Edges and switches, tunnels and bridges

Abstract. Edge casing is a well-known method to improve the readability of drawings of non-planar graphs. A cased drawing orders the edges of each edge crossing and interrupts the lower edge in an appropriate neighborhood of the crossing. Certain orders will lead to a more readable drawing than others. We formulate several optimization criteria that try to capture the concept of a "good" cased drawing. Further, we address the algorithmic question of how to turn a given drawing into an optimal cased drawing. For many of the resulting optimization problems, we either find polynomial time algorithms or NP-hardness results

Orientation-Constrained Rectangular Layouts

We construct partitions of rectangles into smaller rectangles from an input consisting of a planar dual graph of the layout together with restrictions on the orientations of edges and junctions of the layout. Such an orientation-constrained layout, if it exists, may be constructed in polynomial time, and all orientation-constrained layouts may be listed in polynomial time per layout.Comment: To appear at Algorithms and Data Structures Symposium, Banff, Canada, August 2009. 12 pages, 5 figure

Towards Characterizing Graphs with a Sliceable Rectangular Dual

\u3cp\u3eLet G be a plane triangulated graph. A rectangular dual of G is a partition of a rectangle R into a set R of interior-disjoint rectangles, one for each vertex, such that two regions are adjacent if and only if the corresponding vertices are connected by an edge. A rectangular dual is sliceable if it can be recursively subdivided along horizontal or vertical lines. A graph is rectangular if it has a rectangular dual and sliceable if it has a sliceable rectangular dual. There is a clear characterization of rectangular graphs. However, a full characterization of sliceable graphs is still lacking. The currently best result (Yeap and Sarrafzadeh, 1995) proves that all rectangular graphs without a separating 4-cycle are slice- able. In this paper we introduce a recursively defined class of graphs and prove that these graphs are precisely the nonsliceable graphs with exactly one separating 4-cycle.\u3c/p\u3

Recent Surveys in the Forests of Ulu Segama Malua, Sabah, Malaysia, Show That Orang-utans (P. p. morio) Can Be Maintained in Slightly Logged Forests

BACKGROUND: Today the majority of wild great ape populations are found outside of the network of protected areas in both Africa and Asia, therefore determining if these populations are able to survive in forests that are exploited for timber or other extractive uses and how this is managed, is paramount for their conservation. METHODOLOGY/PRINCIPAL FINDINGS: In 2007, the "Kinabatangan Orang-utan Conservation Project" (KOCP) conducted aerial and ground surveys of orang-utan (Pongo pygmaeus morio) nests in the commercial forest reserves of Ulu Segama Malua (USM) in eastern Sabah, Malaysian Borneo. Compared with previous estimates obtained in 2002, our recent data clearly shows that orang-utan populations can be maintained in forests that have been lightly and sustainably logged. However, forests that are heavily logged or subjected to fast, successive coupes that follow conventional extraction methods, exhibit a decline in orang-utan numbers which will eventually result in localized extinction (the rapid extraction of more than 100 m(3) ha(-1) of timber led to the crash of one of the surveyed sub-populations). Nest distribution in the forests of USM indicates that orang-utans leave areas undergoing active disturbance and take momentarily refuge in surrounding forests that are free of human activity, even if these forests are located above 500 m asl. Displaced individuals will then recolonize the old-logged areas after a period of time, depending on availability of food sources in the regenerating areas. CONCLUSION/SIGNIFICANCE: These results indicate that diligent planning prior to timber extraction and the implementation of reduced-impact logging practices can potentially be compatible with great ape conservation

Wooden Geometric Puzzles: Design and Hardness Proofs

We discuss some new geometric puzzles and the complexity of their extension to arbitrary sizes. For gate puzzles and two-layer puzzles we prove NP-completeness of solving them. Not only the solution of puzzles leads to interesting questions, but also puzzle design gives rise to interesting theoretical questions. This leads to the search for instances of partition that use only integers and are uniquely solvable.We show that instances of polynomial size exist with this property. This result also holds for partition into k subsets with the same sum: We construct instances of n integers with subset sum O(nk+1), for fixed k