arXiv subjects

Anita Das

Publications and source records attributed to Anita Das.

9 recordsLinked to original sources

arXiv2026

Graph Burning: Bounds and Hardness

Graph burning is a discrete-time process that models the propagation of information in a network. Given an undirected graph whose vertices are initially unburned, the process evolves in discrete rounds. At each round, an unburned vertex is selected and burned, while any unburned vertex adjacent to a vertex burned in the previous round also becomes burned. The burning number of a graph is the minimum number of steps to burn all its vertices. The BURNING NUMBER PROBLEM asks whether the burning number of an input graph $G$ is at most $k$. In this paper, we investigate the graph burning problem from both algorithmic and structural viewpoints. Although the problem is known to be NP-complete on interval graphs, we strengthen this result by proving that it remains NP-complete even when restricted to connected proper interval graphs. We also study the burning number of $P_k$-free graphs. Motivated by the well-known burning number conjecture, which states that every connected graph of order $n$ has burning number at most $\lceil \sqrt{n}~\rceil$, we establish an improved upper bound for connected $P_k$-free graphs and show that this bound is tight up to an additive constant of $1$. Finally, we study two variants of the problem: edge burning and total burning. We establish fundamental relationships between these variants and the classical burning, and we determine the computational complexity of the corresponding decision problems.

math.CO↗

arXiv2026

Hardness of Burning Number Problem on Regular Graphs

The Burning Number Problem (BNP) models the spread of information or contagion in a network through a discrete-time process on a graph. At each step, one new vertex is selected as a burning source, while fire simultaneously spreads from previously burned vertices to their neighbors. The burning number of a graph is the minimum number of steps required to burn all vertices. The decision version asks whether the burning number is at most a given integer $k$. BNP is known to be NP-complete even on restricted graph classes such as path forests. We study BNP on connected regular graphs, a natural and previously unexplored graph class. We prove that BNP is NP-complete on connected cubic graphs, and moreover APX-hard under this restriction. We further show that BNP remains APX-hard on connected $d$-regular graphs for every fixed $d \geq 4$.

cs.DS↗

arXiv2023

Spanning caterpillar in biconvex bipartite graphs

A bipartite graph $G=(A, B, E)$ is said to be a biconvex bipartite graph if there exist orderings $<_A$ in $A$ and $<_B$ in $B$ such that the neighbors of every vertex in $A$ are consecutive with respect to $<_B$ and the neighbors of every vertex in $B$ are consecutive with respect to $<_A$. A caterpillar is a tree that will result in a path upon deletion of all the leaves. In this note, we prove that there exists a spanning caterpillar in any connected biconvex bipartite graph. Besides being interesting on its own, this structural result has other consequences. For instance, this directly resolves the burning number conjecture for biconvex bipartite graphs.

math.CO↗

arXiv2018

Algorithms and Bounds for Very Strong Rainbow Coloring

A well-studied coloring problem is to assign colors to the edges of a graph $G$ so that, for every pair of vertices, all edges of at least one shortest path between them receive different colors. The minimum number of colors necessary in such a coloring is the strong rainbow connection number ($\src(G)$) of the graph. When proving upper bounds on $\src(G)$, it is natural to prove that a coloring exists where, for \emph{every} shortest path between every pair of vertices in the graph, all edges of the path receive different colors. Therefore, we introduce and formally define this more restricted edge coloring number, which we call \emph{very strong rainbow connection number} ($\vsrc(G)$). In this paper, we give upper bounds on $\vsrc(G)$ for several graph classes, some of which are tight. These immediately imply new upper bounds on $\src(G)$ for these classes, showing that the study of $\vsrc(G)$ enables meaningful progress on bounding $\src(G)$. Then we study the complexity of the problem to compute $\vsrc(G)$, particularly for graphs of bounded treewidth, and show this is an interesting problem in its own right. We prove that $\vsrc(G)$ can be computed in polynomial time on cactus graphs; in contrast, this question is still open for $\src(G)$. We also observe that deciding whether $\vsrc(G) = k$ is fixed-parameter tractable in $k$ and the treewidth of $G$. Finally, on general graphs, we prove that there is no polynomial-time algorithm to decide whether $\vsrc(G) \leq 3$ nor to approximate $\vsrc(G)$ within a factor $n^{1-\varepsilon}$, unless P$=$NP.

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arXiv2013

Rainbow path and color degree in edge colored graphs

Let $G$ be an edge colored graph. A {\it}{rainbow path} in $G$ is a path in which all the edges are colored with distinct colors. Let $d^c(v)$ be the color degree of a vertex $v$ in $G$, i.e. the number of distinct colors present on the edges incident on the vertex $v$. Let $t$ be the maximum length of a rainbow path in $G$. Chen and Li showed that if $d^c \geq k$, for every vertex $v$ of $G$, then $t \geq \left \lceil \frac{3 k}{5}\right \rceil + 1$ (Long heterochromatic paths in edge-colored graphs, The Electronic Journal of Combinatorics 12 (2005), # R33, Pages:1-33.) Unfortunately, proof by Chen and Li is very long and comes to about 23 pages in the journal version. Chen and Li states in their paper that it was conjectured by Akira Saito, that $t \ge \left \lceil \frac {2k} {3} \right \rceil$. They also states in their paper that they believe $t \ge k - c$ for some constant $c$. In this note, we give a short proof to show that $t \ge \left \lceil \frac{3 k}{5}\right \rceil$, using an entirely different method. Our proof is only about 2 pages long. The draw-back is that our bound is less by 1, than the bound given by Chen and Li. We hope that the new approach adopted in this paper would eventually lead to the settlement of the conjectures by Saito and/or Chen and Li.

cs.DM↗

arXiv2012

Isoperimetric Sequences for Infinite Complete Binary Trees, Meta-Fibonacci Sequences and Signed Almost Binary Partitions

In this paper we demonstrate connections between three seemingly unrelated concepts. (1) The discrete isoperimetric problem in the infinite binary tree with all the leaves at the same level, $ {\mathcal T}_{\infty}$: The $n$-th edge isoperimetric number $δ(n)$ is defined to be $\min_{|S|=n, S \subset V({\mathcal T}_{\infty})} |(S,\bar{S})|$, where $(S,\bar{S})$ is the set of edges in the cut defined by $S$. (2) Signed almost binary partitions: This is the special case of the coin-changing problem where the coins are drawn from the set ${\pm (2^d - 1): $d$ is a positive integer}$. The quantity of interest is $τ(n)$, the minimum number of coins necessary to make change for $n$ cents. (3) Certain Meta-Fibonacci sequences: The Tanny sequence is defined by $T(n)=T(n{-}1{-}T(n{-}1))+T(n{-}2{-}T(n{-}2))$ and the Conolly sequence is defined by $C(n)=C(n{-}C(n{-}1))+C(n{-}1{-}C(n{-}2))$, where the initial conditions are $T(1) = C(1) = T(2) = C(2) = 1$. These are well-known "meta-Fibonacci" sequences. The main result that ties these three together is the following: $$ δ(n) = τ(n) = n+ 2 + 2 \min_{1 \le k \le n} (C(k) - T(n-k) - k).$$ Apart from this, we prove several other results which bring out the interconnections between the above three concepts.

math.CO↗

arXiv2010

Rainbow Connection Number and Connected Dominating Sets

Rainbow connection number rc(G) of a connected graph G is the minimum number of colours needed to colour the edges of G, so that every pair of vertices is connected by at least one path in which no two edges are coloured the same. In this paper we show that for every connected graph G, with minimum degree at least 2, the rainbow connection number is upper bounded by γ_c(G) + 2, where γ_c(G) is the connected domination number of G. Bounds of the form diameter(G) \leq rc(G) \leq diameter(G) + c, 1 \leq c \leq 4, for many special graph classes follow as easy corollaries from this result. This includes interval graphs, AT-free graphs, circular arc graphs, threshold graphs, and chain graphs all with minimum degree at least 2 and connected. We also show that every bridge-less chordal graph G has rc(G) \leq 3.radius(G). In most of these cases, we also demonstrate the tightness of the bounds. An extension of this idea to two-step dominating sets is used to show that for every connected graph on n vertices with minimum degree δ, the rainbow connection number is upper bounded by 3n/(δ + 1) + 3. This solves an open problem of Schiermeyer (2009), improving the previously best known bound of 20n/δ by Krivelevich and Yuster (2010). Moreover, this bound is seen to be tight up to additive factors by a construction of Caro et al. (2008).

math.CO↗

arXiv2008

On the cubicity of bipartite graphs

{\it A unit cube in $k$-dimension (or a $k$-cube) is defined as the cartesian product $R_1 \times R_2 \times ... \times R_k$, where each $R_i$ is a closed interval on the real line of the form $[a_i, a_i+1]$. The {\it cubicity} of $G$, denoted as $cub(G)$, is the minimum $k$ such that $G$ is the intersection graph of a collection of $k$-cubes. Many NP-complete graph problems can be solved efficiently or have good approximation ratios in graphs of low cubicity. In most of these cases the first step is to get a low dimensional cube representation of the given graph. It is known that for a graph $G$, $cub(G) \leq \lfloor\frac{2n}{3}\rfloor$. Recently it has been shown that for a graph $G$, $cub(G) \leq 4(Δ+ 1)\ln n$, where $n$ and $Δ$ are the number of vertices and maximum degree of $G$, respectively. In this paper, we show that for a bipartite graph $G = (A \cup B, E)$ with $|A| = n_1$, $|B| = n_2$, $n_1 \leq n_2$, and $Δ' = \min\{Δ_A, Δ_B\}$, where $Δ_A = {max}_{a \in A}d(a)$ and $Δ_B = {max}_{b \in B}d(b)$, $d(a)$ and $d(b)$ being the degree of $a$ and $b$ in $G$ respectively, $cub(G) \leq 2(Δ'+2) \lceil \ln n_2 \rceil$. We also give an efficient randomized algorithm to construct the cube representation of $G$ in $3(Δ'+2)\lceil \ln n_2 \rceil$ dimensions. The reader may note that in general $Δ'$ can be much smaller than $Δ$.}

cs.DM↗

arXiv2007

Cubicity, Boxicity and Vertex Cover

A $k$-dimensional box is the cartesian product $R_1 \times R_2 \times ... \times R_k$ where each $R_i$ is a closed interval on the real line. The {\it boxicity} of a graph $G$, denoted as $box(G)$, is the minimum integer $k$ such that $G$ is the intersection graph of a collection of $k$-dimensional boxes. A unit cube in $k$-dimensional space or a $k$-cube is defined as the cartesian product $R_1 \times R_2 \times ... \times R_k$ where each $R_i$ is a closed interval on the real line of the form $[a_i, a_{i}+1]$. The {\it cubicity} of $G$, denoted as $cub(G)$, is the minimum $k$ such that $G$ is the intersection graph of a collection of $k$-cubes. In this paper we show that $cub(G) \leq t + \left \lceil \log (n - t)\right\rceil - 1$ and $box(G) \leq \left \lfloor\frac{t}{2}\right\rfloor + 1$, where $t$ is the cardinality of the minimum vertex cover of $G$ and $n$ is the number of vertices of $G$. We also show the tightness of these upper bounds. F. S. Roberts in his pioneering paper on boxicity and cubicity had shown that for a graph $G$, $box(G) \leq \left \lfloor\frac{n}{2} \right \rfloor$, where $n$ is the number of vertices of $G$, and this bound is tight. We show that if $G$ is a bipartite graph then $box(G) \leq \left \lceil\frac{n}{4} \right\rceil$ and this bound is tight. We point out that there exist graphs of very high boxicity but with very low chromatic number. For example there exist bipartite (i.e., 2 colorable) graphs with boxicity equal to $\frac{n}{4}$. Interestingly, if boxicity is very close to $\frac{n}{2}$, then chromatic number also has to be very high. In particular, we show that if $box(G) = \frac{n}{2} - s$, $s \geq 0$, then $χ(G) \geq \frac{n}{2s+2}$, where $χ(G)$ is the chromatic number of $G$.

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