arXiv subjects

Shirish Gosavi

Publications and source records attributed to Shirish Gosavi.

4 recordsLinked to original sources

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↗

arXiv2024

Total Domination, Separated Clusters, CD-Coloring: Algorithms and Hardness

Domination and coloring are two classic problems in graph theory. The major focus of this paper is the CD-COLORING problem which combines the flavours of domination and colouring. Let $G$ be an undirected graph. A proper vertex coloring of $G$ is a $cd-coloring$ if each color class has a dominating vertex in $G$. The minimum integer $k$ for which there exists a $cd-coloring$ of $G$ using $k$ colors is called the cd-chromatic number, $χ_{cd}(G)$. A set $S\subseteq V(G)$ is a total dominating set if any vertex in $G$ has a neighbor in $S$. The total domination number, $γ_t(G)$ of $G$ is the minimum integer $k$ such that $G$ has a total dominating set of size $k$. A set $S\subseteq V(G)$ is a $separated-cluster$ if no two vertices in $S$ lie at a distance 2 in $G$. The separated-cluster number, $ω_s(G)$, of $G$ is the maximum integer $k$ such that $G$ has a separated-cluster of size $k$. In this paper, first we explore the connection between CD-COLORING and TOTAL DOMINATION. We prove that CD-COLORING and TOTAL DOMINATION are NP-Complete on triangle-free $d$-regular graphs for each fixed integer $d\geq 3$. We also study the relationship between the parameters $χ_{cd}(G)$ and $ω_s(G)$. Analogous to the well-known notion of `perfectness', here we introduce the notion of `cd-perfectness'. We prove a sufficient condition for a graph $G$ to be cd-perfect (i.e. $χ_{cd}(H)= ω_s(H)$, for any induced subgraph $H$ of $G$) which is also necessary for certain graph classes (like triangle-free graphs). Here, we propose a generalized framework via which we obtain several exciting consequences in the algorithmic complexities of special graph classes. In addition, we settle an open problem by showing that the SEPARATED-CLUSTER is polynomially solvable for interval graphs.

cs.DS↗

arXiv2024

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↗

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↗