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Jasine Babu

Publications and source records attributed to Jasine Babu.

At least 19 recordsLinked to original sources
arXiv2026

Eternal Vertex Cover Problem on Halin Graphs

Eternal vertex cover problem is a graph protection problem which is a dynamic two player game variant of the classical vertex cover problem. In this game, the minimum number of guards required to protect a graph $G$ is called the eternal vertex cover number of $G$, denoted by $evc(G)$. It is known that for any graph $G$, $\ mvc(G) \le evc(G) \le 2mvc(G)$, where $mvc(G)$ is the vertex cover number of $G$, and that these bounds are generally tight. However, no biconnected graph $G$ achieves $evc(G) = 2mvc(G)$ and no better lower bounds are known for them. In this work, we focus on biconnected graphs in graph families. For infinite graph families $\mathcal{F}$, consider the parameter $ρ(\mathcal{F})=\sup\{r \in \mathbb{R}:\text{ for infinitely many graphs }G \in \mathcal{F},\frac{evc(G)}{mvc(G)}\ge r\}$. No class of biconnected graphs $\mathcal{F}$ is known yet, for which $1 < ρ(\mathcal{F})<2$. In this paper, we show that when $\mathcal{F}$ is the family of Halin graphs, $\frac{7}{6} \le ρ(\mathcal{F}) \le \frac{3}{2}$. Halin graphs are $3$-connected and they have treewidth three. To show the lower bound, we construct a family of Halin graphs for which the ratio tends to $\frac{7}{6}$ with increasing graph size. For the upper bound, we give two algorithms. Our first algorithm gives a defense strategy with $\frac{3}{2} mvc(G)$ guards and serves as a $\frac{3}{2}$ factor approximation algorithm to compute the eternal vertex cover number of Halin graphs. This algorithm also gives an upper bound of $\frac{4}{3}$ for $ρ$ for several subclasses of Halin graphs. Our second algorithm attains the upper bound of $\frac{4}{3}$ for caterpillar Halin graphs. Whether computing eternal vertex cover number is NP-hard for Halin graphs remains an open problem, as is the case with treewidth two graphs.

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arXiv2025

A Correct by Construction Fault Tolerant Voter for Input Selection of a Control System

Safety-critical systems use redundant input units to improve their reliability and fault tolerance. A voting logic is then used to select a reliable input from the redundant sources. A fault detection and isolation rules help in selecting input units that can participate in voting. This work deals with the formal requirement formulation, design, verification and synthesis of a generic voting unit for an $N$-modular redundant measurement system used for control applications in avionics systems. The work follows a correct-by-construction approach, using the Rocq theorem prover.

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arXiv2025

Color-Constrained Arborescences in Edge-Colored Digraphs

Given a multigraph $G$ whose edges are colored from the set $[q]:=\{1,2,\ldots,q\}$ (\emph{$q$-colored graph}), and a vector $α=(α_1,\ldots,α_{q}) \in \mathbb{N}^{q}$ (\emph{color-constraint}), a subgraph $H$ of $G$ is called \emph{$α$-colored}, if $H$ has exactly $α_i$ edges of color $i$ for each $i \in[q]$. In this paper, we focus on $α$-colored arborescences (spanning out-trees) in $q$-colored multidigraphs. We study the decision, counting and search versions of this problem. It is known that the decision and search problems are polynomial-time solvable when $q=2$ and that the decision problem is NP-complete when $q$ is arbitrary. However the complexity status of the problem for fixed $q$ was open for $q > 2$. We show that, for a $q$-colored digraph $G$ and a vertex $s$ in $G$, the number of $α$-colored arborescences in $G$ rooted at $s$ for all color-constraints $α\in \mathbb{N}^q$ can be read from the determinant of a symbolic matrix in $q-1$ indeterminates. This result extends Tutte's matrix-tree theorem for directed graphs and gives a polynomial-time algorithm for the counting and decision problems for fixed $q$. We also use it to design an algorithm that finds an $α$-colored arborescence when one exists. Finally, we study the weighted variant of the problem and give a polynomial-time algorithm (when $q$ is fixed) which finds a minimum weight solution.

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arXiv2024

Arborescences and Shortest Path Trees when Colors Matter

Color-constrained subgraph problems are those where we are given an edge-colored (directed or undirected) graph and the task is to find a specific type of subgraph, like a spanning tree, an arborescence, a single-source shortest path tree, a perfect matching etc., with constraints on the number of edges of each color. Some of these problems, like color-constrained spanning tree, have elegant solutions and some of them, like color-constrained perfect matching, are longstanding open questions. In this work, we study color-constrained arborescences and shortest path trees. Computing a color-constrained shortest path tree on weighted digraphs turns out to be NP-hard in general but polynomial-time solvable when all cycles have positive weight. This polynomial-time solvability is due to the fact that the solution space is essentially the set of all color-constrained arborescences of a directed acyclic subgraph of the original graph. While finding color-constrained arborescence of digraphs is NP-hard in general, we give efficient algorithms when the input graph is acyclic. Consequently, a color-constrained shortest path tree on weighted digraphs having only positive weight cycles can be efficiently computed. Our algorithms also generalize to the problem of finding a color-constrained shortest path tree with minimum total weight. En route, we sight nice connections to colored matroids and color-constrained bases.

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arXiv2022

Eternal vertex cover number of maximal outerplanar graphs

Eternal vertex cover problem is a variant of the classical vertex cover problem modeled as a two player attacker-defender game. Computing eternal vertex cover number of graphs is known to be NP-hard in general and the complexity status of the problem for bipartite graphs is open. There is a quadratic complexity algorithm known for this problem for chordal graphs. Maximal outerplanar graphs forms a subclass of chordal graphs, for which no algorithm of sub-quadratic time complexity is known. In this paper, we obtain a recursive algorithm of linear time for computing eternal vertex cover number of maximal outerplanar graphs.

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arXiv2021

Improved Bounds for the Oriented Radius of Mixed Multigraphs

A mixed multigraph is a multigraph which may contain both undirected and directed edges. An orientation of a mixed multigraph $G$ is an assignment of exactly one direction to each undirected edge of $G$. A mixed multigraph $G$ can be oriented to a strongly connected digraph if and only if $G$ is bridgeless and strongly connected [Boesch and Tindell, Am. Math. Mon., 1980]. For each $r \in \mathbb{N}$, let $f(r)$ denote the smallest number such that any strongly connected bridgeless mixed multigraph with radius $r$ can be oriented to a digraph of radius at most $f(r)$. We improve the current best upper bound of $4r^2+4r$ on $f(r)$ [Chung, Garey and Tarjan, Networks, 1985] to $1.5 r^2 + r + 1$. Our upper bound is tight upto a multiplicative factor of $1.5$ since, $\forall r \in \mathbb{N}$, there exists an undirected bridgeless graph of radius $r$ such that every orientation of it has radius at least $r^2 + r$ [Chvátal and Thomassen, J. Comb. Theory. Ser. B., 1978]. We prove a marginally better lower bound, $f(r) \geq r^2 + 3r + 1$, for mixed multigraphs. While this marginal improvement does not help with asymptotic estimates, it clears a natural suspicion that, like undirected graphs, $f(r)$ may be equal to $r^2 + r$ even for mixed multigraphs. En route, we show that if each edge of $G$ lies in a cycle of length at most $η$, then the oriented radius of $G$ is at most $1.5 r η$. All our proofs are constructive and lend themselves to polynomial time algorithms.

math.CO↗
arXiv2020

A local characterization for perfect plane near-triangulations

We derive a local criterion for a plane near-triangulated graph to be perfect. It is shown that a plane near-triangulated graph is perfect if and only if it does not contain either a vertex, an edge or a triangle, the neighbourhood of which has an odd hole as its boundary. The characterization leads to an $O(n^2)$ algorithm for checking perfectness of plane near-triangulations.

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arXiv2020

A new lower bound for eternal vertex cover number

We obtain a new lower bound for the eternal vertex cover number of an arbitrary graph $G$, in terms of the cardinality of a vertex cover of minimum size in $G$ containing all its cut vertices. The consequences of the lower bound includes a quadratic time algorithm for computing the eternal vertex cover number of chordal graphs.

cs.DM↗
arXiv2020

A Linear Time Algorithm for Computing the Eternal Vertex Cover Number of Cactus Graphs

The eternal vertex cover problem is a dynamic variant of the classical vertex cover problem. It is NP-hard to compute the eternal vertex cover number of graphs and known algorithmic results for the problem are very few. This paper presents a linear time recursive algorithm for computing the eternal vertex cover number of cactus graphs. Unlike other graph classes for which polynomial time algorithms for eternal vertex cover number are based on efficient computability of a known lower bound directly derived from minimum vertex cover, we show that it is a certain substructure property that helps the efficient computation of eternal vertex cover number of cactus graphs. An extension of the result to graphs in which each block is an edge, a cycle or a biconnected chordal graph is also presented.

cs.DM↗
arXiv2020

An Improvement to Chvátal and Thomassen's Upper Bound for Oriented Diameter

An orientation of an undirected graph $G$ is an assignment of exactly one direction to each edge of $G$. The oriented diameter of a graph $G$ is the smallest diameter among all the orientations of $G$. The maximum oriented diameter of a family of graphs $\mathscr{F}$ is the maximum oriented diameter among all the graphs in $\mathscr{F}$. Chvátal and Thomassen [JCTB, 1978] gave a lower bound of $\frac{1}{2}d^2+d$ and an upper bound of $2d^2+2d$ for the maximum oriented diameter of the family of $2$-edge connected graphs of diameter $d$. We improve this upper bound to $ 1.373 d^2 + 6.971d-1 $, which outperforms the former upper bound for all values of $d$ greater than or equal to $8$. For the family of $2$-edge connected graphs of diameter $3$, Kwok, Liu and West [JCTB, 2010] obtained improved lower and upper bounds of $9$ and $11$ respectively. For the family of $2$-edge connected graphs of diameter $4$, the bounds provided by Chvátal and Thomassen are $12$ and $40$ and no better bounds were known. By extending the method we used for diameter $d$ graphs, along with an asymmetric extension of a technique used by Chvátal and Thomassen, we have improved this upper bound to $21$.

math.CO↗
arXiv2019

On Graphs whose Eternal Vertex Cover Number and Vertex Cover Number Coincide

The eternal vertex cover problem is a variant of the classical vertex cover problem where a set of guards on the vertices have to be dynamically reconfigured from one vertex cover to another in every round of an attacker-defender game. The minimum number of guards required to protect a graph $G$ from an infinite sequence of attacks is the eternal vertex cover number of $G$, denoted by $evc(G)$. It is known that, given a graph $G$ and an integer $k$, checking whether $evc(G) \le k$ is NP-hard. However, it is unknown whether this problem is in NP or not. Precise value of eternal vertex cover number is known only for certain very basic graph classes like trees, cycles and grids. For any graph $G$, it is known that $mvc(G) \le evc(G) \le 2 mvc(G)$, where $mvc(G)$ is the minimum vertex cover number of $G$. Though a characterization is known for graphs for which $evc(G) = 2 mvc(G)$, a characterization of graphs for which $evc(G) = mvc(G)$ remained open. Here, we achieve such a characterization for a class of graphs that includes chordal graphs and internally triangulated planar graphs. For some graph classes including biconnected chordal graphs, our characterization leads to a polynomial time algorithm to precisely determine $evc(G)$ and to determine a safe strategy of guard movement in each round of the game with $evc(G)$ guards. The characterization also leads to NP-completeness results for the eternal vertex cover problem for some graph classes including biconnected internally triangulated planar graphs. To the best of our knowledge, these are the first NP-completeness results known for the problem for any graph class.

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arXiv2019

On Induced Colourful Paths in Triangle-free Graphs

Given a graph $G=(V,E)$ whose vertices have been properly coloured, we say that a path in $G$ is "colourful" if no two vertices in the path have the same colour. It is a corollary of the Gallai-Roy-Vitaver Theorem that every properly coloured graph contains a colourful path on $χ(G)$ vertices. We explore a conjecture that states that every properly coloured triangle-free graph $G$ contains an induced colourful path on $χ(G)$ vertices and prove its correctness when the girth of $G$ is at least $χ(G)$. Recent work on this conjecture by Gyárfás and Sárközy, and Scott and Seymour has shown the existence of a function $f$ such that if $χ(G)\geq f(k)$, then an induced colourful path on $k$ vertices is guaranteed to exist in any properly coloured triangle-free graph $G$.

math.CO↗
arXiv2017

A fix-point characterization of Herbrand equivalence of expressions in data flow frameworks

The problem of determining Herbrand equivalence of terms at each program point in a data flow framework is a central and well studied question in program analysis. Most of the well-known algorithms for the computation of Herbrand equivalence in data flow frameworks proceed via iterative fix-point computation on some abstract lattice of short expressions relevant to the given flow graph. However the mathematical definition of Herbrand equivalence is based on a meet over all path characterization over the (infinite) set of all possible expressions. The aim of this paper is to develop a lattice theoretic fix-point formulation of Herbrand equivalence on the (infinite) concrete lattice defined over the set of all terms constructible from variables, constants and operators of a program. The present characterization uses an axiomatic formulation of the notion of Herbrand congruence and defines the (infinite) concrete lattice of Herbrand congruences. Transfer functions and non-deterministic assignments are formulated as monotone functions over this concrete lattice. Herbrand equivalence is defined as the maximum fix point of a composite transfer function defined over an appropriate product lattice of the above concrete lattice. A re-formulation of the classical meet-over-all-paths definition of Herbrand equivalence in the above lattice theoretic framework is also presented and is proven to be equivalent to the new lattice theoretic fix-point characterization.

cs.LO↗
arXiv2015

Sublinear Approximation Algorithms for Boxicity and Related Problems

Boxicity of a graph G(V, E) is the minimum integer k such that G can be represented as the intersection graph of axis parallel boxes in $\mathbb{R}^k$. Cubicity is a variant of boxicity, where the axis parallel boxes in the intersection representation are restricted to be of unit length sides. Deciding whether boxicity (resp. cubicity) of a graph is at most k is NP-hard, even for k=2 or 3. Computing these parameters is inapproximable within $O(n^{1 - ε})$-factor, for any $ε>0$ in polynomial time unless NP=ZPP, even for many simple graph classes. In this paper, we give a polynomial time $κ(n)$ factor approximation algorithm for computing boxicity and a $κ(n)\lceil \log \log n\rceil$ factor approximation algorithm for computing the cubicity, where $κ(n) =2\left\lceil\frac{n\sqrt{\log \log n}}{\sqrt{\log n}}\right\rceil$. These o(n) factor approximation algorithms also produce the corresponding box (resp. cube) representations. As a special case, this resolves the question paused by Spinrad about polynomial time construction of o(n) dimensional box representations for boxicity 2 graphs. Other consequences of our approximation algorithm include $O(κ(n))$ factor approximation algorithms for computing the following parameters: the partial order dimension of finite posets, the interval dimension of finite posets, minimum chain cover of bipartite graphs, threshold dimension of split graphs and Ferrer's dimension of digraphs. Each of these parameters is inapproximable within an $O(n^{1 - ε})$-factor, for any $ε>0$ in polynomial time unless NP=ZPP and the algorithms we derive seem to be the first o(n) factor approximation algorithms known for all these problems.

cs.DM↗
arXiv2014

Parameterized and Approximation Algorithms for Boxicity

Boxicity of a graph $G(V,$ $E)$, denoted by $box(G)$, is the minimum integer $k$ such that $G$ can be represented as the intersection graph of axis parallel boxes in $\mathbb{R}^k$. The problem of computing boxicity is inapproximable even for graph classes like bipartite, co-bipartite and split graphs within $O(n^{1 - ε})$-factor, for any $ε>0$ in polynomial time unless $NP=ZPP$. We give FPT approximation algorithms for computing the boxicity of graphs, where the parameter used is the vertex or edge edit distance of the given graph from families of graphs of bounded boxicity. This can be seen as a generalization of the parameterizations discussed in \cite{Adiga2}. Extending the same idea in one of our algorithms, we also get an $O\left(\frac{n\sqrt{\log \log n}}{\sqrt{\log n}}\right)$ factor approximation algorithm for computing boxicity and an $O\left(\frac{n {(\log \log n)}^{\frac{3}{2}}}{\sqrt{\log n}}\right)$ factor approximation algorithm for computing the cubicity. These seem to be the first $o(n)$ factor approximation algorithms known for both boxicity and cubicity. As a consequence of this result, a $o(n)$ factor approximation algorithm for computing the partial order dimension of finite posets and a $o(n)$ factor approximation algorithm for computing the threshold dimension of split graphs would follow.

cs.DS↗
arXiv2014

Approximating the Cubicity of Trees

Cubicity of a graph $G$ is the smallest dimension $d$, for which $G$ is a unit disc graph in ${\mathbb{R}}^d$, under the $l^\infty$ metric, i.e. $G$ can be represented as an intersection graph of $d$-dimensional (axis-parallel) unit hypercubes. We call such an intersection representation a $d$-dimensional cube representation of $G$. Computing cubicity is known to be inapproximable in polynomial time, within an $O(n^{1-ε})$ factor for any $ε>0$, unless NP=ZPP. In this paper, we present a randomized algorithm that runs in polynomial time and computes cube representations of trees, of dimension within a constant factor of the optimum. It is also shown that the cubicity of trees can be approximated within a constant factor in deterministic polynomial time, if the cube representation is not required to be computed. As far as we know, this is the first constant factor approximation algorithm for computing the cubicity of trees. It is not yet clear whether computing the cubicity of trees is NP-hard or not.

cs.DM↗
arXiv2014

2-connecting Outerplanar Graphs without Blowing Up the Pathwidth

Given a connected outerplanar graph G of pathwidth p, we give an algorithm to add edges to G to get a supergraph of G, which is 2-vertex-connected, outerplanar and of pathwidth O(p). This settles an open problem raised by Biedl, in the context of computing minimum height planar straight line drawings of outerplanar graphs, with their vertices placed on a two dimensional grid. In conjunction with the result of this paper, the constant factor approximation algorithm for this problem obtained by Biedl for 2-vertex-connected outerplanar graphs will work for all outer planar graphs.

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