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This paper presents a detailed symbolic approach to the study of self-similar tilings. It uses properties of addresses associated with graph-directed iterated function systems to establish conjugacy properties of tiling spaces. Tiles may be…
We generalize the structure theorem of Robertson and Seymour for graphs excluding a fixed graph $H$ as a minor to graphs excluding $H$ as a topological subgraph. We prove that for a fixed $H$, every graph excluding $H$ as a topological…
This work is the extension of the results by the author in [7] and [6] for low-genus surfaces. Let $S$ be an orientable, connected surface of finite topological type, with genus $g \leq 2$, empty boundary, and complexity at least $2$; as a…
Using a notation of corner between edges when graph has a fixed rotation, i.e. cyclical order of edges around vertices, we define combinatorial objects - combinatorial maps as pairs of permutations, one for vertices and one for faces.…
We describe structural properties of strongly connected finite directed graphs, that are invariants of the topological conjugacy of their Markov-Dyck shifts. For strongly connected finite directed graphs with these properties topological…
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…
In the recent years, several polynomial algorithms of a dynamical nature have been proposed to address the graph isomorphism problem. In this paper we propose a generalization of an approach exposed in cond-mat/0209112 and find that this…
We give a constructive method for realizing an arbitrary directed graph (with no one-cycles) as a heteroclinic or an excitable dynamic network in the phase space of a system of coupled cells of two types. In each case, the system is…
Graph theoretical ideas are highly utilized by computer science fields especially data mining. In this field, a data structure can be designed in the form of tree. Covering is a widely used form of data representation in data mining and…
Many complex questions in biology, physics, and mathematics can be mapped to the graph isomorphism problem and the closely related graph automorphism problem. In particular, these problems appear in the context of network visualization,…
The study of complex systems has captured widespread attention in recent years, emphasizing the exploration of interactions and emergent properties among system units. Network analysis based on graph theory has emerged as a powerful…
We relate the graph isomorphism problem to the solvability of certain systems of linear equations with nonnegative variables. This version replaces the two previous versions of this paper.
We prove that the topological cycles of an arbitrary infinite graph induce a matroid. This matroid in general is neither finitary nor cofinitary.
This work is devoted to the study of the relationships between graph theory and the qualitative analysis of ordinary differential equations, with a special focus on two-dimensional systems. In particular, we reinterpret classical results…
Modern methods of graph theory describe a graph up to isomorphism, which makes it difficult to create mathematical models for visualizing graph drawings on a plane. The topological drawing of the planar part of a graph allows representing…
This paper presents sufficient graph-theoretic conditions for injectivity of collections of differentiable functions on rectangular subsets of R^n. The results have implications for the possibility of multiple fixed points of maps and…
We discuss some properties of linear functionals on topological hyperbolic and topological bicomplex modules. The hyperbolic and bicomplex analogues of the uniform boundedness principle, the open mapping theorem, the closed graph theorem…
It has recently been shown that infinite matroids can be axiomatized in a way that is very similar to finite matroids and permits duality. This was previously thought impossible, since finitary infinite matroids must have non-finitary…
In this paper, we construct a bialgebraic and further a Hopf algebraic structure on top of subgraphs of a given graph. Further, we give the dual structure of this Hopf algebraic structure. We study the algebra morphisms induced by graph…
A Hamiltonian decomposition of a regular graph is a partition of its edge set into Hamiltonian cycles. The problem of finding edge-disjoint Hamiltonian cycles in a given regular graph has many applications in combinatorial optimization and…