Strip Planarity Testing of Embedded Planar Graphs

In this paper we introduce and study the strip planarity testing problem, which takes as an input a planar graph $G(V,E)$ and a function $\gamma:V \rightarrow \{1,2,\dots,k\}$ and asks whether a planar drawing of $G$ exists such that each edge is monotone in the $y$-direction and, for any $u,v\in V$ with $\gamma(u)<\gamma(v)$, it holds $y(u)<y(v)$. The problem has strong relationships with some of the most deeply studied variants of the planarity testing problem, such as clustered planarity, upward planarity, and level planarity. We show that the problem is polynomial-time solvable if $G$ has a fixed planar embedding.Comment: 24 pages, 12 figures, extended version of 'Strip Planarity Testing' (21st International Symposium on Graph Drawing, 2013

Optimal Morphs of Convex Drawings

We give an algorithm to compute a morph between any two convex drawings of the same plane graph. The morph preserves the convexity of the drawing at any time instant and moves each vertex along a piecewise linear curve with linear complexity. The linear bound is asymptotically optimal in the worst case.Comment: To appear in SoCG 201

Relaxing the Constraints of Clustered Planarity

In a drawing of a clustered graph vertices and edges are drawn as points and curves, respectively, while clusters are represented by simple closed regions. A drawing of a clustered graph is c-planar if it has no edge-edge, edge-region, or region-region crossings. Determining the complexity of testing whether a clustered graph admits a c-planar drawing is a long-standing open problem in the Graph Drawing research area. An obvious necessary condition for c-planarity is the planarity of the graph underlying the clustered graph. However, such a condition is not sufficient and the consequences on the problem due to the requirement of not having edge-region and region-region crossings are not yet fully understood. In order to shed light on the c-planarity problem, we consider a relaxed version of it, where some kinds of crossings (either edge-edge, edge-region, or region-region) are allowed even if the underlying graph is planar. We investigate the relationships among the minimum number of edge-edge, edge-region, and region-region crossings for drawings of the same clustered graph. Also, we consider drawings in which only crossings of one kind are admitted. In this setting, we prove that drawings with only edge-edge or with only edge-region crossings always exist, while drawings with only region-region crossings may not. Further, we provide upper and lower bounds for the number of such crossings. Finally, we give a polynomial-time algorithm to test whether a drawing with only region-region crossings exist for biconnected graphs, hence identifying a first non-trivial necessary condition for c-planarity that can be tested in polynomial time for a noticeable class of graphs

On Planar Greedy Drawings of 3-Connected Planar Graphs

A graph drawing is greedy if, for every ordered pair of vertices (x,y), there is a path from x to y such that the Euclidean distance to y decreases monotonically at every vertex of the path. Greedy drawings support a simple geometric routing scheme, in which any node that has to send a packet to a destination "greedily" forwards the packet to any neighbor that is closer to the destination than itself, according to the Euclidean distance in the drawing. In a greedy drawing such a neighbor always exists and hence this routing scheme is guaranteed to succeed. In 2004 Papadimitriou and Ratajczak stated two conjectures related to greedy drawings. The greedy embedding conjecture states that every 3-connected planar graph admits a greedy drawing. The convex greedy embedding conjecture asserts that every 3-connected planar graph admits a planar greedy drawing in which the faces are delimited by convex polygons. In 2008 the greedy embedding conjecture was settled in the positive by Leighton and Moitra. In this paper we prove that every 3-connected planar graph admits a planar greedy drawing. Apart from being a strengthening of Leighton and Moitra\u27s result, this theorem constitutes a natural intermediate step towards a proof of the convex greedy embedding conjecture

Fiscal sustainability and policy implications for the euro area

In this paper we examine the sustainability of euro area public finances against the backdrop of population ageing. We critically assess the widely used projections of the Working Group on Ageing Populations (AWG) of the EU's Economic Policy Committee and argue that ageing costs may be higher than projected in the AWG reference scenario. Taking into account adjusted headline estimates for ageing costs, largely based upon the sensitivity analysis carried out by the AWG, we consider alternative indicators to quantify sustainability gaps for euro area countries. With respect to the policy implications, we assess the appropriateness of different budgetary strategies to restore fiscal sustainability taking into account intergenerational equity. Our stylised analysis based upon the lifetime contribution to the government's primary balance of different generations suggests that an important degree of pre-funding of the ageing costs is necessary to avoid shifting the burden of adjustment in a disproportionate way to future generations. For many euro area countries this implies that the medium-term targets defined in the context of the revised stability and growth pact would ideally need to be revised upwards to significant surpluses. --population ageing,fiscal sustainability,generational accounting,medium-term objectives for fiscal policy

Ultrasound-assisted extraction of olive oil: A preliminary economic analysis

The objective of this work was to perform a preliminary economic analysis of the ultrasound-assisted extraction process of olive oil and compare it with the manufacturing costs of the conventional extraction process. The DWSIM software was used for economic modeling considering data found in the literature; the five cost parameters were considered. The results showed a cost of USD 126.50/kg of olive oil in the proposed process. Although expensive, about 54,350 bottles more of oil could be produced due to the higher yield that can be obtained. In parallel, about USD 11,350,000.00 could be earned in one year of operation. In this sense, more research must be done to demonstrate the economic viability of this type of industry, to improve the operations and the costs of manufacturing until the process becomes profitable