Flavia Bonomo-Braberman

Flavia Bonomo-Braberman, Min Chih Lin, Ignacio Maqueda

A graph is a $(k,\ell)$-graph if its vertex set can be partitioned into $k$ independent sets and $\ell$ cliques. This family simultaneously generalizes split, bipartite, and co-bipartite graphs. While the recognition problem is NP-complete whenever $k\geq 3$ or $\ell\geq 3$, the remaining small cases are polynomial-time solvable. In this paper we revisit the known recognition algorithms for the first nontrivial polynomial cases, namely $(2,1)$-, $(1,2)$-, and $(2,2)$-graphs, and show how to remove specific bottlenecks in their existing recognition procedures. For $(2,1)$-graphs, we show that the extra quadratic enumeration in the algorithm of Brandstädt, Le and Szymczak can be avoided by exploiting the structure of a shortest odd cycle in the relevant residual graph, reducing the running time from $O((n+m)^2)$ to $O(n(n+m))$. By complementation, this yields an $O(n(n+\overline m))$-time recognition algorithm for $(1,2)$-graphs, where $\overline m$ denotes the number of edges of the complement graph. For $(2,2)$-graphs, we refine the sparse-dense partition framework of Feder, Hell, Klein and Motwani by restricting the local-search enumeration to sets that are simultaneously bipartite and co-bipartite, and by using the improved algorithms for $(2,1)$- and $(1,2)$-graphs as preprocessing tools. This gives an $O(n^4(n+\min\{m,\overline m\})^3)$-time recognition algorithm for $(2,2)$-graphs.

Víctor A. Braberman, Flavia Bonomo-Braberman, Yiannis Charalambous et al.

The emergence of prompting as the dominant paradigm for leveraging Large Language Models (LLMs) has led to a proliferation of LLM-native software, where application behavior arises from complex, stochastic data transformations. However, the engineering of such systems remains largely exploratory and ad-hoc, hampered by the absence of conceptual frameworks, ex-ante methodologies, design guidelines, and specialized benchmarks. We argue that a foundational step towards a more disciplined engineering practice is a systematic understanding of the core functional units--generative transformations--and their compositional patterns within LLM-native applications. Focusing on the rich domain of software verification and falsification, we conduct a secondary study of over 100 research proposals to address this gap. We first present a fine-grained taxonomy of generative transformations, abstracting prompt-based interactions into conceptual signatures. This taxonomy serves as a scaffolding to identify recurrent transformation relationship patterns--analogous to software design patterns--that characterize solution approaches in the literature. Our analysis not only validates the utility of the taxonomy but also surfaces strategic gaps and cross-dimensional relationships, offering a structured foundation for future research in modular and compositional LLM application design, benchmarking, and the development of reliable LLM-native systems.

Juan Manuel Baldonado, Flavia Bonomo-Braberman, Víctor Adrián Braberman

Ensuring the reliability and verifiability of large language model (LLM)-enabled systems remains a significant challenge in software engineering. We propose a probabilistic framework for systematically analyzing and improving these systems by modeling and refining distributions over clusters of semantically equivalent outputs. This framework facilitates the evaluation and iterative improvement of Transference Models--key software components that utilize LLMs to transform inputs into outputs for downstream tasks. To illustrate its utility, we apply the framework to the autoformalization problem, where natural language documentation is transformed into formal program specifications. Our case illustrates how distribution-aware analysis enables the identification of weaknesses and guides focused alignment improvements, resulting in more reliable and interpretable outputs. This principled approach offers a foundation for addressing critical challenges in the development of robust LLM-enabled systems.