C-Planarity Testing of Embedded Clustered Graphs with Bounded Dual Carving-Width

For a clustered graph, i.e, a graph whose vertex set is recursively partitioned into clusters, the C-Planarity Testing problem asks whether it is possible to find a planar embedding of the graph and a representation of each cluster as a region homeomorphic to a closed disk such that (1) the subgraph induced by each cluster is drawn in the interior of the corresponding disk, (2) each edge intersects any disk at most once, and (3) the nesting between clusters is reflected by the representation, i.e., child clusters are properly contained in their parent cluster. The computational complexity of this problem, whose study has been central to the theory of graph visualization since its introduction in 1995 [Feng, Cohen, and Eades, Planarity for clustered graphs, ESA’95], has only been recently settled [Fulek and Tóth, Atomic Embeddability, Clustered Planarity, and Thickenability, to appear at SODA’20]. Before such a breakthrough, the complexity question was still unsolved even when the graph has a prescribed planar embedding, i.e, for embedded clustered graphs. We show that the C-Planarity Testing problem admits a single-exponential single-parameter FPT (resp., XP) algorithm for embedded flat (resp., non-flat) clustered graphs, when parameterized by the carving-width of the dual graph of the input. These are the first FPT and XP algorithms for this long-standing open problem with respect to a single notable graph-width parameter. Moreover, the polynomial dependency of our FPT algorithm is smaller than the one of the algorithm by Fulek and Tóth. In particular, our algorithm runs in quadratic time for flat instances of bounded treewidth and bounded face size. To further strengthen the relevance of this result, we show that an algorithm with running time O(r(n)) for flat instances whose underlying graph has pathwidth 1 would result in an algorithm with running time O(r(n)) for flat instances and with running time $$O(r(n^2) + n^2)$$ O ( r ( n 2 ) + n 2 ) for general, possibly non-flat, instances.

Sharp Estimation Type Inequalities for Lipschitzian Mappings in Euclidean Sense on a Disk

Some sharp trapezoid and midpoint type inequalities for Lipschitzian bifunctions defined on a closed disk in Euclidean sense are obtained by the use of polar coordinates. Also, bifunctions whose partial derivative is Lipschitzian are considered. A new presentation of Hermite-Hadamard inequality for convex function defined on a closed disk and its reverse are given. Furthermore, two mappings H t and h t are considered to give some generalized Hermite-Hadamard type inequalities in the case that considered functions are Lipschitzian in Euclidean sense on a disk.

Handel’s fixed point theorem revisited

Michael Handel proved in [A fixed-point theorem for planar homeomorphisms. Topology38 (1999), 235–264] the existence of a fixed point for an orientation-preserving homeomorphism of the open unit disk that can be extended to the closed disk, provided that it has points whose orbits form an oriented cycle of links at infinity. Later, Patrice Le Calvez gave a different proof of this theorem based only on Brouwer theory and plane topology arguments in [Une nouvelle preuve du théorème de point fixe de Handel. Geom. Topol.10(2006), 2299–2349]. These methods improved the result by proving the existence of a simple closed curve of index 1. We give a new, simpler proof of this improved version of the theorem and generalize it to non-oriented cycles of links at infinity.

POLYNOMIAL APPROXIMATION, LOCAL POLYNOMIAL CONVEXITY, AND DEGENERATE CR SINGULARITIES — II

We provide some conditions for the graph of a Hölder-continuous function on [Formula: see text], where [Formula: see text] is a closed disk in ℂ, to be polynomially convex. Almost all sufficient conditions known to date — provided the function (say F) is smooth — arise from versions of the Weierstrass Approximation Theorem on [Formula: see text]. These conditions often fail to yield any conclusion if rank ℝDF is not maximal on a sufficiently large subset of [Formula: see text]. We bypass this difficulty by introducing a technique that relies on the interplay of certain plurisubharmonic functions. This technique also allows us to make some observations on the polynomial hull of a graph in ℂ2 at an isolated complex tangency.

UNIFORM APPROXIMATION BY POLYNOMIAL, RATIONAL AND ANALYTIC FUNCTIONS

Let K and X be compact plane sets such that $K\subseteq X$. Let P(K) be the uniform closure of polynomials on K, let R(K) be the uniform closure of rational functions on K with no poles in K and let A(K) be the space of continuous functions on K which are analytic on int(K). Define P(X,K),R(X,K) and A(X,K) to be the set of functions in C(X) whose restriction to K belongs to P(K),R(K) and A(K), respectively. Let S0(A) denote the set of peak points for the Banach function algebra A on X. Let S and T be compact subsets of X. We extend the Hartogs–Rosenthal theorem by showing that if the symmetric difference SΔT has planar measure zero, then R(X,S)=R(X,T) . Then we show that the following properties are equivalent: (i)R(X,S)=R(X,T) ;(ii)$S\setminus T\subseteq S_0(R(X,S))$ and $T\setminus S\subseteq S_0(R(X,T))$;(iii)R(K)=C(K) for every compact set $K \subseteq S\Delta T$;(iv)$R(X,S \cap \overline {U})=R(X,T \cap \overline {U})$ for every open set U in ℂ ;(v)for every p∈X there exists an open disk Dp with centre p such that We prove an extension of Vitushkin’s theorem by showing that the following properties are equivalent: (i)A(X,S)=R(X,T) ;(ii)$A(X,S \cap \overline {D})=R(X,T \cap \overline {D})$ for every closed disk $\overline {D}$ in ℂ ;(iii)for every p∈X there exists an open disk Dp with centre p such that

Fundamental group of tiles associated to quadratic canonical number systems

If A is a 2 × 2 expanding matrix with integral coefficients, and ⊂ ℤ2 a complete set of coset representatives of ℤ2/Aℤ2 with |det(A)| elements, then the set ℐ defined by Aℐ = ℐ + is a self-affine plane tile of ℝ2, provided that its two-dimensional Lebesgue measure is positive.It was shown by Luo and Thuswaldner that the fundamental group of such a tile is either trivial or uncountable.To a quadratic polynomial x 2 + Ax + B, A, B ∈ ℤ such that B ≥ 2 and −1 ≤ A ≤ B, one can attach a tile ℐ. Akiyama and Thuswaldner proved the triviality of the fundamental group of this tile for 2A < B + 3, by showing that a tile of this class is homeomorphic to a closed disk. The case 2A ≥ B + 3 is treated here by using the criterion given by Luo and Thuswaldner.

The Nymph and Additional Imaginal Description of Epeorus melli new combination from China (Ephemeroptera: Heptageniidae)

The nymph plus additional imaginal characters of Epeorus melli (Ulmer) new combination are described and figured in detail. The nymphs have 2 cerci, gills 1–7 which do not form a closed disk ventrally, and abdominal terga with a median row of setae dorsally but without median tubercle. Originally described in the genus Thalerosphyrus, it is actually a member of the genus Epeorus.

CONTINUOUS EXTENSIONS ARE NOT HÖLDER

Let Q denote a quadratic polynomial and A∞ the super-attracting basin of Q at the point ∞ on the Riemann sphere [Formula: see text]. There exists a unique Riemann mapping Φ from the open disk [Formula: see text] onto A∞ such that Φ(∞)=∞, Φ'(∞)=1 and Φ-1 conjugates Q:A∞→A∞ to the squaring map S:D→D:z↦z2. In this paper, we show if Q is real and infinitely renormalizable of bounded type then the continuous extension of Φ to the closed disk cannot have any Hölder continuity on the boundary circle [Formula: see text].