グラフアルゴリズムの効率化と評価に関する研究

平成13年度-平成14年度科学研究費補助金(基盤研究(C)(2))研究成果報告書,課題番号.1367038

Drawings of Planar Graphs with Few Slopes and Segments

We study straight-line drawings of planar graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2-trees, and planar 3-trees. We prove that every 3-connected plane graph on $n$ vertices has a plane drawing with at most ${5/2}n$ segments and at most $2n$ slopes. We prove that every cubic 3-connected plane graph has a plane drawing with three slopes (and three bends on the outerface). In a companion paper, drawings of non-planar graphs with few slopes are also considered.Comment: This paper is submitted to a journal. A preliminary version appeared as "Really Straight Graph Drawings" in the Graph Drawing 2004 conference. See http://arxiv.org/math/0606446 for a companion pape

構造的グラフに対するアルゴリズムの工学的研究

平成11年度-平成12年度科学研究費補助金(基盤研究(C)(2))研究成果報告書,課題番号.1168033

Steinitz Theorems for Orthogonal Polyhedra

We define a simple orthogonal polyhedron to be a three-dimensional polyhedron with the topology of a sphere in which three mutually-perpendicular edges meet at each vertex. By analogy to Steinitz's theorem characterizing the graphs of convex polyhedra, we find graph-theoretic characterizations of three classes of simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric projection in the plane with only one hidden vertex, xyz polyhedra, in which each axis-parallel line through a vertex contains exactly one other vertex, and arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz polyhedra are exactly the bipartite cubic polyhedral graphs, and every bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of a corner polyhedron. Based on our characterizations we find efficient algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure

The effect of venting configuration on the blast response of an aircraft unit load device

The inclusion of venting areas in aircraft unit load devices (ULDs) as a potential blast mitigation technique is investigated in this work. Damage to the ULD, such as large deflections or container rupture, from an internal explosion threatens to tear the aircraft skin and cause fuselage decompression. The loading within a luggage container was expected to be reduced when the explosive products were vented into the adjacent ULDs. Although previous work has investigated the effect of venting on ULD blast loading, this has only considered a single venting side and not multiple venting configurations. To determine if a multiple-venting system would be beneficial in ULDs, experimental blast testing was performed by subjecting a 1:6 scaled ULD box to representative blast loads with different venting configurations. The blast response of the side of the ULD which would be positioned closest to the fuselage was measured. Numerical simulations were established to provide insight into the blast loading effects not measured experimentally. The loading within the ULD box, in terms of the number and magnitude of blast wave reflections, and internal pressure build-up, was reduced when introducing venting areas. Final deformations were reduced by 11% and 22% when using a single- and double-venting configuration, respectively. Further deformation reduction was expected if more venting area was made available: unconfined blasts tests (demonstrating complete absence of ULD confinement) reduced deformations by 44%. The fully-confined (no venting) blast test resulted in rupture failure when blasted with a 20 g explosive, whereas the vented tests exhibited no tearing when blasted with higher charge masses. The double-venting configuration demonstrated better blast mitigation than the single-venting configuration. However, since both reduced the deformations and rupture probability of the container, the implementation of a multiple-venting system within aircraft ULDs would improve the survivability of the ULD container during a blast event

Experiments with Point Placement Algorithms and Recognition of Line Rigid Graphs

The point placement problem is to determine the position of n distinct points on a line, up to translation and reflection by fewest possible pairwise adversarial distance queries. This masters thesis focusses on two aspects of point placement problem. In one part we focusses on an experimental study of a number of deterministic point placement algorithms and an incremental randomized algorithm, with the goal of obtaining a greater insight into the behavior of these algorithms, particularly of the randomize algorithm. The pairwise distance queries in the point placement problem creates a type of graph, called point placement graph. A point placement graph G is dened as line rigid graph if and only if the vertices of G has unique placement on a line. The other part of this thesis focusses on recognizing line rigid graph of certain class based on structural property of an arbitrarily given graph. Layer graph drawing and rectangular drawing are used as key idea in recognizing line rigid graphs

The Language of Mathematics: Mathematical Terminology Simplified for Classroom Use.

After recognizing the need for a simpler approach to the teaching of mathematical terminology, I concluded it would be valuable to make a unit of simplified terms and describe methods of teaching these terms. In this thesis I have compared the terminology found in the Virginia Standards of Learning objectives to the materials found at each grade level. The units developed are as follows: The Primary Persistence Unit- for grades K-2; The Elementary Expansion Unit- for grades 3-5; and The Middle School Mastery Unit- for grades 6-8