Advances in Graph Representations and Pattern Counting

Recent research has seen significant advancements in the representation of graphs and the counting of permutation patterns. In the realm of graph theory, there is a growing interest in exploring novel connections between graph representations and formal languages. This approach allows for the identification of new graph classes and the study of their properties, such as heredity and decidability. Notably, the use of language theory has enabled the discovery of graph classes that match the efficiency of standard representations, addressing previous limitations in graph storage and retrieval.

In the domain of pattern counting, there has been a notable shift towards developing approximate algorithms for counting permutation patterns. These algorithms, which provide near-linear time approximations, represent a significant leap forward, particularly for patterns of small fixed lengths. This development not only bridges the gap between detection and counting but also introduces new methodologies, such as the Birgé decomposition, into the field of pattern counting.

Noteworthy Papers

• Eulerian orientations and Hadamard codes: A novel connection via counting: This paper establishes a groundbreaking link between Eulerian orientations and Hadamard codes, significantly advancing the understanding of counting problems in graph theory.
• Approximate counting of permutation patterns: This work introduces a near-linear time approximation algorithm for counting permutation patterns, marking a significant step forward in pattern counting methodologies.