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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 (HoTT). This allows us to provide an elementary characterisation of planarity for locally directed finite and connected multigraphs that takes inspirati...
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... nodes. Consider m as a graph map for our house graph induced from Fig. 1 (I). At node 2, the graph map results in the sequence [1,4,3]. This sequence not only lists the adjacent nodes but also specifies a counterclockwise order among the connecting edges. Thus, edge (2, 1) is followed by (2,4) and then by (2,3) in this established order, see Fig. ...
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... The topological graph theory approach inspires our definition of a combinatorial notion of embedding/map in the sphere for graphs [23], referred to as spherical maps in this paper, see Definition 5.4. A graph map can be described by the graph itself and the circular ordering of the edges incident to each vertex [10, §3]. ...
... Given a map M, the faces of M are the regions obtained by the cellular decomposition of the corresponding surface by M. We omit the formal type of faces herein, so as not to distract the reader from the goals of this paper. The type of faces requires proper attention [23]. Put briefly, a face is a cyclic walk in the embedded graph without repeating nodes and without edges inside [10]. ...
... Later, we show an alternative definition for graphs with a node set in Definition 5.5. Given a distinguished face in a connected graph, being spherical for a graph embedding serves to establish elementary planarity criteria for graphs [23]. ...
... The topological graph theory approach inspires our definition of a combinatorial notion of embedding/map in the sphere for graphs [23], referred to as spherical maps in this paper, see Definition 5.4. A graph map can be described by the graph itself and the circular ordering of the edges incident to each vertex [10, §3]. ...
... Given a map M, the faces of M are the regions obtained by the cellular decomposition of the corresponding surface by M. We omit the formal type of faces herein, so as not to distract the reader from the goals of this paper. The type of faces requires proper attention [23]. Put briefly, a face is a cyclic walk in the embedded graph without repeating nodes and without edges inside [10]. ...
... Later, we show an alternative definition for graphs with a node set in Definition 5.5. Given a distinguished face in a connected graph, being spherical for a graph embedding serves to establish elementary planarity criteria for graphs [23]. ...
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.
Introduction: Increasing network communication area has lot of unstructured routing to create complex structures. The communication structure is non-linear to create connective edges to degrade the communication performance. Many non-linear solutions and distance theory models contains maximum non-liability of variables are taken to solve the problems. But structure difference and dynamic variables are constantly applied to make solution which leads errors and complex solutions. To resolve this problem, to propose a Redundant mathematical solution for complex homotopy structures using Multinomial-Cordial Graph Theory (MCGT) based on Bipartite Chromatic Polynomial Distribution Theory (BCPDT) for solving distance problems. To apply neighbor-based distance coverage model with cordial labeling variable structure to reduce the complexity variable structure problems, this paper explores a novel strategy for encapsulating the non-linear complex homotopy in its entirety by employing graph theory and the concept of cordial labeling. By establishing a connection between algebraic topology and graph theoretical constructs, we formulate a redundant solution that illuminates the intricacies of complex homotopy but also provides practical methodologies for solving distance-related issues prevalent in various mathematical and applied fields as well, compared to the previous models.
