Kuratowski's Theorem Research Articles

Removing even crossings

An edge in a drawing of a graph is called even if it intersects every other edge of the graph an even number of times. Pach and Tóth proved that a graph can always be redrawn so that its even edges are not involved in any intersections. We give a new and significantly simpler proof of the stronger statement that the redrawing can be done in such a way that no new odd intersections are introduced. We include two applications of this strengthened result: an easy proof of a theorem of Hanani and Tutte (the only proof we know of not to use Kuratowski's theorem), and the new result that the odd crossing number of a graph equals the crossing number of the graph for values of at most 3. The paper begins with a disarmingly simple proof of a weak (but standard) version of the theorem by Hanani and Tutte.

Extension of functions with small oscillation

A classical theorem of Kuratowski says that every Baire one function on a Gsubspace of a Polish (= separable completely metriz- able) space X can be extended to a Baire one function on X. Kechris and Louveau introduced a finer gradation of Baire one functions into small Baire classes. A Baire one function f is assigned into a class in this heirarchy depending on its oscillation index �(f). We prove a re- finement of Kuratowski's theorem: if Y is a subspace of a metric space X and f is a real-valued function on Y such thatY (f) < ! � , � < !1, then f has an extension F onto X so thatX(F)� ! � . We also show that if f is a continuous real valued function on Y, then f has an exten- sion F onto X so thatX (F) � 3. An example is constructed to show that this result is optimal.

Variations on a theme of Kuratowski

Kuratowski's Theorem characterizes planar graphs in terms of two excluded subgraphs. In this paper we survey variations of Kuratowski's Theorem. We examine both finite and infinite graphs, surfaces and pseudosurfaces, and generalizations of outer-planarity.

Removing Even Crossings

An edge in a drawing of a graph is called $\textit{even}$ if it intersects every other edge of the graph an even number of times. Pach and Tóth proved that a graph can always be redrawn such that its even edges are not involved in any intersections. We give a new, and significantly simpler, proof of a slightly stronger statement. We show two applications of this strengthened result: an easy proof of a theorem of Hanani and Tutte (not using Kuratowski's theorem), and the result that the odd crossing number of a graph equals the crossing number of the graph for values of at most $3$. We begin with a disarmingly simple proof of a weak (but standard) version of the theorem by Hanani and Tutte.

Covering $\mathbb R^{n+1}$ by graphs of $n$-ary functions and long linear orderings of Turing degrees

A point (x 0 ,...,x n ) ∈ X n+1 is covered by a function f: X n → X iff there is a permutation σ of n + 1 such that x σ(0) = f(x σ(1) ,...,x σ(n) ). By a theorem of Kuratowski, for every infinite cardinal κ exactly κ n-ary functions are needed to cover all of (K +n ) n+1 . We show that for arbitrarily large uncountable κ it is consistent that the size of the continuum is C +n and R n+1 is covered by κ n-ary continuous functions. We study other cardinal invariants of the σ-ideal on R n+1 generated by continuous n-ary functions and finally relate the question of how many continuous functions are necessary to cover R 2 to the least size of a set of parameters such that the Turing degrees relative to this set of parameters are linearly ordered.

Partitioning large vector spaces

The theme of this paper is the generalization of theorems about partitions of the sets of points and lines of finite-dimensional Euclidean spaces ℝdto vector spaces over ℝ of arbitrary dimension and, more generally still, to arbitrary vector spaces over other fields so long as these fields are not too big. These theorems have their origins in the following striking theorem of Sierpiński [12] which appeared a half century ago.Sierpiński's Theorem.The Continuum Hypothesis is equivalent to: There is a partition{X, Y, Z}ofℝ3such that if ℓ is a line parallel to the x-axis[respectively:y-axis, z-axis]then X∩ℓ[respectively:Y∩ℓ, Z∩ℓ]is finite.The history of this theorem and some of its subsequent developments are discussed in the very interesting article by Simms [13]. Sierpiński's Theorem was generalized by Kuratowski [9] to partitions of ℝn+2inton+ 2 sets obtaining an equivalence with. The geometric character that Sierpiński's Theorem and its generalization by Kuratowski appear to have is bogus, since the lines parallel to coordinate axes are essentially combinatorial, rather than geometric, objects. The following version of Kuratowski's theorem emphasizes its combinatorial character.Kuratowski's Theorem.Let n &lt; ω and A be any set. Then∣A∣ ≤ ℵnif and only if there is a partition P:An+2→n+ 2such that if i≤n+ 1and ℓ is a line parallel to the i-th coordinate axis, then{x∈ℓ:P(x) =i}is finite.

On a Theorem of Banach and Kuratowski and $K$-Lusin Sets

In a paper of 1929, Banach and Kuratowski proved-assuming the continuum hypothesis-a combinatorial theorem which implies that there is no non-vanishing σ-additive finite measure μ on R which is defined for every set of reals. It will be shown that the combinatorial theorem is equivalent to the existence of a K-Lusin set of size 2 N 0 and that the existence of such sets is independent of ZFC + ¬CH.

A Kuratowski Theorem for some Pseudosurfaces

A Kuratowski Theorem for some Pseudosurfaces

ACHIRALITY AND PLANARITY

An embedding of a space Y into the 3-sphere S3 is said to be strictly achiral if its image is pointwise fixed by an orientation reversing homeomorphism of S3. A space Y is said to be abstractly planar if it can be embedded into the 2-sphere S2. We first extend Kuratowski Theorem into a criterion for abstract planarity of polyhedra, then show that a polyhedron has a strictly achiral polyhedral embedding into S3 if and only if it is abstractly planar. Some related higher dimensional examples are also discussed.

On scrambled sets and a theorem of Kuratowski on independent sets

The measure of scrambled sets of interval self-maps f : I = [ 0 , 1 ] → I f:I=[0,1] \to I was studied by many authors, including Smítal, Misiurewicz, Bruckner and Hu, and Xiong and Yang. In this note, first we introduce the notion of “ ∗ \ast -chaos" which is related to chaos in the sense of Li-Yorke, and we prove a general theorem which is an improvement of a theorem of Kuratowski on independent sets. Second, we apply the result to scrambled sets of higher dimensional cases. In particular, we show that if a map f : I k → I k ( k ≥ 1 ) f: I^{k} \to I^{k}~(k\geq 1) of the unit k k -cube I k I^k is ∗ \ast -chaotic on I k I^{k} , then for any ϵ &gt; 0 \epsilon &gt; 0 there is a map g : I k → I k g: I^{k} \to I^{k} such that f f and g g are topologically conjugate, d ( f , g ) &gt; ϵ d(f,g) &gt; \epsilon and g g has a scrambled set which has Lebesgue measure 1, and hence if k ≥ 2 k \geq 2 , then there is a homeomorphism f : I k → I k f: I^{k} \to I^{k} with a scrambled set S S satisfying that S S is an F σ F_{\sigma } -set in I k I^k and S S has Lebesgue measure 1.

A Simpler Proof of the Excluded Minor Theorem for Higher Surfaces

We give a simple proof of the fact (which follows from the Robertson–Seymour theory) that a graph which is minimal of genusgcannot contain a subdivision of a large grid. Combining this with the tree-width theorem and the quasi-wellordering of graphs of bounded tree-width in the Robertson–Seymour theory, we obtain a simpler proof of the generalized Kuratowski theorem for each fixed surface. The proof requires no previous knowledge of graph embeddings.

A Characterization of Almost-Planar Graphs

Kuratowski's Theorem, perhaps the most famous result in graph theory, states that K5 and K3,3 are the only non-planar graphs for which both G\e, the deletion of the edge e, and G/e, the contraction of the edge e, are planar for all edges e of G. We characterize the almost-planar graphs, those non-planar graphs for which G\e or G/e is planar for all edges e of G. This paper gives two characterizations of the almost-planar graphs: an explicit description of the structure of almost-planar graphs; and an excluded minor criterion. We also give a best possible bound on the number of edges of an almost-planar graph.

The order of points on the second convex hull of a simple polygon

Given a setV of points on the plane, if {q 1,...,q n } is the set of points on the second convex hull ofV, the order in which these points are visited in anyV-gon is characterized. This order must verify two similar conditions to those of Kuratowski's theorem for planar graphs. Moreover, the number of possible orders that verify these conditions is obtained. It isO(5 n ).

Fuzzy strongly semiopen sets and fuzzy strong semicontinuity

In this paper we first introduce and study the fuzzy strongly semiopen and fuzzy strongly semiclosed sets in fuzzy space. Then we introduce the strong semi-interior and strong semiclosure of a fuzzy set, and Kuratowski's 14-sets theorem of general topology is generalized. Finally we introduce the fuzzy strongly semicontinuous, fuzzy strongly semiopen and fuzzy strongly semiclosed mappings on fuzzy topological spaces, and also establish some of their characteristic properties, and discuss relations between fuzzy continuous, fuzzy semicontinuous and fuzzy strongly semicontinuous mappings.

The Jordan-Schonflies Theorem and the Classification of Surface

(1992). The Jordan-Schonflies Theorem and the Classification of Surfaces. The American Mathematical Monthly: Vol. 99, No. 2, pp. 116-130.

The 2 and 3 representative projective planar embeddings

A graph embedded on a surface is n-representative if every nontrivial closed curve in the surface which does not intersect edges of the embedding must contain at least n vertices of the graph. The property of being n-representative on a surface is closed upward under minor inclusion; hence, by the results of N. Robertson and P. D. Seymour (Graph minors. VIII. A Kuratowski theorem for general surfaces, submitted for publication), the set of minor minimal n-representative embeddings on a surface is finite up to isomorphism. The property of being minor minimal n-representative is invariant under Y- Δ operations. The set of minor minimal 2 and 3 representative embeddings on the projective plane are found. These embeddings are used to produce the topologically minimal 2 and 3 representative projective embeddings.

Graph minors. VIII. A kuratowski theorem for general surfaces

We prove that for any infinite set of graphs of bounded genus, some member of the set is isomorphic to a minor of another. As a consequence, for any surface Σ there is a finite list of graphs, such that a general graph may be drawn in Σ if an only if it topologically contains none of the graphs in the list.

Interpretation of Kuratowski's theorem in graph theory as both a topological abstraction and a chemical reality

Interpretation of Kuratowski's theorem in graph theory as both a topological abstraction and a chemical reality

A Kuratowski theorem for nonorientable surfaces

Let Σ denote a surface. A graph G is irreducible for Σ provided that G does not embed in Σ, but any proper subgraph does so embed. Let I( Σ) denote the set of graphs without degree two vertices which are irreducible for Σ. Observe that a graph embeds in Σ if and only if it does not contain a subgraph homeomorphic to a member of I( Σ). For example, Kuratowski's theorem shows that I( Σ) = { K 3,3, K 5} when Σ is the sphere. In this paper we prove that the set I( Σ) is finite for each nonorientable surface, setting in part a conjecture of Erdös from the 1930s.

A new approach to the linearity of testing planarity of graphs

In testing planarity of graphs, there are many criteria. The earliest one as known is the Kuratowski's theorem, then Whitney's, Maclane's, and so forth. Since the early sixties, people have begun researches on algorithms. Up to 1974, Hopcroft and Tarjan found an algorithm with a computing time being a linear function of the order of a graph. This is the linearity concerned here.