Disjoint edges in topological graphs and the tangled-thrackle conjecture

It is shown that for a constant $t\in \mathbb{N}$, every simple topological graph on $n$ vertices has $O(n)$ edges if it has no two sets of $t$ edges such that every edge in one set is disjoint from all edges of the other set (i.e., the complement of the intersection graph of the edges is $K_{t,t}$-free). As an application, we settle the \emph{tangled-thrackle} conjecture formulated by Pach, Radoi\v{c}i\'c, and T\'oth: Every $n$-vertex graph drawn in the plane such that every pair of edges have precisely one point in common, where this point is either a common endpoint, a crossing, or a point of tangency, has at most $O(n)$ edges

The Szemeredi-Trotter Theorem in the Complex Plane

It is shown that $n$ points and $e$ lines in the complex Euclidean plane ${\mathbb C}^2$ determine $O(n^{2/3}e^{2/3}+n+e)$ point-line incidences. This bound is the best possible, and it generalizes the celebrated theorem by Szemer\'edi and Trotter about point-line incidences in the real Euclidean plane ${\mathbb R}^2$.Comment: 24 pages, 5 figures, to appear in Combinatoric

Untangling polygons and graphs

Untangling is a process in which some vertices of a planar graph are moved to obtain a straight-line plane drawing. The aim is to move as few vertices as possible. We present an algorithm that untangles the cycle graph C_n while keeping at least \Omega(n^{2/3}) vertices fixed. For any graph G, we also present an upper bound on the number of fixed vertices in the worst case. The bound is a function of the number of vertices, maximum degree and diameter of G. One of its consequences is the upper bound O((n log n)^{2/3}) for all 3-vertex-connected planar graphs.Comment: 11 pages, 3 figure

Grid-Obstacle Representations with Connections to Staircase Guarding

In this paper, we study grid-obstacle representations of graphs where we assign grid-points to vertices and define obstacles such that an edge exists if and only if an $xy$-monotone grid path connects the two endpoints without hitting an obstacle or another vertex. It was previously argued that all planar graphs have a grid-obstacle representation in 2D, and all graphs have a grid-obstacle representation in 3D. In this paper, we show that such constructions are possible with significantly smaller grid-size than previously achieved. Then we study the variant where vertices are not blocking, and show that then grid-obstacle representations exist for bipartite graphs. The latter has applications in so-called staircase guarding of orthogonal polygons; using our grid-obstacle representations, we show that staircase guarding is \textsc{NP}-hard in 2D.Comment: To appear in the proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Blocking Coloured Point Sets

This paper studies problems related to visibility among points in the plane. A point $x$ \emph{blocks} two points $v$ and $w$ if $x$ is in the interior of the line segment $\bar{vw}$. A set of points $P$ is \emph{$k$-blocked} if each point in $P$ is assigned one of $k$ colours, such that distinct points $v,w\in P$ are assigned the same colour if and only if some other point in $P$ blocks $v$ and $w$. The focus of this paper is the conjecture that each $k$-blocked set has bounded size (as a function of $k$). Results in the literature imply that every 2-blocked set has at most 3 points, and every 3-blocked set has at most 6 points. We prove that every 4-blocked set has at most 12 points, and that this bound is tight. In fact, we characterise all sets $\{n_1,n_2,n_3,n_4\}$ such that some 4-blocked set has exactly $n_i$ points in the $i$-th colour class. Amongst other results, for infinitely many values of $k$, we construct $k$-blocked sets with $k^{1.79...}$ points

Decomposition of Multiple Coverings into More Parts

We prove that for every centrally symmetric convex polygon Q, there exists a constant alpha such that any alpha*k-fold covering of the plane by translates of Q can be decomposed into k coverings. This improves on a quadratic upper bound proved by Pach and Toth (SoCG'07). The question is motivated by a sensor network problem, in which a region has to be monitored by sensors with limited battery lifetime

The Health Belief Model And Factors Relating To Potential Use Of A Vaccine For Shigellosis In Kaeng Koi District, Saraburi Province, Thailand

Shigellosis is an important cause of morbidity and mortality throughout the world. Approximately, 1.1 million deaths occur a year due to this disease, making it the fourth leading cause of mortality worldwide. This paper explores local interest in and potential use of a vaccine for shigellosis in Thailand where Shigella poses an important public-health concern. Data for this study were collected during June- November 2002 from 522 subjects surveyed using a sociobehavioural questionnaire in Kaeng Koi district in central Thailand. The community demand and likely use of a vaccine were examined in relation to the Health Belief Model, which provides analytical constructs for investigating the multiple issues of local readiness to accept and access a new vaccine. As the key outcome variable, most respondents showed interest in receiving a vaccine against dysentery which they thought would provide useful protection against the disease. However, there was only a moderate number who perceived dysentery as serious and themselves as susceptible to it, although it was perceived to cause some burden to and additional expense for families. Most people identified a number of groups who were thought to be especially vulnerable to dysentery, such as the elderly, pre-school, and school-age children, and poor labourers. Other outcomes of the study included the identification of acceptable and convenient sites for its delivery, such as government health clinics and private clinics, and respected sources for information about the vaccine, such as health clinic personnel and community health volunteers. This information suggests that components of the Health Belief Model may be useful in identifying community acceptance of a vaccine and the means of introducing it. This health information is important for planning and implementing vaccine programmes. Key words: Dysentery, Bacillary; Shigella; Bacterial vaccines; Health Belief Model; Perceptions; Cross-sectional studies; Thailan

On the Recognition of Fan-Planar and Maximal Outer-Fan-Planar Graphs

Fan-planar graphs were recently introduced as a generalization of 1-planar graphs. A graph is fan-planar if it can be embedded in the plane, such that each edge that is crossed more than once, is crossed by a bundle of two or more edges incident to a common vertex. A graph is outer-fan-planar if it has a fan-planar embedding in which every vertex is on the outer face. If, in addition, the insertion of an edge destroys its outer-fan-planarity, then it is maximal outer-fan-planar. In this paper, we present a polynomial-time algorithm to test whether a given graph is maximal outer-fan-planar. The algorithm can also be employed to produce an outer-fan-planar embedding, if one exists. On the negative side, we show that testing fan-planarity of a graph is NP-hard, for the case where the rotation system (i.e., the cyclic order of the edges around each vertex) is given