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Adarsh Srinivasan

Bio: Adarsh Srinivasan is an academic researcher from Indian Institute of Science. The author has contributed to research in topics: Domination analysis & Image (category theory). The author has an hindex of 1, co-authored 2 publications receiving 2 citations. Previous affiliations of Adarsh Srinivasan include Indian Institute of Science Education and Research, Pune.

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TL;DR: A new lower bound is proved of $\left\lceil\frac{mn+2\ lceil \frac{\min \{m,n\}}{3}\rceil}{3} \right\ rceil$ for arbitrary $m, n \geq 4$.
Abstract: Closed form expressions for the domination number of an $n \times m$ grid have attracted significant attention, and an exact expression has been obtained in 2011 by Goncalves et al. In this paper, we present our results on obtaining new lower bounds on the connected domination number of an $n \times m$ grid. The problem has been solved for grids with up to $4$ rows and with $6$ rows by Tolouse et al and the best currently known lower bound for arbitrary $m,n$ is $\lceil\frac{mn}{3}\rceil$. Fujie came up with a general construction for a connected dominating set of an $n \times m$ grid of size $\min \left\{2n+(m-4)+\lfloor\frac{m-4}{3}\rfloor(n-2), 2m+(n-4)+\lfloor\frac{n-4}{3}\rfloor(m-2) \right\}$ . In this paper, we investigate whether this construction is indeed optimum. We prove a new lower bound of $\left\lceil\frac{mn+2\lceil\frac{\min \{m,n\}}{3}\rceil}{3} \right\rceil$ for arbitrary $m,n \geq 4$.

2 citations

Book ChapterDOI
11 Feb 2021
TL;DR: In this article, a new lower bound for Open image in new window for arbitrary m,n,n \ge 4 was obtained for grids with up to 4 rows and with 6 rows.
Abstract: Closed form expressions for the domination number of an \(n \times m\) grid have attracted significant attention, and an exact expression has been obtained in 2011 [7]. In this paper, we present our results on obtaining new lower bounds on the connected domination number of an \(n \times m\) grid. The problem has been solved for grids with up to 4 rows and with 6 rows and the best currently known lower bound for arbitrary m, n is Open image in new window [11]. Fujie [4] came up with a general construction for a connected dominating set of an \(n \times m\) grid. In this paper, we investigate whether this construction is indeed optimum. We prove a new lower bound of Open image in new window for arbitrary \(m,n \ge 4\).

1 citations

Journal ArticleDOI
TL;DR: In this article , it was shown that any depth reduction in the explanation incurs unbounded loss in the k-means and k-median cost, even when the input points are in the Euclidean plane.
Abstract: Over the last few years Explainable Clustering has gathered a lot of attention. Dasgupta et al. [ICML'20] initiated the study of explainable k-means and k-median clustering problems where the explanation is captured by a threshold decision tree which partitions the space at each node using axis parallel hyperplanes. Recently, Laber et al. [Pattern Recognition'23] made a case to consider the depth of the decision tree as an additional complexity measure of interest. In this work, we prove that even when the input points are in the Euclidean plane, then any depth reduction in the explanation incurs unbounded loss in the k-means and k-median cost. Formally, we show that there exists a data set X in the Euclidean plane, for which there is a decision tree of depth k-1 whose k-means/k-median cost matches the optimal clustering cost of X, but every decision tree of depth less than k-1 has unbounded cost w.r.t. the optimal cost of clustering. We extend our results to the k-center objective as well, albeit with weaker guarantees.

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TL;DR: In this article, the authors introduced a variant of partial domination called the connected α-partial domination, which is the smallest cardinality of a connected αpartial dominating set in a graph.
Abstract: This paper introduces and investigates a variant of partial domination called the connected α-partial domination. For any graph G = (V (G), E(G)) and α ∈ (0, 1], a set S ⊆ V (G) is an α-partial dominating set in G if |N[S]| ≥ α |V (G)|. An α-partial dominating set S ⊆ V (G) is a connected α-partial dominating set in G if ⟨S⟩, the subgraph induced by S, is connected. The connected α-partial domination number of G, denoted by ∂Cα(G), is the smallest cardinality of a connected α-partial dominating set in G. In this paper, we characterize the connected α-partial dominating sets in the join and lexicographic product of graphs for any α ∈ (0, 1] and determine the corresponding connected α-partial domination numbers of graphs resulting from the said binary operations. Moreover, we establish sharp bounds for the connected α-partial domination numbers of the corona and Cartesian product of graphs. Furthermore, we determine ∂Cα(G) of some special graphs when α =1/2. Several realization problems are also generated in this paper.

1 citations

Posted Content
TL;DR: In this article, the minimum cardinality of a connected dominating set, called the connected domination number (CDP), of an undirected simple graph was determined for any $m \times n$ grid graph.
Abstract: Given an undirected simple graph, a subset of the vertices of the graph is a {\em dominating set} if every vertex not in the subset is adjacent to at least one vertex in the subset. A subset of the vertices of the graph is a {\em connected dominating set} if the subset is a dominating set and the subgraph induced by the subset is connected. In this paper, we determine the minimum cardinality of a connected dominating set, called the {\em connected domination number}, of an $m \times n$ grid graph for any $m$ and $n$.
20 Apr 2023
TL;DR: The second page of the Mayer-Vietoris spectral sequence with respect to anti-star covers can be identified with another homological invariant of simplicial complexes: the $0$-degree ''uberhomology'' as mentioned in this paper .
Abstract: We prove that the second page of the Mayer-Vietoris spectral sequence, with respect to anti-star covers, can be identified with another homological invariant of simplicial complexes: the $0$-degree \"uberhomology. Consequently, we obtain a combinatorial interpretation of the second page of the Mayer-Vietoris sequence in this context. This interpretation is then used to extend the computations of bold homology, which categorifies the connected domination polynomial at $-1$.