We work with combinatorial maps to represent graph embeddings into surfaces up to isotopy. The surface in which the graph is embedded is left implicit in this approach. The constructions herein are proof-relevant and stated with a subset of the language of homotopy type theory.This article presents a refinement of one characterisation of embeddings in the sphere, called spherical maps, of connected and directed multigraphs with discrete node sets. A combinatorial notion of homotopy for walks and the normal form of walks under a reduction relation is introduced. The first characterisation of spherical maps states that a graph can be embedded in the sphere if any pair of walks with the same endpoints are merely walk-homotopic. The refinement of this definition filters out any walk with inner cycles. As we prove in one of the lemmas, if a spherical map is given for a graph with a discrete node set, then any walk in the graph is merely walk-homotopic to a normal form.The proof assistant Agda contributed to formalising the results recorded in this article.

This paper discusses the so-called missing data problem, i.e. the problem of imputing missing values in information systems. A new algorithm, called the ARSI algorithm, is proposed to address the imputation problem of missing values on categorical databases using the framework of rough set theory. This algorithm can be seen as a refinement of the ROUSTIDA algorithm and combines the approach of a generalized non-symmetric similarity relation with a generalized discernibility matrix to predict the missing values on incomplete information systems. Computational experiments show that the proposed algorithm is as efficient and competitive as other imputation algorithms.

In this paper, we present a constructive and proof-relevant development of graph theory, including the notion of maps, their faces, and maps of graphs embedded in the sphere, in homotopy type theory. This allows us to provide an elementary characterisation of planarity for locally directed finite and connected multigraphs that takes inspiration from topological graph theory, particularly from combinatorial embeddings of graphs into surfaces. A graph is planar if it has a map and an outer face with which any walk in the embedded graph is walk-homotopic to another. A result is that this type of planar maps forms a homotopy set for a graph. As a way to construct examples of planar graphs inductively, extensions of planar maps are introduced. We formalise the essential parts of this work in the proof assistant Agda with support for homotopy type theory.CCS Concepts: • Theory of computation → Constructive mathematics; Type theory; • Mathematics of computing → Graphs and surfaces.

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