Search the repository

A+ | A- | | SLO | ENG

Abstract: This article discusses mutual-visibility in graphs through a game-based version of the problem. Two players, Builder and Blocker, alternately select an unmarked vertex on a graph keeping the property that the set of marked vertices forms a mutual-visibility set. The game ends when no such selection is possible. The goal of Builder is to create a largest possible mutual-visibility set, Blocker's goal is the opposite. The central problem here is to determine the number of vertices selected during the game assuming that both players played optimally. Bounds on this number are proved and several general properties of the game derived. Special attention is paid to complete multipartite graphs and Hamming graphs.
Keywords: mutual-visibility set, games on graphs, complete multipartite graph, Hamming graph
Published in DiRROS: 02.04.2026; Views: 210; Downloads: 127
Full text (342,16 KB)
This document has many files! More...

Abstract: Building on the success of large language models (LLMs), LLM-based representations have dominated the document representation landscape, achieving strong performance on document embedding benchmarks. However, high-dimensional, computationally expensive LLM embeddings can be too generic or inefficient for domain-specific and resource-scarce applications. To address these limitations, we introduce FuDoBa—a Bayesian optimisation-based representation learning method that integrates LLM embeddings with domain-specific structured knowledge, sourced both locally and from external repositories such as WikiData. This fusion produces low-dimensional, task-relevant representations while reducing training complexity and yielding interpretable early-fusion weights for improved classification performance. We demonstrate the effectiveness of our approach on six datasets across two domains, showing that when paired with robust AutoML-based classifiers, our method performs on par with, or surpasses, proprietary LLM-only embedding baselines, while offering modality-wise interpretability and a smaller dimensional footprint.
Keywords: document classification, Bayesian optimisation, representation learning, knowledge graphs
Published in DiRROS: 13.03.2026; Views: 230; Downloads: 235
Full text (7,33 MB)
This document has many files! More...

Abstract: A set of vertices $X\subseteq V(G)$ is a $d$-distance dominating set if for every $u\in V(G)\setminus X$ there exists $x\in X$ such that $d(u,x) \le d$, and $X$ is a $p$-packing if $d(u,v) \ge p+1$ for every different $u,v\in X$. The $d$-distance $p$-packing domination number $\gamma_d^p(G)$ of $G$ is the minimum size of a set of vertices of $G$ which is both a $d$-distance dominating set and a $p$-packing. It is proved that for every two fixed integers $d$ and $p$ with $2 \le d$ and $0 \le p \leq 2d-1$, the decision problem whether $\gamma_d^p(G) \leq k$ holds is NP-complete for bipartite planar graphs. A necessary and sufficient condition for the existence of a $d$-distance $p$-packing dominating set in $C_n$ is obtained and $\gamma_d^p(C_n)$ determined for every $d$, $p$, and $n$. For a tree $T$ on $n$ vertices with $\ell$ leaves and $s$ support vertices it is proved that (i) $\gamma_2^0(T) \geq \frac{n-\ell-s+4}{5}$, (ii) $\left \lceil \frac{n-\ell-s+4}{5} \right \rceil \leq \gamma_2^2(T) \leq \left \lfloor \frac{n+3s-1}{5} \right \rfloor$, and if $d \geq 2$, then (iii) $\gamma_d^2(T) \leq \frac{n-2\sqrt{n}+d+1}{d}$. Inequality (i) improves an earlier bound due to Meierling and Volkmann, and independently Raczek, Lema\'nska, and Cyman, while (iii) extends an earlier result for $\gamma_2^2(T)$ due to Henning. Sharpness of the bounds is discussed and established in most cases. It is also proved that every connected graph $G$ contains a spanning tree $T$ such that $\gamma_2^2(T) \leq \gamma_2^2(G)$.
Keywords: d-distance dominating set, p-packing set, dominating set, trees, planar graphs
Published in DiRROS: 23.02.2026; Views: 288; Downloads: 183
Full text (634,30 KB)
This document has many files! More...

Abstract: The ▫$\sigma$▫-irregularity, a variant of the well-established Albertson irregularity, is a topological invariant defined for a graph ▫$G=(V,E)$▫ as ▫$\sigma (G) = \sum_{u,v \in E}(d(u) - d(v))^2$▫, where ▫$d(u)$▫ and ▫$d(v)$▫ denote the degrees of vertices ▫$u$▫ and ▫$v$▫, respectively. Recent research has successfully characterized chemical trees with the maximum ▫$\sigma$▫-irregularity. In this paper, we expand upon this research by establishing several structural properties of maximal trees with prescribed maximum degree ▫$\Delta$▫. Application of these properties enables us to characterize maximal trees with ▫$\Delta = 5$▫. We establish that extremal trees contain only vertices of degrees ▫$1$▫, ▫$2$▫ and ▫$\Delta$▫. Moreover, the number of edges with both end-vertices having the degree ▫$2$▫ or ▫$\Delta$▫ is very small, so almost all edges have the (second) maximum possible contribution to ▫$\sigma$▫-irregularity. We believe this property or similar should extend to maximal trees for any value of ▫$\Delta$▫, so this is an interesting direction for further research.
Keywords: regular graph, trees, maximum degree, [sigma]-irregularity, maximal graphs, graph measure, topological index
Published in DiRROS: 05.02.2026; Views: 632; Downloads: 274
Full text (679,47 KB)
This document has many files! More...

Abstract: This paper explores the applications of Intuitionistic Fuzzy Graphs (ℐ ℱ � ) representing uncertainty and impre cision in complex systems through the analysis of correlation and regression coefficients (� ℛ� �) with focus on the maximal product. The study examines the relationships between the edges of the graph by analysing the line graph derived from ℐ ℱ � , facilitating a deeper understanding of the network’s dynamics. The construction of adjacency matrices that incorporate both membership and non-membership values enables the calculation of energy and weight scores, quantifying the strength and predictive correlations among variables. Furthermore, the study discusses the complement of Intuitionistic Fuzzy Line Graphs (ℐ ℱ ℒ � ), using maximal product anal ysis to uncover concealed relationships within the network. MATLAB is used to generate heatmaps that visually represent the importance of correlation to critical network characteristics. The practical importance is demon strated in a healthcare context, particularly in predicting diabetes risk by modelling factors of glucose levels, body mass index (BMI), and insulin. Heatmaps can be effectively visualized to show interrelationships between these features, aiding in the interpretation of network patterns.
Keywords: intuitionistic fuzzy graphs, intuitionistic fuzzy line graphs, maximal product, adjacency matrices, correlation and regression coefficients
Published in DiRROS: 04.02.2026; Views: 618; Downloads: 262
Full text (3,20 MB)
This document has many files! More...