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Juan Gutiérrez

Publications and source records attributed to Juan Gutiérrez.

At least 19 recordsLinked to original sources
arXiv2026

The Minimum Subgraph Complementation Problem

Subgraph complementation is an operation that toggles all adjacencies inside a selected vertex set. Given a graph \(G\) and a target class \(\mathcal{C}\), the Minimum Subgraph Complementation problem asks for a minimum-size vertex set \(S\) such that complementing the subgraph induced by \(S\) transforms \(G\) into a graph belonging to \(\mathcal{C}\). While the decision version of Subgraph Complementation has been extensively studied and is NP-complete for many graph classes, the algorithmic complexity of its optimization variant has remained largely unexplored. In this paper, we study MSC from an algorithmic perspective. We present polynomial-time algorithms for MSC in several nontrivial settings. Our results include polynomial-time solvability for transforming graphs between bipartite, co-bipartite, and split graphs, as well as for complementing bipartite regular graphs into chordal graphs. We also show that MSC to the class of graphs of fixed degeneracy can be solved in polynomial time when the input graph is a forest. Moreover, we investigate MSC with respect to connectivity and prove that MSC to the class of disconnected graphs and to the class of 2-connected graphs can be solved in polynomial time for arbitrary inputs.

cs.DS↗
arXiv2026

Improved Domination--Packing Bounds in Claw-Free Cubic Graphs and Unit Disk Graphs

Given a graph $G$, the domination number $γ(G)$ is the minimum cardinality of a dominating set in $G$, and the packing number $ρ(G)$ is the maximum cardinality of a set of vertices that are pairwise at distance at least $3$. The ratio between these parameters has been widely studied in several graph classes. It is known that $γ(G) \le 2ρ(G)$ for claw-free subcubic graphs, up to finitely many exceptions, and that $γ(G) \le 32ρ(G)$ for unit disk graphs. In this paper, we improve the latter bound by showing that $γ(G) \le 16ρ(G)$ for a unit disk graph $G$. For the former bound, we show that it can be improved in the cubic bridgeless setting; more precisely, every bridgeless claw-free cubic graph $G$ satisfies $γ(G) \le \frac{7}{4}ρ(G) + \frac{5}{6}$. These results are not tight. In fact, we give example of an infinite family of bridgeless cubic graphs $G$ with $γ(G) = 5ρ(G)/4$ and an infnite family of unit disk graphs $G$ in which $γ(G) = 3ρ(G)$.

math.CO↗
arXiv2026

Separating Matchings in Cubic Graphs

We study separating matchings in graphs, that is, matchings whose removal increases the number of connected components, and focus on determining the maximum size of such a matching in a graph $G$, denoted by $\mathrm{mms}(G)$. We show that every subcubic graph admits a separating matching, except for exactly eight graphs, which allows us to focus on bounding $\mathrm{mms}(G)$ for cubic graphs. Our main results show that every cubic graph $G$ on $n$ vertices that admits a separating matching satisfies $\mathrm{mms}(G) \ge n/2 - 2$. For bipartite cubic graphs, assuming a conjecture of Funk, the problem reduces to a recursively defined class $\mathcal{F}$, for which we prove that $\mathrm{mms}(G) \ge n/2 - 1$, up to four exceptional graphs. In contrast, we show that every claw-free cubic graph satisfies $\mathrm{mms}(G) = n/2$. These results extend previous work on matching cuts and disconnected $2$-factors, and provide the first systematic study of maximum separating matchings.

math.CO↗
arXiv2026

PANC: Prior-Aware Normalized Cut via Anchor-Augmented Token Graphs

Unsupervised segmentation from self-supervised ViT patches holds promise but lacks robustness: multi-object scenes confound saliency cues, and low-semantic images weaken patch relevance, both leading to erratic masks. To address this, we present Prior-Aware Normalized Cut (PANC), a training-free method that data-efficiently produces consistent, user-steerable segmentations. PANC extends the Normalized Cut algorithm by connecting labeled prior tokens to foreground/background anchors, forming an anchor-augmented generalized eigenproblem that steers low-frequency partitions toward the target class while preserving global spectral structure. With prior-aware eigenvector orientation and thresholding, our approach yields stable masks. Spectral diagnostics confirm that injected priors widen eigengaps and stabilize partitions, consistent with our analytical hypotheses. PANC outperforms strong unsupervised and weakly supervised baselines, achieving mIoU improvements of +2.3% on DUTS-TE, +2.8% on DUT-OMRON, and +8.7% on low-semantic CrackForest datasets.

cs.CV↗
arXiv2026

An Evaluation of Hybrid Annotation Workflows on High-Ambiguity Spatiotemporal Video Footage

Manual annotation remains the gold standard for high-quality, dense temporal video datasets, yet it is inherently time-consuming. Vision-language models can aid human annotators and expedite this process. We report on the impact of automatic Pre-Annotations from a tuned encoder on a Human-in-the-Loop labeling workflow for video footage. Quantitative analysis in a study of a single-iteration test involving 18 volunteers demonstrates that our workflow reduced annotation time by 35% for the majority (72%) of the participants. Beyond efficiency, we provide a rigorous framework for benchmarking AI-assisted workflows that quantifies trade-offs between algorithmic speed and the integrity of human verification.

cs.CV↗
arXiv2025

Strong-lensing rates of massive black hole binaries in LISA

Similarly to electromagnetic (EM) signals, gravitational lensing by intervening galaxies can also affect gravitational waves (GWs). In this paper, we estimate the strong-lensing rate of massive black hole mergers observed with LISA. Given the uncertainties in the source populations as well as in the population of galaxies at high redshift, we consider: six different source population models, including light and heavy seeds, as well as three lens population models, including redshift-independent and redshift-dependent evolution properties. Among all the scenarios explored, the expected number of strong lensed events detected in a 4-year observation time in LISA ranges between 0.13-231 with most of them having two (one) images detectable in the heavy (light) seed scenarios. The event numbers obtained correspond to 0.2%-0.9% of all detected unlensed events. Out of all the detectable strong-lensed events, up to 61% (in the light-seed scenario) and 1% (in the heavy-seed scenario) of them are above the detectability threshold solely due to strong lensing effects and would otherwise be undetectable. For detectable pairs of strong-lensed events by galaxy lenses, we also find between 72%-81% of them to have time delays from 1 week to 1 year.

astro-ph.CO↗
arXiv2025

Towards a Dual Version of Woodall's Conjecture for Partial 3-Trees

A dual version of a conjecture by Woodall asserts that, in a planar digraph, the length of a shortest dicycle equals the maximum number of pairwise disjoint feedback arc sets. We verify this conjecture for the case where the underlying graph is a 3-tree or a partial 3-tree with girth $3$. Additionally, we show that every 3-tree has a feedback arc set of size at most~$m/3-1$, where~$m$ is the number of arcs of the digraph, and this bound is tight. We further establish an upper bound on the size of a minimum feedback arc set in $k$-trees. Finally, we discuss some open problems and conjectures.

math.CO↗
arXiv2025

Open-Source System for Multilingual Translation and Cloned Speech Synthesis

We present an open-source system designed for multilingual translation and speech regeneration, addressing challenges in communication and accessibility across diverse linguistic contexts. The system integrates Whisper for speech recognition with Voice Activity Detection (VAD) to identify speaking intervals, followed by a pipeline of Large Language Models (LLMs). For multilingual applications, the first LLM segments speech into coherent, complete sentences, which a second LLM then translates. For speech regeneration, the system uses a text-to-speech (TTS) module with voice cloning capabilities to replicate the original speaker's voice, maintaining naturalness and speaker identity. The system's open-source components can operate locally or via APIs, offering cost-effective deployment across various use cases. These include real-time multilingual translation in Zoom sessions, speech regeneration for public broadcasts, and Bluetooth-enabled multilingual playback through personal devices. By preserving the speaker's voice, the system ensures a seamless and immersive experience, whether translating or regenerating speech. This open-source project is shared with the community to foster innovation and accessibility. We provide a detailed system performance analysis, including latency and word accuracy, demonstrating its potential to enable inclusive, adaptable communication solutions in real-world multilingual scenarios.

eess.AS↗
arXiv2024

Domination and packing in graphs

Given a graph~$G$, the domination number, denoted by~$γ(G)$, is the minimum cardinality of a dominating set in~$G$. Dual to the notion of domination number is the packing number of a graph. A packing of~$G$ is a set of vertices whose pairwise distance is at least three. The packing number~$ρ(G)$ of~$G$ is the maximum cardinality of one such set. Furthermore, the inequality~$ρ(G) \leq γ(G)$ is well-known. Henning et al.\ conjectured that~$γ(G) \leq 2ρ(G)+1$ if~$G$ is subcubic. In this paper, we progress towards this conjecture by showing that~${γ(G) \leq \frac{120}{49}ρ(G)}$ if~$G$ is a bipartite cubic graph. We also show that $γ(G) \leq 3ρ(G)$ if~$G$ is a maximal outerplanar graph, and that~$γ(G) \leq 2ρ(G)$ if~$G$ is a biconvex graph. Moreover, in the last case, we show that this upper bound is tight.

math.CO↗
arXiv2023

On two conjectures about the intersection of longest paths and cycles

A conjecture attributed to Smith states that every pair of longest cycles in a $k$-connected graph intersect each other in at least $k$ vertices. In this paper, we show that every pair of longest cycles in a~$k$-connected graph on $n$ vertices intersect each other in at least~$\min\{n,8k-n-16\}$ vertices, which confirms Smith's conjecture when $k\geq (n+16)/7$. An analog conjecture for paths instead of cycles was stated by Hippchen. By a simple reduction, we relate both conjectures, showing that Hippchen's conjecture is valid when either $k \leq 6$ or $k \geq (n+9)/7$.

math.CO↗
arXiv2023

Independent dominating sets in planar triangulations

In 1996, Matheson and Tarjan proved that every near planar triangulation on $n$ vertices contains a dominating set of size at most $n/3$, and conjectured that this upper bound can be reduced to $n/4$ for planar triangulations when $n$ is sufficiently large. In this paper, we consider the analogous problem for independent dominating sets: What is the minimum $ε$ for which every near planar triangulation on $n$ vertices contains an independent dominating set of size at most $εn$? We prove that $2/7 \leq ε\leq 5/12$. Moreover, this upper bound can be improved to $3/8$ for planar triangulations, and to $1/3$ for planar triangulations with minimum degree 5.

math.CO↗
arXiv2023

On Tuza's conjecture in co-chain graphs

In 1981, Tuza conjectured that the cardinality of a minimum set of edges that intersects every triangle of a graph is at most twice the cardinality of a maximum set of edge-disjoint triangles. This conjecture have been proved for several important graph classes, as planar graphs, tripartite graphs, among others. However, it remains open on other important classes of graphs, as chordal graphs. Furthermore, it remains open for main subclasses of chordal graphs, as split graphs and interval graphs. In this paper, we show that Tuza's conjecture is valid for co-chain graphs with even number of vertices in both sides of the partition, a known subclass of interval graphs.

math.CO↗
arXiv2022

Path eccentricity of graphs

Let $G$ be a connected graph. The eccentricity of a path $P$, denoted by ecc$_G(P)$, is the maximum distance from $P$ to any vertex in $G$. In the \textsc{Central path} (CP) problem our aim is to find a path of minimum eccentricity. This problem was introduced by Cockayne et al., in 1981, in the study of different centrality measures on graphs. They showed that CP can be solved in linear time in trees, but it is known to be NP-hard in many classes of graphs such as chordal bipartite graphs, planar 3-connected graphs, split graphs, etc. We investigate the path eccentricity of a connected graph~$G$ as a parameter. Let pe$(G)$ denote the value of ecc$_G(P)$ for a central path $P$ of $G$. We obtain tight upper bounds for pe$(G)$ in some graph classes. We show that pe$(G) \leq 1$ on biconvex graphs and that pe$(G) \leq 2$ on bipartite convex graphs. Moreover, we design algorithms that find such a path in linear time. On the other hand, by investigating the longest paths of a graph, we obtain tight upper bounds for pe$(G)$ on general graphs and $k$-connected graphs. Finally, we study the relation between a central path and a longest path in a graph. We show that on trees, and bipartite permutation graphs, a longest path is also a central path. Furthermore, for superclasses of these graphs, we exhibit counterexamples for this property.

math.CO↗
arXiv2021

Bi-traceable graphs, the intersection of three longest paths and Hippchen's conjecture

Let $P,Q$ be longest paths in a simple graph. We analyze the possible connections between the components of $P\cup Q\setminus (V(P)\cap V(Q))$ and introduce the notion of a bi-traceable graph. We use the results for all the possible configurations of the intersection points when $\#V(P)\cap V(Q)\le 5$ in order to prove that if the intersection of three longest paths $P,Q,R$ is empty, then $\#(V(P)\cap V(Q))\ge 6$. We also prove Hippchen's conjecture for $k\le 6$: If a graph $G$ is $k$-connected for $k\le 6$, and $P$ and $Q$ are longest paths in $G$, then $\#(V(P)\cap V(Q))\ge 6$.

math.CO↗
arXiv2021

Minimal graph in which the intersection of two longest paths is not a separator

We prove that for a connected simple graph $G$ with $n\le 10$ vertices, and two longest paths $C$ and $D$ in $G$, the intersection of vertex sets $V(C)\cap V(D)$ is a separator. This shows that the graph found previously with $n=11$, in which the complement of the intersection of vertex sets $V(C)\cap V(D)$ of two longest paths is connected, is minimal.

math.CO↗
arXiv2020

On the intersection of two longest paths in $k$-connected graphs

We show that every pair of longest paths in a $k$-connected graph on $n$ vertices intersect each other in at least $(8k-n+2)/5$ vertices. We also show that, in a 4-connected graph, every pair of longest paths intersect each other in at least four vertices. This confirms a conjecture of Hippchen for $k$-connected graphs when $k\leq 4$ or $k\geq (n-2)/3$.

math.CO↗
arXiv2020

On Tuza's conjecture for triangulations and graphs with small treewidth

Tuza (1981) conjectured that the size $τ(G)$ of a minimum set of edges that intersects every triangle of a graph $G$ is at most twice the size $ν(G)$ of a maximum set of edge-disjoint triangles of $G$. In this paper we present three results regarding Tuza's Conjecture. We verify it for graphs with treewidth at most $6$; we show that $τ(G)\leq \frac{3}{2}\,ν(G)$ for every planar triangulation $G$ different from $K_4$; and that $τ(G)\leq\frac{9}{5}\,ν(G) + \frac{1}{5}$ if $G$ is a maximal graph with treewidth 3. Our first result strengthens a result of Tuza, implying that $τ(G) \leq 2\,ν(G)$ for every $K_8$-free chordal graph $G$.

math.CO↗