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Abstract Let X be a local dendrite and let f : X → X be a monotone map. Denote by P (f) and Ω (f) the sets of periodic points and nonwandering points of f , respectively. We show that Ω (f) = P (f) ‾ , whenever P (f) is nonempty and Ω (f) is the unique minimal set included in a circle which is either a Cantor set or a circle, whenever P (f) is empty. In the case where the set of endpoints of X is countable, we show that Ω (f) = P (f) whenever P (f) is nonempty. [ABSTRACT FROM AUTHOR]

Labyrinth fractals are self-similar fractals that were introduced and studied in recent work [2,3] . In the present paper we define and study more general objects, called mixed labyrinth fractals , that are in general not self-similar and are constructed by using sequences of labyrinth patterns . We show that mixed labyrinth fractals are dendrites and study properties of the paths in the graphs associated to prefractals, and of arcs in the fractal, e.g., the path length, and the box counting dimension and length of arcs. We also consider more general objects related to mixed labyrinth fractals, formulate two conjectures about arc lengths, and establish connections to recent results on generalised Sierpiński carpets. [ABSTRACT FROM AUTHOR]

Abstract: The notion of an inverse limit using set-valued functions was introduced by Mahavier in [3], and was further explored by Mahavier and Ingram in [2]. Illanes has shown that a circle cannot be expressed as an inverse limit of an arc indexed by the natural numbers using a single set-valued bonding map [1], and Nall has shown that we cannot use a similar construction to produce any finite graph (other than an arc) [8], or an n-cell for [6]. In this paper, we will explore inverse limits using the integers as an index set. We will present an example of a 2-cell obtained under this index. We will then show that the only finite graph that can be obtained as an inverse limit of arcs using this index is again an arc. [Copyright &y& Elsevier]

Abstract: We give some fixed point theorems in the setting of metric spaces or partial metric spaces endowed with a finite number of graphs. The presented results extend and improve several well-known results in the literature. In particular, we discuss a Caristi type fixed point theorem in the setting of partial metric spaces, which has a close relation to Ekelandʼs principle. [Copyright &y& Elsevier]

Abstract: E. Akin, E. Glasner, and B. Weiss had constructed the special homeomorphism that has a dense conjugacy class in the space of all Cantor homeomorphisms. M. Hochman showed that the universal odometer is the special homeomorphism in the space of all topologically transitive Cantor homeomorphism. Following the approach of E. Akin, E. Glasner, and B. Weiss, we show that the universal odometer is the special homeomorphism in the space of all chain transitive Cantor systems. We extend this result to the space of chain transitive systems that are restricted by a periodic spectrum. Further, we construct the special homeomorphism in the space of all chain recurrent systems. In doing so, every 0-dimensional system is described as the inverse limit of a sequence of finite directed graphs and graph homomorphisms. In the previous paper, we had shown that a certain periodic condition determines whether a Cantor system approximates a chain mixing Cantor system by topological conjugacies. We shall extend this result to the chain transitive case. These conditions are described in terms of sequences of finite directed graphs and graph homomorphisms. [Copyright &y& Elsevier]

Abstract: In this paper we give, in response to a question of Espinoza, Gartside and Mamatelashvili, an example of an n-arc connected (metric) continuum which is not -arc connected for every . [Copyright &y& Elsevier]

Abstract: Let be a countable family of rational functions of two variables with real coefficients. Each rational function can be thought as a continuous function taking values in the projective line and defined on a cofinite subset of the torus . Then the family determines a continuous vector-function defined on the dense -set of . The closure of its graph in is called the graphoid of the family . We prove that the graphoid has topological dimension . If the family contains all linear fractional transformations for , then the graphoid has cohomological dimension for any non-trivial 2-divisible abelian group G. Hence the space is a natural example of a compact space that is not dimensionally full-valued and by this property resembles the famous Pontryagin surface. [Copyright &y& Elsevier]

Abstract: In this paper, we decide the exact value of the color number of a fixed point free homeomorphism on a connected locally finite graph. We prove that for every fixed-point free homeomorphism from a connected locally finite graph into itself, the greatest common divisor of all period for its map is equal to one or three if and only if its color number is 4. [Copyright &y& Elsevier]

Abstract: A fixed point detection theorem for a family of maps defined on the once punctured torus is proved. As a consequence, we produce an example of a homotopy class [f] of self-maps on the once punctured torus that illustrates the following: (i) there is a map in the homotopy class that has no fixed points, and (ii) if the image of f lies in a 1-complex that embeds as a homotopy equivalence, then f must have a fixed point. [Copyright &y& Elsevier]

Abstract: Given a metric continuum X, let and denote the hyperspaces of all nonempty closed subsets and subcontinua, respectively. For we say that B does not block A if and the union of all subcontinua of X intersecting A and contained in is dense in X. In this paper we study some sets of blockers for several kinds of continua. In particular, we determine their Borel classes and, for a large class of locally connected continua X, we recognize them as cap-sets. [Copyright &y& Elsevier]

Abstract: Let X be a compact metric space, and be the group consisting of all homeomorphisms from X to X. A subgroup H of is said to be transitive if there exists a point such that is dense in X. In this paper we show that, if is a connected graph, then the following five conditions are equivalent: (1) has a transitive commutative subgroup; (2) G admits a transitive -action; (3) G admits an edge-transitive commutative group action; (4) G admits an edge-transitive -action; (5) G is a circle, or a k-fold loop with , or a k-fold polygon with , or a k-fold complete bigraph with . As a corollary of this result, we show that a finite connected simple graph whose automorphism group contains an edge-transitive commutative subgroup is either a cycle or a complete bigraph. [Copyright &y& Elsevier]

Abstract: Generalised Sierpiński carpets are planar sets that generalise the well-known Sierpiński carpet and are defined by means of sequences of patterns. We present necessary and sufficient conditions, under which generalised Sierpiński carpets are connected, with respect to Euclidean topology. [Copyright &y& Elsevier]

Abstract: Let Γ be a countable locally finite graph and let and denote the homeomorphism group of Γ with the compact-open topology and its identity component. These groups can be embedded into the space of all closed sets of with the Fell topology, which is compact. Taking closure, we have natural compactifications and . In this paper, we completely determine the topological type of the pair and give a necessary and sufficient condition for this pair to be a -manifold. The pair is also considered for simple examples, and in particular, we find that has homotopy type of . In this investigation we point out a certain inaccuracy in Sakai–Uehara''s preceding results on for finite graphs Γ. [Copyright &y& Elsevier]

Abstract: Locally connected continua which admit monotone maps onto graphs are characterized. A notion of a G-structure is introduced for any graph G as a generalization of a linear or circular chain (of arbitrary finite length) cover of a continuum by its subcontinua. It is proved that a locally connected continuum X has a G-structure iff G is X-like. We show that any nondegenerate locally connected continuum has an arc-structure or a circle-structure. We find some invariants of the likeness relation. [Copyright &y& Elsevier]

In this paper, we give a lower bound for the maximum and minimum genus of a multibranched surface by the first Betti number and the minimum and maximum genus of the boundary of the neighborhood of it, respectively. As its application, we show that the maximum and minimum genus of G × S 1 is equal to twice of the maximum and minimum genus of G for a graph G , respectively. This provides an interplay between graph theory and 3-manifold theory. [ABSTRACT FROM AUTHOR]

Abstract: Generalizing results by J. Ford, J. W. Rogers, Jr. and H. Kato we prove that (1) a map f from a G-like continuum onto a graph G is refinable iff f is monotone; (2) a graph G is an arc or a simple closed curve iff every G-like continuum that contains no nonboundary indecomposable subcontinuum admits a monotone map onto G. We prove that if bonding maps in the inverse sequence of compact spaces are refinable then the projections of the inverse limit onto factor spaces are refinable. We use this fact to show that refinable maps do not preserve completely regular or totally regular continua. [Copyright &y& Elsevier]

Abstract: We investigate connected partitions of continua into compacta. In particular, we consider continua with property that every connected partition into compacta is compact. We characterize graphs which have this property as the trees and the simple closed curve. Dendrites are shown to have the property. An example of a nonlocally connected continuum with the property is also given. [Copyright &y& Elsevier]

Abstract: A Lefschetz formula is given that relates loops in a regular finite graph to traces of a certain representation. As an application the vanishing orders of the Ihara/Bass zeta function are expressed as dimensions of global section spaces of locally constant sheaves. [Copyright &y& Elsevier]

A graph has an increasing ear decomposition if it can be constructed from a simple closed curve by attaching arcs in stages with the endpoints of each arc attached to different points so that at least one new branch point is formed at each stage. A reduced 2-to-1 map is a 2-to-1 map that does not have a restriction that is 2-to-1. A 2-to-1 cut set of a graph G is a finite subset B such that G&z.drule;B has at least 2&z.sfnc;B&z.sfnc; components. A graph has an increasing ear decomposition if and only if it does not have a 2-to-1 cut set, and a graph is the image of a reduced 2-to-1 map if and only if it does not have a 2-to-1 cut set. [Copyright &y& Elsevier]

We introduce new formulas that are Vassiliev invariants of flat vertex isotopy classes of 2-bouquet graphs. Although any Gauss diagram formula of Vassiliev invariants of 2-bouquet graphs in a 3-space has been unknown explicitly, this paper gives the first and simple example. [ABSTRACT FROM AUTHOR]

We consider the problem of realizing abstract graph symmetries by symmetries of the 3-sphere through spatial embeddings. The techniques usually used in preceding studies are based on the placement of vertices and edges, and the problem has been solved for several well-known families of graphs. This paper provides a technique which is applicable to any graph, and gives an affirmative answer to the problem for abstract graph symmetries given by finite subgroups of O (4) with some property on their singular sets. [ABSTRACT FROM AUTHOR]

Abstract: We present a simple way to obtain all graphs with a given disconnection number if we know all the graphs with smaller disconnection number. Using this method we answer a question of Sam B. Nadler, i.e. we prove that there exist precisely 26 continua with disconnection number four. Some known results concerning disconnection number are also obtained as simple corollaries. Further we give some estimates on the number of distinct continua with a fixed disconnection number which concerns a problem of Sam B. Nadler. [ABSTRACT FROM AUTHOR]