arXiv subjects

Sajith Padinhatteeri

Publications and source records attributed to Sajith Padinhatteeri.

6 recordsLinked to original sources

arXiv2024

Gallai's Path Decomposition of Levi Graph

Gallai's path decomposition conjecture states that for a connected graph $G$ on $n$ vertices, there exists a path decomposition of size $\lceil \frac{n}{2} \rceil$. The Levi graph of order one, denoted by $L_{1}(m,k)$, is a bipartite graph with vertex partition $(A,B)$, where $A$ is the collection of all $(k-1)$-element subsets of $[m]$, and $B$ is the collection of all $k$-element subsets of $[m]$. In this graph, a $(k-1)$-element subset is adjacent to a $k$-element subset if and only if it is properly contained within the $k$-element subset. The path number of a graph $G$ is the minimum size of its path decomposition. Gallai's conjecture can be seen as a conjecture on the upper bound of the path number of a connected graph. In this work, we prove the conjecture for $L_{1}(m,k)$ for all $m \ge 2 $ and $2 \le k \le m$. Moreover, we determine the path number of $L_{1}(m,2)$ for all $m$.

math.CO↗

arXiv2022

s-Club Cluster Vertex Deletion on Interval and Well-Partitioned Chordal Graphs

In this paper, we study the computational complexity of \textsc{$s$-Club Cluster Vertex Deletion}. Given a graph, \textsc{$s$-Club Cluster Vertex Deletion ($s$-CVD)} aims to delete the minimum number of vertices from the graph so that each connected component of the resulting graph has a diameter at most $s$. When $s=1$, the corresponding problem is popularly known as \sloppy \textsc{Cluster Vertex Deletion (CVD)}. We provide a faster algorithm for \textsc{$s$-CVD} on \emph{interval graphs}. For each $s\geq 1$, we give an $O(n(n+m))$-time algorithm for \textsc{$s$-CVD} on interval graphs with $n$ vertices and $m$ edges. In the case of $s=1$, our algorithm is a slight improvement over the $O(n^3)$-time algorithm of Cao \etal (Theor. Comput. Sci., 2018) and for $s \geq 2$, it significantly improves the state-of-the-art running time $\left(O\left(n^4\right)\right)$. We also give a polynomial-time algorithm to solve \textsc{CVD} on \emph{well-partitioned chordal graphs}, a graph class introduced by Ahn \etal (\textsc{WG 2020}) as a tool for narrowing down complexity gaps for problems that are hard on chordal graphs, and easy on split graphs. Our algorithm relies on a characterisation of the optimal solution and on solving polynomially many instances of the \textsc{Weighted Bipartite Vertex Cover}. This generalises a result of Cao \etal (Theor. Comput. Sci., 2018) on split graphs. We also show that for any even integer $s\geq 2$, \textsc{$s$-CVD} is NP-hard on well-partitioned chordal graphs.

cs.DS↗

arXiv2017

Vertex transitive graphs $G$ with $χ_D(G) > χ(G)$ and small automorphism group

For a graph $G$ and a positive integer $k$, a vertex labelling $f:V(G)\to\{1,2\ldots,k\}$ is said to be $k$-distinguishing if no non-trivial automorphism of $G$ preserves the sets $f^{-1}(i)$ for each $i\in\{1,\ldots,k\}$. The distinguishing chromatic number of a graph $G$, denoted $χ_D(G)$, is defined as the minimum $k$ such that there is a $k$-distinguishing labelling of $V(G)$ which is also a proper coloring of the vertices of $G$. In this paper, we prove the following theorem: Given $k\in\mathbb{N}$, there exists an infinite sequence of vertex-transitive graphs $G_{i}=(V_i,E_i)$ such that $χ_D(G_i)>χ(G_i)>k$ and $|\mathrm{Aut}(G_i)|=O_k(|V_i|)$, where $\mathrm{Aut}(G_i)$ denotes the full automorphism group of $G_i$. In particular, this answers a problem raised in the paper $χ_D(G)$, $|\mathrm{Aut}(G)|$ and a variant of the Motion lemma.

math.CO↗

arXiv2016

Distinguishing Chromatic Number of Random Cayley graphs

math.CO↗

arXiv2016

The List Distinguishing Number of Kneser Graphs

A graph $G$ is said to be $k$-distinguishable if the vertex set can be colored using $k$ colors such that no non-trivial automorphism fixes every color class, and the distinguishing number $D(G)$ is the least integer $k$ for which $G$ is $k$-distinguishable. If for each $v\in V(G)$ we have a list $L(v)$ of colors, and we stipulate that the color assigned to vertex $v$ comes from its list $L(v)$ then $G$ is said to be $\mathcal{L}$-distinguishable where $\mathcal{L} =\{L(v)\}_{v\in V(G)}$. The list distinguishing number of a graph, denoted $D_l(G)$, is the minimum integer $k$ such that every collection of lists $\mathcal{L}$ with $|L(v)|=k$ admits an $\mathcal{L}$-distinguishing coloring. In this paper, we prove that $D_l(G)=D(G)$ when $G$ is a Kneser graph.

math.CO↗

arXiv2015

$χ_D(G)$, $|Aut(G)|$, and a variant of the Motion Lemma

The \textit{Distinguishing Chromatic Number} of a graph $G$, denoted $χ_D(G)$, was first defined in \cite{collins} as the minimum number of colors needed to properly color $G$ such that no non-trivial automorphism $ϕ$ of the graph $G$ fixes each color class of $G$. In this paper, 1. We prove a lemma that may be considered a variant of the Motion lemma of \cite{RS} and use this to give examples of several families of graphs which satisfy $χ_D(G)=χ(G)+1$. 2.We give an example of families of graphs that admit large automorphism groups in which every proper coloring is distinguishing. We also describe families of graphs with (relatively) very small automorphism groups which satisfy $χ_D(G)=χ(G)+1$, for arbitrarily large values of $χ(G)$. 3. We describe non-trivial families of bipartite graphs that satisfy $χ_D(G)>r$ for any positive integer $r$.

math.CO↗