records attributed to R. B. Sandeep.

At least 19 recordsLinked to original sources

arXiv2026

Hardness of Vertex Splitting: Cographs, Chordal Graphs, and Beyond

Vertex splitting replaces a vertex (v) by two nonadjacent vertices whose neighborhoods together equal (N(v)). A split is \emph{exclusive} if these neighborhoods are disjoint and \emph{shallow} if no newly created vertex is split again. For a graph property (\Pi), \textsc{(\Pi)-Vertex Splitting} asks whether at most (k) splits can transform a graph (G) into one satisfying (\Pi). We continue the systematic study of this operation and settle several open problems. First, we prove that \textsc{Cograph Vertex Splitting} is \textsf{NP}-complete, even on graphs of girth at least 5, resolving a question of Firbas and Sorge (ISAAC 2024). More generally, \textsc{(P_t)-free Vertex Splitting} is \textsf{NP}-complete for every fixed (t\geq 4). We also prove that \textsc{Chordal Vertex Splitting} and \textsc{Unit-Interval Vertex Splitting} are \textsf{NP}-complete, resolving two questions of Abu-Khzam, Chakraborty, Isenmann, and Oijid (IWOCA 2026). Our hardness results extend to the exclusive and shallow variants. Assuming the Exponential Time Hypothesis, none of these problems admits an algorithm running in (2^{o(k)}n^{O(1)}) time; moreover, except for the unit-interval cases, none admits an algorithm running in (2^{o(n)}) time.

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arXiv2026

A proof of Seymour's second neighborhood conjecture for oriented graphs with minimum out-degree equal to 7

We prove Seymour's second neighborhood conjecture on oriented graphs whose minimum out-degree is equal to $7$. This gives, to our knowledge, the first improvement of the minimum out-degree threshold in two decades, since the work of Kaneko and Locke in 2001, who resolved the conjecture for oriented graphs whose minimum out-degree is at most $6$. The proof is partially computer-assisted: after a sequence of local reductions, the remaining finite obstruction models are eliminated by reproducible OR-Tools CP-SAT infeasibility checks.

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arXiv2026

Tight Upper Bounds on Color Reversal by Local Inversions

A bicoloration of a graph $G=(V,E)$ is a map $\beta:V\to\{-1,1\}$. A local inversion at a vertex $v$ complements the subgraph induced by the neighbors of $v$ and simultaneously reverses the colors of all neighbors of $v$. Sabidussi (Discrete Mathematics, 1987) showed that every bicolored graph on $n$ vertices without isolated vertices admits a color reversal using at most $6n+3$ local inversions, and that any two bicolorings of such a graph can be transformed into each other using at most $9n$ local inversions. Recently, Porte, Sandeep, and Santra (CALDAM 2026) improved these bounds to $4n-3$ and $\lfloor(11n-3)/2\rfloor$, respectively. We prove the tight bound $3n$ by showing that, for every graph on $n$ vertices without isolated vertices, any bicoloring can be transformed into any other bicoloring using at most $3n$ local inversions. We also show that this bound is best possible: for complete graphs and stars on $n$ vertices, at least $3n$ local inversions are required to reverse the colors of all vertices. Moreover, the proof of the upper bound is constructive: given two bicolorings, it produces, in polynomial time, a sequence of at most $3n$ local inversions transforming one into the other.

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arXiv2026

On the complexity of edge subdivision to $H$-free graphs

Subdividing an edge $uv$ in a graph replaces it by a path $u w v$ with one new vertex. For a graph $H$, the \textsc{$H$-free Subdivision} problem asks whether, given a graph $G$ and an integer $k$, one can destroy all induced copies of $H$ in $G$ by at most $k$ edge subdivisions. We show that the problem is polynomial-time solvable when every component of $H$ is a subdivided star or a subdivided bistar, and at most one component is a subdivided bistar. On the other hand, we prove that \textsc{$H$-free Subdivision} is NP-complete and, assuming the Exponential Time Hypothesis, admits no $2^{o(k)} n^{O(1)}$-time algorithm whenever $H$ satisfies any of the following conditions: \begin{itemize} \item $H$ has minimum degree at least $2$, and the neighborhood of every degree-$2$ vertex induces a $K_2$; \item the vertices of degree at least $3$ in $H$ induce a graph with at least two edges; \item $H$ has a triangle with two vertices of degree at least $3$; \item $H$ contains, as an induced subgraph, the graph obtained from two vertex-disjoint triangles by adding one edge between them; \item $H$ contains exactly one triangle; \item $H$ has girth at least $4$; \item $H$ is a tree with exactly two vertices of degree at least $3$ at distance $2$ or at least $4$. \end{itemize} A simple bounded search-tree algorithm for the problem runs in $2^{O(k)} n^{O(1)}$ time. Thus, for all hardness cases above, this running time is essentially optimal under ETH.

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arXiv2026

Parameterized algorithms for $k$-Inversion

Inversion of a directed graph $D$ with respect to a vertex subset $Y$ is the directed graph obtained from $D$ by reversing the direction of every arc whose endpoints both lie in $Y$. More generally, the inversion of $D$ with respect to a tuple $(Y_1, Y_2, \ldots, Y_\ell)$ of vertex subsets is defined as the directed graph obtained by successively applying inversions with respect to $Y_1, Y_2, \ldots, Y_\ell$. Such a tuple is called a \emph{decycling family} of $D$ if the resulting graph is acyclic. In the \textsc{$k$-Inversion} problem, the input consists of a directed graph $D$ and an integer $k$, and the task is to decide whether $D$ admits a decycling family of size at most $k$. Alon et al.\ (SIAM J.\ Discrete Math., 2024) proved that the problem is NP-complete for every fixed value of $k$, thereby ruling out XP algorithms, and presented a fixed-parameter tractable (FPT) algorithm parameterized by $k$ for tournament inputs. In this paper, we generalize their algorithm to a broader variant of the problem on tournaments and subsequently use this result to obtain an FPT algorithm for \textsc{$k$-Inversion} when the underlying undirected graph of the input is a block graph. Furthermore, we obtain an algorithm for \textsc{$k$-Inversion} on general directed graphs with running time $2^{O(\mathrm{tw}(k + \mathrm{tw}))} \cdot n^{O(1)}$, where $\mathrm{tw}$ denotes the treewidth of the underlying graph.

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arXiv2025

Algorithms and complexity for monitoring edge-geodetic sets in graphs

A monitoring edge-geodetic set of a graph is a subset $M$ of its vertices such that for every edge $e$ in the graph, deleting $e$ increases the distance between at least one pair of vertices in $M$. We study the following computational problem \textsc{MEG-set}: given a graph $G$ and an integer $k$, decide whether $G$ has a monitoring edge geodetic set of size at most $k$. We prove that the problem is NP-hard even for 2-apex 3-degenerate graphs, improving a result by Haslegrave (Discrete Applied Mathematics 2023). Additionally, we prove that the problem cannot be solved in subexponential-time, assuming the Exponential-Time Hypothesis, even for 3-degenerate graphs. Further, we prove that the optimization version of the problem is APX-hard, even for 4-degenerate graphs. Complementing these hardness results, we prove that the problem admits a polynomial-time algorithm for interval graphs, a fixed-parameter tractable algorithm for general graphs with clique-width plus diameter as the parameter, and a fixed-parameter tractable algorithm for chordal graphs with treewidth as the parameter. We also provide an approximation algorithm with factor $\ln m\cdot OPT$ and $\sqrt{n\ln m}$ for the optimization version of the problem, where $m$ is the number of edges, $n$ the number of vertices, and $OPT$ is the size of a minimum monitoring edge-geodetic set of the input graph.

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arXiv2025

Isometric and induced path partitions: a new upper bound and a characterization of some extremal graphs

An \textit{isometric path} is a shortest path between two vertices. An \textit{isometric path partition} (IPP) of a graph $G$ is a set $\mathcal{I}$ of vertex-disjoint isometric paths in $G$ that partition the vertices of $G$. The \textit{isometric path partition number} of $G$, denoted by $\text{ipp}(G)$, is the minimum cardinality of an IPP of~$G$. An \textit{induced path partition} (IndPP) of a graph $G$ is a set $\mathcal{I}$ of vertex-disjoint induced paths in~$G$ that partition the vertices of $G$. The \textit{induced path partition number} of $G$, denoted by $\text{indpp}(G)$, is the minimum cardinality of an IndPP of $G$. In this article, we study both these parameters and observe that every graph $G$ satisfies $\text{indpp}(G) \leq \text{ipp}(G) \leq |V(G)| - \nu(G)$, where $\nu(G)$ is the matching number of $G$. We further prove that a connected graph $G$ is extremal with respect to this upper bound, i.e.\ satisfies $\text{ipp}(G) = |V(G)| - \nu(G)$, (resp.\ $\text{indpp}(G) = |V(G)| - \nu(G)$), if and only if either (i) all blocks of $G$ are odd complete graphs, or (ii) all blocks of $G$ except one are odd complete graphs, and the unique block $B$ of $G$ that is not an odd complete graph is even and satisfies $\text{ipp}(B) = |V(B)| - \nu(B)$ (resp.\ $\text{indpp}(B) = |V(B)| - \nu(B)$). As corollaries of these results, we obtain a full structural characterization of all connected odd graphs that are extremal with respect to our upper bound, as well as of all extremal block graphs.

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arXiv2025

Erdős-Gyárfás conjecture on graphs without long induced paths

Erdős and Gyárfás conjectured in 1994 that every graph with minimum degree at least 3 has a cycle of length a power of 2. In 2022, Gao and Shan (Graphs and Combinatorics) proved that the conjecture is true for $P_8$-free graphs, i.e., graphs without any induced copies of a path on 8 vertices. In 2024, Hu and Shen (Discrete Mathematics) improved this result by proving that the conjecture is true for $P_{10}$ -free graphs. With the aid of a computer search, we improve this further by proving that the conjecture is true for $P_{13}$ -free graphs.

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arXiv2025

Bounds and extremal graphs for monitoring edge-geodetic sets in graphs

A monitoring edge-geodetic set, or simply an MEG-set, of a graph $G$ is a vertex subset $M \subseteq V(G)$ such that given any edge $e$ of $G$, $e$ lies on every shortest $u$-$v$ path of $G$, for some $u,v \in M$. The monitoring edge-geodetic number of $G$, denoted by $meg(G)$, is the minimum cardinality of such an MEG-set. This notion provides a graph theoretic model of the network monitoring problem. In this article, we compare $meg(G)$ with some other graph theoretic parameters stemming from the network monitoring problem and provide examples of graphs having prescribed values for each of these parameters. We also characterize graphs $G$ that have $V(G)$ as their minimum MEG-set, which settles an open problem due to Foucaud \textit{et al.} (CALDAM 2023), and prove that some classes of graphs fall within this characterization. We also provide a general upper bound for $meg(G)$ for sparse graphs in terms of their girth, and later refine the upper bound using the chromatic number of $G$. We examine the change in $meg(G)$ with respect to two fundamental graph operations: clique-sum and subdivisions. In both cases, we provide a lower and an upper bound of the possible amount of changes and provide (almost) tight examples.

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arXiv2024

Switching Classes: Characterization and Computation

In a graph, the switching operation reverses adjacencies between a subset of vertices and the others. For a hereditary graph class $\mathcal{G}$, we are concerned with the maximum subclass and the minimum superclass of $\mathcal{G}$ that are closed under switching. We characterize the maximum subclass for many important classes $\mathcal{G}$, and prove that it is finite when $\mathcal{G}$ is minor-closed and omits at least one graph. For several graph classes, we develop polynomial-time algorithms to recognize the minimum superclass. We also show that the recognition of the superclass is NP-complete for $H$-free graphs when $H$ is a sufficiently long path or cycle, and it cannot be solved in subexponential time assuming the Exponential Time Hypothesis.

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arXiv2023

Contracting edges to destroy a pattern: A complexity study

Given a graph G and an integer k, the objective of the $Π$-Contraction problem is to check whether there exists at most k edges in G such that contracting them in G results in a graph satisfying the property $Π$. We investigate the problem where $Π$ is `H-free' (without any induced copies of H). It is trivial that H-free Contraction is polynomial-time solvable if H is a complete graph of at most two vertices. We prove that, in all other cases, the problem is NP-complete. We then investigate the fixed-parameter tractability of these problems. We prove that whenever H is a tree, except for seven trees, H-free Contraction is W[2]-hard. This result along with the known results leaves behind three unknown cases among trees.

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arXiv2023

Algorithms for subgraph complementation to some classes of graphs

For a class $\mathcal{G}$ of graphs, the objective of \textsc{Subgraph Complementation to} $\mathcal{G}$ is to find whether there exists a subset $S$ of vertices of the input graph $G$ such that modifying $G$ by complementing the subgraph induced by $S$ results in a graph in $\mathcal{G}$. We obtain a polynomial-time algorithm for the problem when $\mathcal{G}$ is the class of graphs with minimum degree at least $k$, for a constant $k$, answering an open problem by Fomin et al. (Algorithmica, 2020). When $\mathcal{G}$ is the class of graphs without any induced copies of the star graph on $t+1$ vertices (for any constant $t\geq 3$) and diamond, we obtain a polynomial-time algorithm for the problem. This is in contrast with a result by Antony et al. (Algorithmica, 2022) that the problem is NP-complete and cannot be solved in subexponential-time (assuming the Exponential Time Hypothesis) when $\mathcal{G}$ is the class of graphs without any induced copies of the star graph on $t+1$ vertices, for every constant $t\geq 5$.

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arXiv2022

Cutting a tree with Subgraph Complementation is hard, except for some small trees

For a graph property $Π$, Subgraph Complementation to $Π$ is the problem to find whether there is a subset $S$ of vertices of the input graph $G$ such that modifying $G$ by complementing the subgraph induced by $S$ results in a graph satisfying the property $Π$. We prove that the problem of Subgraph Complementation to $T$-free graphs is NP-Complete, for $T$ being a tree, except for 41 trees of at most 13 vertices (a graph is $T$-free if it does not contain any induced copies of $T$). This result, along with the 4 known polynomial-time solvable cases (when $T$ is a path on at most 4 vertices), leaves behind 37 open cases. Further, we prove that these hard problems do not admit any subexponential-time algorithms, assuming the Exponential Time Hypothesis. As an additional result, we obtain that Subgraph Complementation to paw-free graphs can be solved in polynomial-time.

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arXiv2021

Incompressibility of H-free edge modification problems: Towards a dichotomy

Given a graph $G$ and an integer $k$, the $H$-free Edge Editing problem is to find whether there exists at most $k$ pairs of vertices in $G$ such that changing the adjacency of the pairs in $G$ results in a graph without any induced copy of $H$. The existence of polynomial kernels for $H$-free Edge Editing received significant attention in the parameterized complexity literature. Nontrivial polynomial kernels are known to exist for some graphs $H$ with at most 4 vertices, but starting from 5 vertices, polynomial kernels are known only if $H$ is either complete or empty. This suggests the conjecture that there is no other $H$ with at least 5 vertices were $H$-free Edge Editing admits a polynomial kernel. Towards this goal, we obtain a set $\mathcal{H}$ of nine 5-vertex graphs such that if for every $H\in\mathcal{H}$, $H$-free Edge Editing is incompressible and the complexity assumption $NP \not\subseteq coNP/poly$ holds, then $H$-free Edge Editing is incompressible for every graph $H$ with at least five vertices that is neither complete nor empty. That is, proving incompressibility for these nine graphs would give a complete classification of the kernelization complexity of $H$-free Edge Editing for every $H$ with at least 5 vertices. We obtain similar result also for $H$-free Edge Deletion. Here the picture is more complicated due to the existence of another infinite family of graphs $H$ where the problem is trivial (graphs with exactly one edge). We obtain a larger set $\mathcal{H}$ of nineteen graphs whose incompressibility would give a complete classification of the kernelization complexity of $H$-free Edge Deletion for every graph $H$ with at least 5 vertices. Analogous results follow also for the $H$-free Edge Completion problem by simple complementation.

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arXiv2021

On subgraph complementation to H-free graphs

For a class $\mathcal{G}$ of graphs, the problem SUBGRAPH COMPLEMENT TO $\mathcal{G}$ asks whether one can find a subset $S$ of vertices of the input graph $G$ such that complementing the subgraph induced by $S$ in $G$ results in a graph in $\mathcal{G}$. We investigate the complexity of the problem when $\mathcal{G}$ is $H$-free for $H$ being a complete graph, a star, a path, or a cycle. We obtain the following results: - When $H$ is a $K_t$ (a complete graph on $t$ vertices) for any fixed $t\geq 1$, the problem is solvable in polynomial-time. This applies even when $\mathcal{G}$ is a subclass of $K_t$-free graphs recognizable in polynomial-time, for example, the class of $(t-2)$-degenerate graphs. - When $H$ is a $K_{1,t}$ (a star graph on $t+1$ vertices), we obtain that the problem is NP-complete for every $t\geq 5$. This, along with known results, leaves only two unresolved cases - $K_{1,3}$ and $K_{1,4}$. - When $H$ is a $P_t$ (a path on $t$ vertices), we obtain that the problem is NP-complete for every $t\geq 7$, leaving behind only two unresolved cases - $P_5$ and $P_6$. - When $H$ is a $C_t$ (a cycle on $t$ vertices), we obtain that the problem is NP-complete for every $t\geq 8$, leaving behind four unresolved cases - $C_4, C_5, C_6,$ and $C_7$. Further, we prove that these hard problems do not admit subexponential-time algorithms (algorithms running in time $2^{o(|V(G)|)}$), assuming the Exponential Time Hypothesis. A simple complementation argument implies that results for $\mathcal{G}$ are applicable for $\overline{\mathcal{G}}$, thereby obtaining similar results for $H$ being the complement of a complete graph, a star, a path, or a cycle. Our results generalize two main results and resolve one open question by Fomin et al. (Algorithmica, 2020).

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arXiv2018

A Polynomial Kernel for Diamond-Free Editing

An $H$-free editing problem asks whether we can edit at most $k$ edges to make a graph contain no induced copy of the fixed graph $H$. We obtain a polynomial kernel for this problem when $H$ is a diamond. The incompressibility dichotomy for $H$ being a 3-connected graph and the classical complexity dichotomy suggest that except for $H$ being a complete/empty graph, $H$-free editing problems admit polynomial kernels only for a few small graphs $H$. Therefore, we believe that our result is an essential step toward a complete dichotomy on the compressibility of $H$-free editing. Additionally, we give a cubic-vertex kernel for the diamond-free edge deletion problem, which is far simpler than the previous kernel of the same size for the problem.

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arXiv2016

Minimum Fill-In: Inapproximability and Almost Tight Lower Bounds

Given an $n*n$ sparse symmetric matrix with $m$ nonzero entries, performing Gaussian elimination may turn some zeroes into nonzero values. To maintain the matrix sparse, we would like to minimize the number $k$ of these changes, hence called the minimum fill-in problem. Agrawal et al.~[FOCS'90] developed the first approximation algorithm, based on early heuristics by George [SIAM J Numer Anal 10] and by Lipton et al.~[SIAM J Numer Anal 16]. The objective function they used is $m+k$, the number of nonzero elements after elimination. An approximation algorithm using $k$ as the objective function was presented by Natanzon et al.~[STOC'98]. These two versions are incomparable in terms of approximation. Parameterized algorithms for the problem was first studied by Kaplan et al.~[FOCS'94]. Fomin & Villanger [SODA'12] recently gave an algorithm running in time $2^{O(\sqrt{k} \log k)}+n^{O(1)}$. Hardness results of this problem are surprisingly scarce, and the few known ones are either weak or have to use nonstandard complexity conjectures. The only inapproximability result by Wu et al.~[IJCAI'15] applies to only the objective function $m+k$, and is grounded on the Small Set Expansion Conjecture. The only nontrivial parameterized lower bounds, by Bliznets et al.~[SODA'16], include a very weak one based on ETH, and a strong one based on hardness of subexponential-time approximation of the minimum bisection problem on regular graphs. For both versions of the problem, we exclude the existence of PTASs, assuming P$\ne$NP, and the existence of $2^{O(n^{1-δ})}$-time approximation schemes for any positive $δ$, assuming ETH. It also implies a $2^{O(k^{1/2-δ})} n^{O(1)}$ parameterized lower bound. Behind these results is a new reduction from vertex cover, which might be of its own interest: All previous reductions for similar problems are from some kind of graph layout problems.

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arXiv2015

A cubic vertex kernel for Diamond-free Edge Deletion and more

A diamond is a graph obtained by removing an edge from a complete graph on four vertices. A graph is diamond-free if it does not contain an induced diamond. The Diamond-free Edge Deletion problem asks whether there exist at most $k$ edges in the input graph $G$ whose deletion results in a diamond-free graph. For this problem, a polynomial kernel of $O(k^4$) vertices was found by Fellows et. al. (Discrete Optimization, 2011). In this paper, we give an improved kernel of $O(k^3)$ vertices for Diamond-free Edge Deletion. Further, we give an $O(k^2)$ vertex kernel for a related problem {Diamond,K_t}-free Edge Deletion, where $t\geq 4$ is any fixed integer. To complement our results, we prove that these problems are NP-complete even for $K_4$-free graphs and can be solved neither in subexponential time (i.e., $2^{o(|G|)}$) nor in parameterized subexponential time (i.e., $2^{o(k)}\cdot |G|^{O(1)}$), unless Exponential Time Hypothesis fails. Our reduction implies the hardness and lower bound for a general class of problems, where these problems come as a special case.

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