On Hardness of the Joint Crossing Number

The Joint Crossing Number problem asks for a simultaneous embedding of two disjoint graphs into one surface such that the number of edge crossings (between the two graphs) is minimized. It was introduced by Negami in 2001 in connection with diagonal flips in triangulations of surfaces, and subsequently investigated in a general form for small-genus surfaces. We prove that all of the commonly considered variants of this problem are NP-hard already in the orientable surface of genus 6, by a reduction from a special variant of the anchored crossing number problem of Cabello and Mohar

Composite planar coverings of graphs

AbstractWe shall prove that a connected graph G is projective-planar if and only if it has a 2n-fold planar connected covering obtained as a composition of an n-fold covering and a double covering for some n⩾1 and show that every planar regular covering of a nonplanar graph is such a composite covering

Generating families of surface triangulations. The case of punctured surfaces with inner degree at least 4

We present two versions of a method for generating all triangulations of any punctured surface in each of these two families: (1) triangulations with inner vertices of degree at least 4 and boundary vertices of degree at least 3 and (2) triangulations with all vertices of degree at least 4. The method is based on a series of reversible operations, termed reductions, which lead to a minimal set of triangulations in each family. Throughout the process the triangulations remain within the corresponding family. Moreover, for the family (1) these operations reduce to the well-known edge contractions and removals of octahedra. The main results are proved by an exhaustive analysis of all possible local configurations which admit a reduction.Comment: This work has been partially supported by PAI FQM-164; PAI FQM-189; MTM 2010-2044

Weighted zeta functions for quotients of regular coverings of graphs

AbstractLet G be a connected graph. We reformulate Stark and Terras' Galois Theory for a quotient H of a regular covering K of a graph G by using voltage assignments. As applications, we show that the weighted Bartholdi L-function of H associated to the representation of the covering transformation group of H is equal to that of G associated to its induced representation in the covering transformation group of K. Furthermore, we express the weighted Bartholdi zeta function of H as a product of weighted Bartholdi L-functions of G associated to irreducible representations of the covering transformation group of K. We generalize Stark and Terras' Galois Theory to digraphs, and apply to weighted Bartholdi L-functions of digraphs

Recursive Formulas for Beans Functions of Graphs

In this paper, we regard each edge of a connected graph $G$ as a line segment having a unit length, and focus on not only the vertices but also any point lying along such a line segment. So we can define the distance between two points on $G$ as the length of a shortest curve joining them along $G$. The beans function $B_G(x)$ of a connected graph $G$ is defined as the maximum number of points on $G$ such that any pair of points have distance at least $x\u3e0$. We shall show a recursive formula for $B_G(x)$ which enables us to determine the value of $B_G(x)$ for all $x \leq 1$ by evaluating it only for $1/2 \u3c x \leq 1$. As applications of this recursive formula, we shall propose an algorithm for computing $B_G(x)$ for a given value of $x\leq 1$, and determine the beans functions of the complete graphs $K_n$

Long-Moody construction of braid representations and Katz middle convolution

The Long-Moody construction is a method to obtain representations of braid groups introduced by Long and Moody. Also the Katz middle convolution is known to be a method to construct local systems on $\mathbb{C}\backslash\{n\text{-points}\}$ introduced by Katz. In this paper, we explain that these two methods are naturally unified and define a new functor which we call the Katz-Long-Moody functor. This functor extends the framework of Katz algorithm to categories of local systems on various topological spaces, for example, $B_{n}$-bundles associated with simple Weierstrass polynomials, complements of hyperplane arrangements of fiber-type, link complements in the solid torus, and so on.Comment: 50 page

Finite planar emulators for K_{4,5} - 4K_2 and K_{1,2,2,2} and Fellows' Conjecture

In 1988 Fellows conjectured that if a finite, connected graph admits a finite planar emulator, then it admits a finite planar cover. We construct a finite planar emulator for K_{4,5} - 4K_2. Archdeacon showed that K_{4,5} - 4K_2 does not admit a finite planar cover; thus K_{4,5} - 4K_2 provides a counterexample to Fellows' Conjecture. It is known that Negami's Planar Cover Conjecture is true if and only if K_{1,2,2,2} admits no finite planar cover. We construct a finite planar emulator for K_{1,2,2,2}. The existence of a finite planar cover for K_{1,2,2,2} is still open.Comment: Final version. To appear in European Journal of Combinatoric

Diagonal Flips of Triangulations on Closed Surfaces Preserving Specified Properties

AbstractConsider a class P of triangulations on a closed surfaceF2, closed under vertex splitting. We shall show that any two triangulations with the same and sufficiently large number of vertices which belong to P can be transformed into each other, up to homeomorphism, by a finite sequence of diagonal flips through P. Moreover, if P is closed under homeomorphism, then the condition “up to homeomorphism” can be replaced with “up to isotopy.

Investigating positive and threat-based awe in natural and built environments

Staggeringly immense or beautiful awe-inspiring structures, such as religious monumental architecture, have long been important to human culture and society. With the emerging psychological literature on awe, a nascent avenue of research is beginning to uncover specific psychosocial and physiological effects of feeling awe through architecture. Most psychological work relies on nature imagery to evoke awe; yet architecturally-induced awe, which is studied very little, has enormous implications for how awe-eliciting architecture—such as cultural and religious sites—facilitate their sociocultural functions through built form. Besides the awe-inducing stimulus, the specific type of awe elicited also has potential to produce different effects. Many positive effects associated with feeling awe have been demonstrated empirically, including increased prosocial behavior, increased feelings of connection to others, and enhanced physical health through lower levels of proinflammatory cytokines. Recent work has turned to the darker side of awe, investigating effects of feeling threat-based awe, or awe elicited through a threatening stimulus. Although both positive and threat-based awe result in a smaller sense of self, threat-based awe is associated with greater feelings of powerlessness and fear than positive awe. Thus, we hypothesize that awe-inspiring environments may have distinct effects based on whether they induce positive awe or threat-based awe, as well as whether the environment is natural or built. Specifically, we predict that while positive awe will facilitate feelings of universality and integration into larger groups (e.g., the world), threat-based awe will promote social connection to smaller social groups (e.g., one’s community). We further predict that this effect will be more pronounced for architectural environments, which have inherent social meaning, compared to natural environments. Across three online studies, we explore effects of positive and threat-based awe elicited through nature and architecture. Study 1 (N = 116) uses videos of natural phenomena used in previous work to replicate previous findings on positive and threat-based awe: We show that threat-based awe leads to greater feelings of powerlessness and fear than positive awe, and that both positive and threat-based awe result in a smaller perceived self-size than no awe. Study 2 (N = 100) extends these findings to architectural environments chosen to elicit positive and threat-based awe. While both awe conditions in Study 2 led to a smaller perceived self-size than the control condition as predicted, the architectural video meant to elicit threat-based awe elicited positive awe for most participants. Because we failed to elicit threat-based awe with architectural stimuli in Study 2, Study 3 (N = 85) compared only effects of positive awe elicited through natural and architectural environments on feelings of universality and identification with others. We find that both natural and architecturally induced positive awe similarly promote feelings of universality and connection with people all over the world, compared to a control condition. This research expands our understanding of how we respond to beautiful and threatening awe-evoking environments, from ancient monumental structures and natural phenomena to the supertall skyscrapers and natural disasters that are becoming increasingly common. This research furthermore helps us understand what awe-related effects demonstrated in cognitive science will have implications for architectural design