arXiv subjects

Soura Sena Das

Publications and source records attributed to Soura Sena Das.

5 recordsLinked to original sources

arXiv2026

Optimal $(1,\le \ell)$-locating-dominating codes in infinite triangular grid

We study two variants of location-domination in infinite triangular grids which allow locating up to $\ell$ vertices simultaneously. A locating-dominating code $S$ is a dominating vertex set such that every vertex outside of it has a unique neighborhood within the code $S$. As this neighborhood is unique, we may use it as an identifier for locating the considered vertex outside of the vertex set. With the variants we study, we may also locate sets of vertices instead of only single vertices. In particular, we obtain the exact minimum densities of these types of sets in the infinite triangular grid for every value of $\ell$.

math.CO↗

arXiv2026

On identifying codes on oriented graphs

This article studies identifying codes in oriented graphs from a computational complexity perspective. We investigate the $\mathcal{F}$-Id Code problem, where given a simple graph $G$ and a vertex subset $C$, which induces a subgraph in the family $\mathcal{F}$, as inputs and ask whether it is possible to orient $G$ in such a way that $C$ becomes its oriented identifying code. Focusing on the family $\mathcal{F}_d$ of $d$-regular graphs, we establish a complete dichotomy by proving that the problem is polynomial-time solvable for $d\leq1$ and NP-complete for all $d\geq2$.

cs.DM↗

arXiv2026

On multiplicativity of directed graphs

A graph category is a category with a set of graphs or similar structures (such as, directed graphs, signed graphs, etc.) playing the role of objects, and an appropriate notion of homomorphism playing the role of morphisms. The characterization of multiplicative objects are important open problems in categories of undirected and directed graphs. While the recent disproving of the Hedetniemi's conjecture due to Shitov (Ann. Math. 2019), which claimed that all complete graphs are multiplicative, provided a breakthrough in the study of multiplicative undirected graphs, the characterization of multiplicative undirected graphs remains known only for cycles, circular cliques $K_{n/k}$ where ${n/k} \in (2,4]$, complete graphs, and graphs whose each edge is part of at most one $4$-cycle. Similarly, whether a given directed graph is multiplicative or not is known only for some oriented paths, oriented cycles, and transitive tournaments. We study multiplicative graphs in the category of directed graphs where pushable homomorphism plays the role of morphism. We provide full multiplicativity characterization for directed bipartite graphs, oriented cycles, and transitive tournaments. As a consequence we find new (infinite) classes of non-multiplicative directed graphs in the usual directed graphs category. We also resolve an open question posed by Das \textit{et al.} (CALDAM 2026) related to the existence of exponential directed graphs with respect to pushable homomorphisms, and use our solution as a tool for our proofs.

cs.DM↗

arXiv2026

On homomorphism related parameters of oriented triangle-free planar graphs

The first major contribution of this work is proving that the oriented relative clique number of oriented triangle-free planar graphs is $10$, which completely answers and closes an open problem posed by Sopena (Discrete Mathematics 2016) in the most recent survey on oriented colorings. The second major contribution of the paper is to prove that if all oriented triangle-free planar graphs admit a homomorphism to a particular oriented graph $\overrightarrow{T}$, then its underlying graph $T$ must have minimum degree at least $10$. This result implies that, for the family of oriented triangle-free planar graphs, the lower bounds of the parameters oriented chromatic number, pushable chromatic number, $2$-dipath $L(p,1)$-labeling span, and oriented $L(p,1)$-labeling span are at least $11$, $6$, $p+8$, and $2p+8$, respectively, where $p \geq 1$. That is, we are able to obtain improved lower bounds of a number of other parameters restricted to the family of oriented triangle-free planar graphs using our second major contribution.

cs.DM↗

arXiv2023

A theoretical expansion of the Sprout game

Sprout is a two-player pen and paper game which starts with $n$ vertices, and the players take turns to join two pre-existing dots by a subdivided edge while keeping the graph sub-cubic planar at all times. The first player not being able to move loses. A major conjecture claims that Player 1 has a winning strategy if and only if $n \equiv 3,4,5$ ($\bmod~6$). The conjecture is verified until $44$, and a few isolated values of $n$, usually with the help of a computer. However, to the best of our understanding, not too much progress could be made towards finding a theoretical proof of the conjecture till now. In this article, we try to take a bottom-up approach and start building a theory around the problem. We start by expanding a related game called Brussels Sprout (where dots are replaced by crosses) introduced by Conway, possibly to help the understanding of Sprout. In particular, we introduce and study a generalized version of Brussels Sprout where crosses are replaced by a dot having an arbitrary number of ``partial edges'' (say, general cross) coming out, and planar graphs are replaced by any (pre-decided) hereditary class of graphs. We study the game for forests, graphs on surfaces, and sparse planar graphs. We also do a nimber characterization of the game when the hereditary class is taken to be triangle-free planar graphs, and we have started the game with two arbitrary generalized crosses. Moreover, while studying this particular case, we naturally stumble upon a circular version of the same game and solve a difficult nimber characterization using the method of structural induction. The above mentioned proof may potentially be one approach to solving the Sprout conjecture.

math.CO↗