arXiv subjects

Subhabrata Paul

Publications and source records attributed to Subhabrata Paul.

22 records · Page 2Linked to original sources

arXiv2019

On vertex-edge and independent vertex-edge domination

Given a graph $G = (V,E)$, a vertex $u \in V$ ve-dominates all edges incident to any vertex of $N_G[u]$. A set $S \subseteq V$ is a ve-dominating set if for all edges $e\in E$, there exists a vertex $u \in S$ such that $u$ ve-dominates $e$. Lewis [Ph.D. thesis, 2007] proposed a linear time algorithm for ve-domination problem for trees. In this paper, first we have constructed an example where the proposed algorithm fails. Then we have proposed a linear time algorithm for ve-domination problem in block graphs, which is a superclass of trees. We have also proved that finding minimum ve-dominating set is NP-complete for undirected path graphs. Finally, we have characterized the trees with equal ve-domination and independent ve-domination number.

cs.DM↗

arXiv2015

Uniformity of point samples in metric spaces using gap ratio

Teramoto et al. defined a new measure called the gap ratio that measures the uniformity of a finite point set sampled from $\cal S$, a bounded subset of $\mathbb{R}^2$. We generalize this definition of measure over all metric spaces by appealing to covering and packing radius. The definition of gap ratio needs only a metric unlike discrepancy, a widely used uniformity measure, that depends on the notion of a range space and its volume. We also show some interesting connections of gap ratio to Delaunay triangulation and discrepancy in the Euclidean plane. The major focus of this work is on solving optimization related questions about selecting uniform point samples from metric spaces; the uniformity being measured using gap ratio. We consider discrete spaces like graph and set of points in the Euclidean space and continuous spaces like the unit square and path connected spaces. We deduce lower bounds, prove hardness and approximation hardness results. We show that a general approximation algorithm framework gives different approximation ratios for different metric spaces based on the lower bound we deduce. Apart from the above, we show existence of coresets for sampling uniform points from the Euclidean space -- for both the static and the streaming case. This leads to a $\left( 1+ε\right)$-approximation algorithm for uniform sampling from the Euclidean space.

cs.CG↗

arXiv2014

$(1,j)$-set problem in graphs

A subset $D \subseteq V $of a graph $G = (V, E)$ is a $(1, j)$-set if every vertex $v \in V \setminus D$ is adjacent to at least $1$ but not more than $j$ vertices in D. The cardinality of a minimum $(1, j)$-set of $G$, denoted as $γ_{(1,j)} (G)$, is called the $(1, j)$-domination number of $G$. Given a graph $G = (V, E)$ and an integer $k$, the decision version of the $(1, j)$-set problem is to decide whether $G$ has a $(1, j)$-set of cardinality at most $k$. In this paper, we first obtain an upper bound on $γ_{(1,j)} (G)$ using probabilistic methods, for bounded minimum and maximum degree graphs. Our bound is constructive, by the randomized algorithm of Moser and Tardos [MT10], We also show that the $(1, j)$- set problem is NP-complete for chordal graphs. Finally, we design two algorithms for finding $γ_{(1,j)} (G)$ of a tree and a split graph, for any fixed $j$, which answers an open question posed in [CHHM13].

cs.DM↗

arXiv2014

Linear kernels for k-tuple and liar's domination in bounded genus graphs

A set $D\subseteq V$ is called a $k$-tuple dominating set of a graph $G=(V,E)$ if $\left| N_G[v] \cap D \right| \geq k$ for all $v \in V$, where $N_G[v]$ denotes the closed neighborhood of $v$. A set $D \subseteq V$ is called a liar's dominating set of a graph $G=(V,E)$ if (i) $\left| N_G[v] \cap D \right| \geq 2$ for all $v\in V$ and (ii) for every pair of distinct vertices $u, v\in V$, $\left| (N_G[u] \cup N_G[v]) \cap D \right| \geq 3$. Given a graph $G$, the decision versions of $k$-Tuple Domination Problem and the Liar's Domination Problem are to check whether there exists a $k$-tuple dominating set and a liar's dominating set of $G$ of a given cardinality, respectively. These two problems are known to be NP-complete \cite{LiaoChang2003, Slater2009}. In this paper, we study the parameterized complexity of these problems. We show that the $k$-Tuple Domination Problem and the Liar's Domination Problem are $\mathsf{W}[2]$-hard for general graphs but they admit linear kernels for graphs with bounded genus.

cs.CC↗