Edge crossings in drawings of bipartite graphs

Systems engineers have recently shown interest in algorithms for drawing directed graphs so that they are easy to understand and remember. Each of the commonly used methods has a step which aims to adjust the drawing to decrease the number of arc crossings. We show that the most popular strategy involves an NP-complete problem regarding the minimization of the number of arcs in crossings in a bipartite graph. The performance of the commonly employed barycenter heuristic for this problem is analyzed. An alternative method, the median heuristic, is proposed and analyzed. The new method is shown to compare favorably with the old in terms of performance guarantees. As a bonus, we show that the median heuristic performs well with regard to the total length of the arcs in the drawing.     

Finding and defining the generators of genus-n objects for constructing topological and cut graphs

The topology of an object is commonly represented through a topological graph or a cut graph (polygonal scheme). Over the past few years, many studies have focused on extracting the topological and cut graphs of complex freeform objects that are represented by meshes. For an object with genus-n, the topological graph has n cycles, while the cut graph contains 2n cycles. These loops, however, do not always explicitly represent the holes in the objects. That is, a cycle in the graph can be a cycle around a solid (meridian), a cycle around a hole (longitude), or almost any combination of the two. The task of classifying the cycles (generators) as cycles around holes (longitude) and cycles around solids (meridians) on the mesh is not straightforward. Every closed orientable 2-manifold with genus-n can be seen as a collection of n toruses stitched together, so that each hole in the object can be referred to as a torus with two generators. This paper proposes a method that extracts the generators from which the longitudes and the meridians are found. The topological graph is defined by the longitudes and by a spanning tree constructed between them. The cut graph is constructed using the same concept. The advantage of the proposed method over other methods is that each loop in the topological graph explicitly represents a hole in the object.     

Cyclic twill-woven objects

Classical (or biaxial) twill is a textile weave in which the weft threads pass over and under two or more warp threads, with an offset between adjacent weft threads to give an appearance of diagonal lines. This paper introduces a theoretical framework for constructing twill-woven objects, i.e., cyclic twill-weavings on arbitrary surfaces, and it provides methods to convert polygonal meshes into twill-woven objects. It also develops a general technique to obtain exact triaxial-woven objects from an arbitrary polygonal mesh surface.     

Applications of the crossing number

LetG be a graph ofn vertices that can be drawn in the plane by straight-line segments so that nok+1 of them are pairwise crossing. We show thatG has at mostc _k nlog^2k–2 n edges. This gives a partial answer to a dual version of a well-known problem of Avital-Hanani, Erdós, Kupitz, Perles, and others. We also construct two point sets {p ₁,,p _n }, {q ₁,,q _n } in the plane such that any piecewise linear one-to-one mappingfR ²R ² withf(pi)=qi (1in) is composed of at least (n ²) linear pieces. It follows from a recent result of Souvaine and Wenger that this bound is asymptotically tight. Both proofs are based on a relation between the crossing number and the bisection width of a graph.The first author was supported by NSF Grant CCR-91-22103, PSC-CUNY Research Award 663472, and OTKA-4269. An extended abstract of this paper was presented at the 10th Annual ACM Symposium on Computational Geometry, Stony Brook, NY, 1994.     

On the stability number of the edge intersection of two graphs

Let G be the graph obtained as the edge intersection of two graphs G₁, G₂ on the same vertex set V. We show that if at least one of G₁, G₂ is perfect, then α(G)?α(G₁)·α(G₂), where α() is the cardinality of the largest stable set. Moreover, for general G₁ and G₂, we show that α(G)?R(α(G₁)+1,α(G₂)+1)−1, where R(k,?) is the Ramsey number.     

On the structure of certain intersection graphs

Recently Gavril introduced a new class of intersection graphs called interval-filament graphs. These include co-comparability graphs and polygon-circle graphs (the intersection graphs of polygons inscribed in a circle), which include circular-arc graphs (the intersection graphs of arcs of a circle), circle graphs (the intersection graphs of chords of a circle), chordal graphs, and outerplanar graphs. We give a structural property of polygon-circle graphs. We prove a bound on the clique-covering number for interval-filament graphs in terms of the size of a largest independent set of nodes in the graph. We prove that complements of interval-filament graphs are 4-divisible in the sense of Hoàng and McDiarmid. Similar results are obtained for complements of other intersection graphs introduced by Gavril.     

On the adjacent vertex distinguishing edge colourings of graphs

A k-adjacent vertex distinguishing edge colouring or a k-avd-colouring of a graph G is a proper k-edge colouring of G such that no pair of adjacent vertices meets the same set of colours. The avd-chromatic number, denoted by χ′_a(G), is the minimum number of colours needed in an avd-colouring of G. It is proved that for any connected 3-colourable Hamiltonian graph G, we have χ′_a(G)≤Δ+3.     

On the size of critical graphs with small maximum degree

In this paper, we give some new properties of edge chromatic critical graphs, and give new lower bounds for the average degree of Δ-critical graphs with Δ=11, 12 by the use of these properties.     

The oriented chromatic number of Halin graphs

The oriented chromatic number of an oriented graph G is the minimum order of an oriented graph H such that G admits a homomorphism to H. The oriented chromatic number of an unoriented graph G is the maximal chromatic number over all possible orientations of G. In this paper, we prove that every Halin graph has oriented chromatic number at most 8, improving a previous bound by Hosseini Dolama and Sopena, and confirming the conjecture given by Vignal.     

Defining, contouring, and visualizing scalar functions on point-sampled surfaces

This paper addresses the definition, contouring, and visualization of scalar functions on unorganized point sets, which are sampled from a surface in 3D space; the proposed framework builds on moving least-squares techniques and implicit modeling. Given a scalar function f:P→R, defined on a point set P, the idea behind our approach is to exploit the local connectivity structure of the k-nearest neighbor graph of P and mimic the contouring of scalar functions defined on triangle meshes. Moving least-squares and implicit modeling techniques are used to extend f from P to the surface M underlying P. To this end, we compute an analytical approximation of f that allows us to provide an exact differential analysis of , draw its iso-contours, visualize its behavior on and around M, and approximate its critical points. We also compare moving least-squares and implicit techniques for the definition of the scalar function underlying f and discuss their numerical stability and approximation accuracy. Finally, the proposed framework is a starting point to extend those processing techniques that build on the analysis of scalar functions on 2-manifold surfaces to point sets.     

两种典型图布局算法的实验性对比研究

对两种典型图布局算法FR模型和LinLog模型进行了实验性的对比,实验中用自构造数据集和社会网络数据集在对称性和聚类特性两方面进行了测试,实验结果显示FR在对称性方面显示的效果比LinLog好,而LinLog显示的聚类效果比FR好.     

Vertex-neighbour-integrity of composition graphs of paths and cycles

A vertex subversion strategy of a graph G is a set of vertices X? V(G) whose closed neighbourhood is deleted from G. The survival subgraph is denoted by G/X. The vertex-neighbour-integrity of G is defined to be VNI(G)=min{|X|+τ(G/X):X? V(G)}, where τ(G/X) is the maximum order of the components of G/X. This graph parameter was introduced by Cozzens and Wu to measure the vulnerability of spy networks. Gambrell proved that the decision problem of computing the vertex-neighbour-integrity of a graph is 𝒩𝒫-complete. In this paper we evaluate the vertex-neighbour-integrity of the composition graphs of paths and cycles.     

Oriented vertex and arc colorings of outerplanar graphs

A homomorphism from an oriented graph G to an oriented graph H is an arc-preserving mapping φ from V(G) to V(H), that is φ(x)φ(y) is an arc in H whenever xy is an arc in G. The oriented chromatic number of G is the minimum order of an oriented graph H such that G has a homomorphism to H. The oriented chromatic index of G is the minimum order of an oriented graph H such that the line-digraph of G has a homomorphism to H.In this paper, we determine for every k?3 the oriented chromatic number and the oriented chromatic index of the class of oriented outerplanar graphs with girth at least k.     

Minimal simple closed 18-surfaces and a topological preservation of 3D surfaces

The aim of this paper is to investigate a map which preserves digital topological properties of a digital 3D surface, such as the topological number, digital k-contractibility and so on. Furthermore, the two types of minimal simple closed 18-surfaces MSS₁₈ and are established in Z³ in relation with 18-contractibility, which are shown to be different from each other up to digital 18-homotopy. Moreover, it is proved that MSS₁₈ is the minimal simple closed 18-surface without 18-contractibility and is the minimal Malgouyres’ simple closed 18-surface with 18-contractibility. Finally, we show that a digital (k₀,k₁)-homeomorphism preserves a simple k₀-surface to a simple k₁-surface and vice versa.