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17 pages, 340 KB

Open AccessFeature PaperArticle

Efficient Direct Reconstruction of Bipartite (Multi)Graphs from Their Line Graphs Through a Characterization of Their Edges

by Drago Bokal and Janja Jerebic

Mathematics 2025, 13(17), 2876; https://doi.org/10.3390/math13172876 - 5 Sep 2025

Viewed by 1248

Abstract

We study the line graphs of bipartite multigraphs, which naturally arise in combinatorics, game theory, and applications such as scheduling and motion planning. We introduce a new characterization of these graphs via valid partial assignments of the edges of the underlying bipartite multigraph to the vertices of its line graph. We show that an empty assignment extends to a complete one precisely when the graph is a line graph of a bipartite multigraph. Based on this, we design an

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algorithm that incrementally constructs such assignments. The algorithm also provides a data structure supporting efficient solutions to problems of maximum clique, maximum weighted clique, minimum clique cover, chromatic number, and independence number. For line graphs of bipartite simple graphs these problems become solvable in linear time, improving on previously known polynomial-time results. For general bipartite multigraphs, our method enhances the

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recognition algorithm of Peterson and builds on the results of Demaine et al., Hedetniemi, Cook et al., and Gurvich and Temkin. Full article

(This article belongs to the Special Issue New Perspectives of Graph Theory and Combinatorics)

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32 pages, 12366 KB

Open AccessArticle

Process Discovery for Event Logs with Multi-Occurrence Event Types

by László Kovács and Ali Jlidi

Algorithms 2025, 18(2), 83; https://doi.org/10.3390/a18020083 - 4 Feb 2025

Viewed by 2635

Abstract

One of the most actively researched areas in the field of process mining is process discovery, which aims to construct a schema that aligns with existing event trace sequences. Current standard industrial workflow schema induction methods impose certain limitations on the system being examined. To address the shortcomings, this article proposes a novel solution that employs graph neural networks and convolutional neural networks to perform schema discovery. In the first phase of schema generation, we perform equivalence prediction, implemented as an edge prediction task. From the obtained equivalence network, we identify the target schema nodes, which correspond to the maximal quasi-cliques of this network. The results of the performed efficiency tests demonstrate that the proposed method can manage such complex cases that are not covered by standard process discovery methods, and it provides more compact and more precise schema graphs. Full article

(This article belongs to the Section Combinatorial Optimization, Graph, and Network Algorithms)

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26 pages, 602 KB

Open AccessFeature PaperArticle

Spectral Applications of Vertex-Clique Incidence Matrices Associated with a Graph

by Shaun Fallat and Seyed Ahmad Mojallal

Mathematics 2023, 11(16), 3595; https://doi.org/10.3390/math11163595 - 19 Aug 2023

Cited by 1 | Viewed by 2373

Abstract

Using the notions of clique partitions and edge clique covers of graphs, we consider the corresponding incidence structures. This connection furnishes lower bounds on the negative eigenvalues and their multiplicities associated with the adjacency matrix, bounds on the incidence energy, and on the [...] Read more.

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of all real symmetric matrices associated with a graph G, we apply an extended version of an incidence matrix tied to an edge clique cover to establish several classes of graphs that allow two distinct eigenvalues. Full article

(This article belongs to the Special Issue Spectral Graph Theory and the Inverse Eigenvalue Problem of a Graph)

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33 pages, 806 KB

Open AccessArticle

q-Rung Orthopair Fuzzy Competition Graphs with Application in the Soil Ecosystem

by Amna Habib, Muhammad Akram and Adeel Farooq

Mathematics 2019, 7(1), 91; https://doi.org/10.3390/math7010091 - 16 Jan 2019

Cited by 67 | Viewed by 6558

Abstract

The q-rung orthopair fuzzy set is a powerful tool for depicting fuzziness and uncertainty, as compared to the Pythagorean fuzzy model. The aim of this paper is to present q-rung orthopair fuzzy competition graphs (q-ROFCGs) and their generalizations, including q-rung orthopair fuzzy k-competition graphs, p-competition q-rung orthopair fuzzy graphs and m-step q-rung orthopair fuzzy competition graphs with several important properties. The study proposes the novel concepts of q-rung orthopair fuzzy cliques and triangulated q-rung orthopair fuzzy graphs with real-life characterizations. In particular, the present work evolves the notion of competition number and m-step competition number of q-rung picture fuzzy graphs with algorithms and explores their bounds in connection with the size of the smallest q-rung orthopair fuzzy edge clique cover. In addition, an application is illustrated in the soil ecosystem with an algorithm to highlight the contributions of this research article in practical applications. Full article

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