arXiv subjects

Pablo Torres

Publications and source records attributed to Pablo Torres.

16 recordsLinked to original sources

arXiv2026

On the spectrum and expansion of graph associahedra

In this article, we contribute to the spectral analysis of graph associahedra by providing a lower bound for the second largest eigenvalue of $\mathcal{A}(G)$. Furthermore, using equitable partitions, we analyze the spectrum of the stellohedron $\mathcal{A}(K_{1,n})$. Specifically, we prove the existence of an eigenvalue in each interval $(n-i, n-i+1]$ for $1 \leq i \leq 5$, establish the presence of an eigenvalue with high multiplicity in $(n - \frac{3}{n} + \frac{2}{n^2-n}, n)$, and identify two additional small eigenvalues.

math.CO↗

arXiv2025

Grundy double domination number: bounds, graph operations, and efficient computation for $P_4$-tidy graphs

Inspired by graph domination games, various domination-type vertex sequences have been introduced, including the Grundy double dominating sequence (GDDS) of a graph and its associated parameter, the Grundy double domination number (GDDN). The decision version of the problem of computing the GDDN is known to be NP-complete, even when restricted to split graphs and bipartite graphs. In this paper, we establish general tight bounds for the GDDN. We also describe GDDSs for vertex-removed graphs and for the join of two graphs. Applying these results, we prove that computing the GDDN is linear for $P_4$-tidy graphs, thereby solving an open problem previously posed for cographs by B. Bre\v{s}ar et al. in [Bre\v{s}ar, B., Pandey, A., and Sharma, G. (2022). Computational aspects of some vertex sequences of grundy domination-type. Indian J. Discrete Math., 8:21-38].

math.CO↗

arXiv2024

Effect of graph operations on graph associahedra

Given a graph $G$, we determine the structure of the rotation graph of a graph obtained by applying certain operations to $G$. Specifically, we consider the operations of adding a simplicial vertex, adding a true twin to a vertex, and the two closely related operations of deleting the set of edges from a subgraph induced by a set of true twins, and adding a false twin to a vertex. We describe how applying these operations to a graph affects the structure of its rotation graph. Furthermore, by using this description, we study chromatic number, distance, and diameter in rotation graphs. In particular, we establish conditions under which the chromatic number of the rotation graphs is preserved. As an interesting consequence, we obtain that the chromatic number of the rotation graphs of threshold graphs (which includes complete split graphs and star graphs) and of complete bipartite graphs is 3. We also provide a new lower bound for $\text{diam}(\mathcal{R}(G-S))$ in terms of $\text{diam}(\mathcal{R}(G))$, where $S$ is the set of edges of the subgraph of $G$ induced by a set of true twins. As a consequence, we improve the known lower bound for the diameter of the rotation graph of balanced complete bipartite graphs, allowing us to compute the exact value of $\text{diam}(\mathcal{R}(K_{2,q}))$ for $q\in\{3,4,5,6,7,8\}$.

math.CO↗

arXiv2023

$k$-tuple domination on Kneser graphs

This paper considers multiple domination on Kneser graphs. We focus on $k$-tuple dominating sets, $2$-packings and the associated graph parameters $k$-tuple domination number and $2$-packing number. In particular, we compute the $2$-packing number of Kneser graphs $K(3r-2,r)$ and in odd graphs we obtain minimum $k$-tuple dominating sets of $K(7,3)$ and $K(11,5)$ for every $k$. Besides, we determine the Kneser graphs $K(n,r)$ with $k$-tuple domination number exactly $k+r$ and find all the minimum $k$-tuple dominating sets for these graphs, which generalize results for domination on Kneser graphs. Finally, we give a characterization of the $k$-tuple dominating sets of $K(n,2)$ in terms of the occurrences of the elements in $[n]$, which allows us to obtain minimum sized $k$-tuple dominating sets of $K(n,2)$ for $n\geq \Omega(\sqrt{k})$. Keywords: Kneser graphs, multiple domination, $k$-tuple domination, $2$-packings.

math.CO↗

arXiv2023

Which Argumentative Aspects of Hate Speech in Social Media can be reliably identified?

With the increasing diversity of use cases of large language models, a more informative treatment of texts seems necessary. An argumentative analysis could foster a more reasoned usage of chatbots, text completion mechanisms or other applications. However, it is unclear which aspects of argumentation can be reliably identified and integrated in language models. In this paper, we present an empirical assessment of the reliability with which different argumentative aspects can be automatically identified in hate speech in social media. We have enriched the Hateval corpus (Basile et al. 2019) with a manual annotation of some argumentative components, adapted from Wagemans (2016)'s Periodic Table of Arguments. We show that some components can be identified with reasonable reliability. For those that present a high error ratio, we analyze the patterns of disagreement between expert annotators and errors in automatic procedures, and we propose adaptations of those categories that can be more reliably reproduced.

cs.CL↗

arXiv2022

Parsimonious Argument Annotations for Hate Speech Counter-narratives

We present an enrichment of the Hateval corpus of hate speech tweets (Basile et. al 2019) aimed to facilitate automated counter-narrative generation. Comparably to previous work (Chung et. al. 2019), manually written counter-narratives are associated to tweets. However, this information alone seems insufficient to obtain satisfactory language models for counter-narrative generation. That is why we have also annotated tweets with argumentative information based on Wagemanns (2016), that we believe can help in building convincing and effective counter-narratives for hate speech against particular groups. We discuss adequacies and difficulties of this annotation process and present several baselines for automatic detection of the annotated elements. Preliminary results show that automatic annotators perform close to human annotators to detect some aspects of argumentation, while others only reach low or moderate level of inter-annotator agreement.

cs.CL↗

arXiv2022

High-resolution large-eddy simulations of simplified urban flows

High-fidelity large-eddy simulations of the flow around two rectangular obstacles are carried out at a Reynolds number of 10,000 based on the free-stream velocity and the obstacle height. The incoming flow is a developed turbulent boundary layer. Mean-velocity components, turbulence fluctuations, and the terms of the turbulent-kinetic-energy budget are analyzed for three flow regimes: skimming flow, wake interference, and isolated roughness. Three regions are identified where the flow undergoes the most significant changes: the first obstacle's wake, the region in front of the second obstacle, and that around the second obstacle. In the skimming-flow case, turbulence activity in the cavity between the obstacles is limited and mainly occurs in a small region in front of the second obstacle. In the wake-interference case, there is a strong interaction between the free-stream flow that penetrates the cavity and the wake of the first obstacle. This interaction results in more intense turbulent fluctuations between the obstacles. In the isolated-roughness case, the wake of the first obstacle is in good agreement with that of an isolated obstacle. Separation bubbles with strong turbulent fluctuations appear around the second obstacle.

physics.flu-dyn↗

arXiv2021

Aim in Climate Change and City Pollution

The sustainability of urban environments is an increasingly relevant problem. Air pollution plays a key role in the degradation of the environment as well as the health of the citizens exposed to it. In this chapter we provide a review of the methods available to model air pollution, focusing on the application of machine-learning methods. In fact, machine-learning methods have proved to importantly increase the accuracy of traditional air-pollution approaches while limiting the development cost of the models. Machine-learning tools have opened new approaches to study air pollution, such as flow-dynamics modelling or remote-sensing methodologies.

cs.LG↗

arXiv2021

On the diameter of Schrijver graphs

For $k \geq 1$ and $n \geq 2k$, the well known Kneser graph $\operatorname{KG}(n,k)$ has all $k$-element subsets of an $n$-element set as vertices; two such subsets are adjacent if they are disjoint. Schrijver constructed a vertex-critical subgraph $\operatorname{SG}(n,k)$ of $\operatorname{KG}(n,k)$ with the same chromatic number. In this paper, we compute the diameter of the graph $\operatorname{SG}(2k+r,k)$ with $r \geq 1$. We obtain an exact value of the diameter of $\operatorname{SG}(2k+r,k)$ when $r \in \{1,2\}$ or when $r \geq k-3$. For the remained cases, when $3 \leq r \leq k-4$, we obtain that the diameter of $\operatorname{SG}(2k+r,k)$ belongs to the integer interval $[4..k-r-1]$.

math.CO↗

arXiv2021

Compact and Effective Representations for Sketch-based Image Retrieval

Sketch-based image retrieval (SBIR) has undergone an increasing interest in the community of computer vision bringing high impact in real applications. For instance, SBIR brings an increased benefit to eCommerce search engines because it allows users to formulate a query just by drawing what they need to buy. However, current methods showing high precision in retrieval work in a high dimensional space, which negatively affects aspects like memory consumption and time processing. Although some authors have also proposed compact representations, these drastically degrade the performance in a low dimension. Therefore in this work, we present different results of evaluating methods for producing compact embeddings in the context of sketch-based image retrieval. Our main interest is in strategies aiming to keep the local structure of the original space. The recent unsupervised local-topology preserving dimension reduction method UMAP fits our requirements and shows outstanding performance, improving even the precision achieved by SOTA methods. We evaluate six methods in two different datasets. We use Flickr15K and eCommerce datasets; the latter is another contribution of this work. We show that UMAP allows us to have feature vectors of 16 bytes improving precision by more than 35%.

cs.CV↗

arXiv2018

Grundy dominating sequences on $X$-join product

In this paper we study the Grundy domination number on the $X$-join product $G\hookleftarrow \mathcal R$ of a graph $G$ and a family of graphs $\mathcal R=\{G_v: v\in V(G)\}$. The results led us to extend the few known families of graphs where this parameter can be efficiently computed. We prove that if, for all $v\in V(G)$, the Grundy domination number of $G_v$ is given, and $G$ is a power of a cycle, a power of a path, or a split graph, computing the Grundy domination number of $G\hookleftarrow \mathcal R$ can be done in polynomial time. In particular, the results for power of cycles and paths are derived from a polynomial reduction to the Maximum Weight Independent Set problem on these graphs. As a consequence, we derive closed formulas to compute the Grundy domination number of the lexicographic product $G\circ H$ when $G$ is a power of a cycle, a power of a path or a split graph, generalizing the results on cycles and paths given by Bresar et al. in 2016. Moreover, the results on the $X$-join product when $G$ is a split graph also provide polynomial-time algorithms to compute the Grundy domination number for $(q,q-4)$ graphs, partner limited graphs and extended $P_4$-laden graphs, graph classes which are high in the hierarchy of few $P_4$'s graphs.

math.CO↗

arXiv2016

Total dominating sequences in trees, split graphs, and under modular decomposition

A sequence of vertices in a graph $G$ with no isolated vertices is called a total dominating sequence if every vertex in the sequence totally dominates at least one vertex that was not totally dominated by preceding vertices in the sequence, and, at the end all vertices of $G$ are totally dominated (by definition a vertex totally dominates its neighbors). The maximum length of a total dominating sequence is called the Grundy total domination number, $\gamma_{\rm gr}^t(G)$, of $G$, as introduced in [B. Bre\v{s}ar, M.A. Henning, and D. F. Rall, Total dominating sequences in graphs, Discrete Math. 339 (2016), 1165--1676]. In this paper we continue the investigation of this concept, mainly from the algorithmic point of view. While it was known that the decision version of the problem is NP-complete in bipartite graphs, we show that this is also true if we restrict to split graphs. A linear time algorithm for determining the Grundy total domination number of an arbitrary tree $T$ is presented, based on the formula $\gamma_{\rm gr}^t(T)=2\tau(T)$, where $\tau(T)$ is the vertex cover number of $T$. A similar efficient algorithm is presented for bipartite distance-hereditary graphs. Using the modular decomposition of a graph, we present a frame for obtaining polynomial algorithms for this problem in classes of graphs having relatively simple modular subgraphs. In particular, a linear algorithm for determining the Grundy total domination number of $P_4$-tidy graphs is presented. In addition, we prove a realization result by exhibiting a family of graphs $G_k$ such that $\gamma_{\rm gr}^t(G_k)=k$, for any $k\in{\mathbb{Z}^+}\setminus\{1,3\}$, and showing that there are no graphs $G$ with $\gamma_{\rm gr}^t(G)\in \{1,3\}$. We also present such a family, which has minimum possible order and size among all graphs with Grundy total domination number equal to $k$.

math.CO↗

arXiv2016

Shifts of the Stable Kneser Graphs and Hom-Idempotence

A graph $G$ is said to be {\em hom-idempotent} if there is a homomorphism from $G^2$ to $G$, and {\em weakly hom-idempotent} if for some $n \geq 1$ there is a homomorphism from $G^{n+1}$ to $G^n$. Larose et al. [{\em Eur. J. Comb. 19:867-881, 1998}] proved that Kneser graphs $\operatorname{KG}(n,k)$ are not weakly hom-idempotent for $n \geq 2k+1$, $k\geq 2$. For $s \geq 2$, we characterize all the shifts (i.e., automorphisms of the graph that map every vertex to one of its neighbors) of $s$-stable Kneser graphs $\operatorname{KG}(n,k)_{s-\operatorname{stab}}$ and we show that $2$-stable Kneser graphs are not weakly hom-idempotent, for $n \geq 2k+2$, $k \geq 2$. Moreover, for $s,k\geq 2$, we prove that $s$-stable Kneser graphs $\operatorname{KG}(ks+1,k)_{s-\operatorname{stab}}$ are circulant graphs and so hom-idempotent graphs. Finally, for $s \geq 3$, we show that $s$-stable Kneser graphs $\operatorname{KG}(2s+2,2)_{s-\operatorname{stab}}$ are cores, not $\chi$-critical, not hom-idempotent and their chromatic number is equal to $s+2$.

math.CO↗

arXiv2015

On the Packing Chromatic Number on Hamming Graphs and General Graphs

The packing chromatic number $\chi_\rho(G)$ of a graph $G$ is the smallest integer $k$ needed to proper color the vertices of $G$ in such a way the distance between any two vertices having color $i$ be at least $i+1$. We obtain $\chi_\rho(H_{q,m})$ for $m=3$, where $H_{q,m}$ is the Hamming graph of words of length $m$ and alphabet with $q$ symbols, and tabulate bounds of them for $m \geq 4$ up to 10000 vertices. We also give a polynomial reduction from the problem of finding $\chi_\rho(G)$ to the Maximum Stable Set problem.

cs.DM↗

arXiv2015

The automorphism group of the $s$-stable Kneser graphs

For $k,s\geq2$, the $s$-stable Kneser graphs are the graphs with vertex set the $k$-subsets $S$ of $\{1,\ldots,n\}$ such that the circular distance between any two elements in $S$ is at least $s$ and two vertices are adjacent if and only if the corresponding $k$-subset are disjoint. Braun showed that for $n\geq 2k+1$ the automorphism group of the $2$-stable Kneser graphs (Schrijver graphs) is isomorphic to the dihedral group of order $2n$. In this paper we generalize this result by proving that for $s\geq 2$ and $n\geq sk+1$ the automorphism group of the $s$-stable Kneser graphs also is isomorphic to the dihedral group of order $2n$.

math.CO↗

arXiv2013

The Identifying Code problem on $P_4$-tidy graphs

In this paper we first analyze the behaviour of the identifying code number under union and join operations in graphs. This study forced us to analyze three new parameters related to identifying codes, dominating sets and total dominating sets of the complementary graph. We obtain closed formulas for these parameters on spider and quasi-spider graphs. These results easily derive in a linear dynamic programming-based algorithm for the Identifying Code problem for $P_4$-tidy graphs.

math.CO↗