Computing Cartograms with Optimal Complexity

In a rectilinear dual of a planar graph vertices are represented by simple rectilinear polygons, while edges are represented by side-contact between the corresponding polygons. A rectilinear dual is called a cartogram if the area of each region is equal to a pre-specified weight. The complexity of a cartogram is determined by the maximum number of corners (or sides) required for any polygon. In a series of papers the polygonal complexity of such representations for maximal planar graphs has been reduced from the initial 40 to 34, then to 12 and very recently to the currently best known 10. Here we describe a construction with 8-sided polygons, which is optimal in terms of polygonal complexity as 8-sided polygons are sometimes necessary. Specifically, we show how to compute the combinatorial structure and how to refine it into an area-universal rectangular layout in linear time. The exact cartogram can be computed from the area-universal layout with numerical iteration, or can be approximated with a hill-climbing heuristic. We also describe an alternative construction of cartograms for Hamiltonian maximal planar graphs, which allows us to directly compute the cartograms in linear time. Moreover, we prove that even for Hamiltonian graphs 8-sided rectilinear polygons are necessary, by constructing a non-trivial lower bound example. The complexity of the cartograms can be reduced to 6 if the Hamiltonian path has the extra property that it is one-legged, as in outer-planar graphs. Thus, we have optimal representations (in terms of both polygonal complexity and running time) for Hamiltonian maximal planar and maximal outer-planar graphs. Finally we address the problem of constructing small-complexity cartograms for 4-connected graphs (which are Hamiltonian). We first disprove a conjecture, posed by two set of authors, that any 4-connected maximal planar graph has a one-legged Hamiltonian cycle, thereby invalidating an attempt to achieve a polygonal complexity 6 in cartograms for this graph class. We also prove that it is NP-hard to decide whether a given 4-connected plane graph admits a cartogram with respect to a given weight function.     

Graph parameters measuring neighbourhoods in graphs—Bounds and applications

The neighbourhood-width of a graph G=(V,E) is introduced in [F. Gurski, Linear layouts measuring neighbourhoods in graphs, Discrete Math. 306 (15) (2006) 1637-1650.] as the smallest integer k such that there is a linear layout ?:V→{1,…,|V|} such that for every 1?i<|V| the vertices u with ?(u)?i can be divided into at most k subsets each members having the same neighbours with respect to the vertices v with ?(v)>i.In this paper we show first bounds for the neighbourhood-width of general graphs, caterpillars, trees and grid graphs and give applications of the layout parameter neighbourhood-width in graph drawing and VLSI design.     

Bandwidth of bipartite permutation graphs in polynomial time

We give the first polynomial-time algorithm that computes the bandwidth of bipartite permutation graphs. Bandwidth is an NP-complete graph layout problem that is notorious for its difficulty even on small graph classes. For example, it remains NP-complete on caterpillars of hair length at most 3, a very restricted subclass of trees. Much attention has been given to designing approximation algorithms for computing the bandwidth, as it is NP-hard to approximate the bandwidth of general graphs with a constant factor guarantee. The problem is considered important even for approximation on restricted classes, with several distinguished results in this direction. Prior to our work, polynomial-time algorithms for exact computation of bandwidth were known only for caterpillars of hair length at most 2, chain graphs, cographs, and most interestingly, interval graphs.     

On bipartite matchings of minimum density

We study the complexity of bipartite matchings between points on two horizontal lines when the density (i.e., the number of edges of the matching that cross any vertical cut) is to be minimized. We show that finding density-minimizing matchings is considerably harder than finding weight-minimizing matchings by proving that the problem is NP-hard for general bipartite graphs. For a number of variants we present a new combinatorial characterization of the optimal density that leads to efficient and elegant algorithms. The problem of finding a matching of minimum density has applications in the area of circuit layout.     

Special graph representation and visualization of semantic networks

A visibility representation of graphs in which each vertex is mapped to a horizontal segment was originally proposed in 1980s in the context of the VLSI layout construction problem. In this paper, we present an up-to-date survey on this representation and propose a way of using it in visualization of semantic networks.     

一个去除边标签重叠的扩充张力模型

在图的最优可视化过程中,当图的边和节点都包含文字或图形标签时,显示这些标签必须保证它们互相不重叠. 这项工作可以融入初始布局的一部分,或作为后处理步骤. 去除重叠的核心问题在于保持布局中固有的结构信息,最大限度地减 少所需的额外面积,并保持边尽可能地直. 提出了一种同时去除节点和边的标签重叠的计算方法. 该算法基于最小化一个目标函数, 使得图的布局尽少改变,并保持边的平直.     

Energy and NEPS of graphs

The energy of a graph is the sum of the absolute values of the eigenvalues of the graph. We study the energy of the noncomplete extended p-sum (NEPS) of the graphs, a very general composition of the graphs in which the special case is the product of graphs. We show that the energy of the product of graphs is the product of the energy of graphs, and how this result may be used to construct arbitrarily large families of noncospectral connected graphs having the same number of vertices and the same energy. Further, unlike the product, we show that the energy of any other NEPS of the graphs cannot be represented as a function of the energy of starting graphs.     

Maximum independent sets in 3- and 4-regular Hamiltonian graphs

Smooth 4-regular Hamiltonian graphs are generalizations of cycle-plus-triangles graphs. While the latter have been shown to be 3-choosable, 3-colorability of the former is NP-complete. In this paper we first show that the independent set problem for 3-regular Hamiltonian planar graphs is NP-complete, and using this result we show that this problem is also NP-complete for smooth 4-regular Hamiltonian graphs. We also show that this problem remains NP-complete if we restrict the problem to the existence of large independent sets (i.e., independent sets whose size is at least one third of the order of the graphs).     

Oriented bipartite graphs and the Goldbach graph

In this paper, we study oriented bipartite graphs. In particular, we introduce “bitransitive” graphs. Several characterizations of bitransitive bitournaments are obtained. We show that bitransitive bitounaments are equivalent to acyclic bitournaments. As applications, we characterize acyclic bitournaments with Hamiltonian paths, determine the number of non-isomorphic acyclic bitournaments of a given order, and solve the graph-isomorphism problem in linear time for acyclic bitournaments. Next, we prove the well-known Caccetta-Häggkvist Conjecture for oriented bipartite graphs in some cases for which it is unsolved, in general, for oriented graphs. We also introduce the concept of undirected as well as oriented “odd-even” graphs. We characterize bipartite graphs and acyclic oriented bipartite graphs in terms of them. In fact, we show that any bipartite graph (acyclic oriented bipartite graph) can be represented by some odd-even graph (oriented odd-even graph). We obtain some conditions for connectedness of odd-even graphs. This study of odd-even graphs and their connectedness is motivated by a special family of odd-even graphs which we call “Goldbach graphs”. We show that the famous Goldbach's conjecture is equivalent to the connectedness of Goldbach graphs. Several other number theoretic conjectures (e.g., the twin prime conjecture) are related to various parameters of Goldbach graphs, motivating us to study the nature of vertex-degrees and independent sets of these graphs. Finally, we observe Hamiltonian properties of some odd-even graphs related to Goldbach graphs for a small number of vertices.     

关于图的同构判定方法的探讨

对于两图的同构的判定方法进行较深入的探讨,给出判定两图同构和判定两图不同构的几种方法,并对其判定方法的优劣进行比较.     

Queue layouts of iterated line directed graphs

In this paper, we study queue layouts of iterated line directed graphs. A k-queue layout of a directed graph consists of a linear ordering of the vertices and an assignment of each arc to exactly one of the k queues so that any two arcs assigned to the same queue do not nest. The queuenumber of a directed graph is the minimum number of queues required for a queue layout of the directed graph.We present upper and lower bounds on the queuenumber of an iterated line directed graph L^k(G) of a directed graph G. Our upper bound depends only on G and is independent of the number of iterations k. Queue layouts can be applied to three-dimensional drawings. From the results on the queuenumber of L^k(G), it is shown that for any fixed directed graph G, L^k(G) has a three-dimensional drawing with O(n) volume, where n is the number of vertices in L^k(G). These results are also applied to specific families of iterated line directed graphs such as de Bruijn, Kautz, butterfly, and wrapped butterfly directed graphs. In particular, the queuenumber of k-ary butterfly directed graphs is determined if k is odd.     

Geometric aspects of 2-walk-regular graphs

A t-walk-regular graph is a graph for which the number of walks of given length between two vertices depends only on the distance between these two vertices, as long as this distance is at most t. Such graphs generalize distance-regular graphs and t-arc-transitive graphs. In this paper, we will focus on 1- and in particular 2-walk-regular graphs, and study analogues of certain results that are important for distance-regular graphs. We will generalize Delsarte?s clique bound to 1-walk-regular graphs, Godsil?s multiplicity bound and Terwilliger?s analysis of the local structure to 2-walk-regular graphs. We will show that 2-walk-regular graphs have a much richer combinatorial structure than 1-walk-regular graphs, for example by proving that there are finitely many non-geometric 2-walk-regular graphs with given smallest eigenvalue and given diameter (a geometric graph is the point graph of a special partial linear space); a result that is analogous to a result on distance-regular graphs. Such a result does not hold for 1-walk-regular graphs, as our construction methods will show.     

On betweenness-uniform graphs

The betweenness centrality of a vertex of a graph is the fraction of shortest paths between all pairs of vertices passing through that vertex. In this paper, we study properties and constructions of graphs whose vertices have the same value of betweenness centrality (betweenness-uniform graphs); we show that this property holds for distanceregular graphs (which include strongly regular graphs) and various graphs obtained by graph cloning and local join operation. In addition, we show that, for sufficiently large n, there are superpolynomially many betweenness-uniform graphs on n vertices, and explore the structure of betweenness-uniform graphs having a universal or sub-universal vertex.     

Independent packings in structured graphs

Packing a maximum number of disjoint triangles into a given graph G is NP-hard, even for most classes of structured graphs. In contrast, we show that packing a maximum number of independent (that is, disjoint and nonadjacent) triangles is polynomial-time solvable for many classes of structured graphs, including weakly chordal graphs, asteroidal triple-free graphs, polygon-circle graphs, and interval-filament graphs. These classes contain other well-known classes such as chordal graphs, cocomparability graphs, circle graphs, circular-arc graphs, and outerplanar graphs. Our results apply more generally to independent packings by members of any family of connected graphs. Research of both authors is supported by the Natural Sciences and Engineering Research Council of Canada.     

Coloring intersection graphs of x-monotone curves in the plane

A class of graphs G is χ-bounded if the chromatic number of the graphs in G is bounded by some function of their clique number. We show that the class of intersection graphs of simple families of x-monotone curves in the plane intersecting a vertical line is χ-bounded. As a corollary, we show that the class of intersection graphs of rays in the plane is χ-bounded, and the class of intersection graphs of unit segments in the plane is χ-bounded.     

On the Complexity of Cover-Incomparability Graphs of Posets

In this paper we show that the recognition problem for C-I graphs of posets is NP-complete. On the other hand, we prove that induced subgraphs of C-I graphs are exactly complements of comparability graphs, and hence the recognition problem for induced subgraphs of C-I graphs of posets is polynomial.     

Homomorphism–homogeneous graphs

We answer two open questions posed by Cameron and Nesetril concerning homomorphism–homogeneous graphs. In particular we show, by giving a characterization of these graphs, that extendability to monomorphism or to homomorphism leads to the same class of graphs when defining homomorphism–homogeneity. Further, we show that there are homomorphism–homogeneous graphs that do not contain the Rado graph as a spanning subgraph answering the second open question. We also treat the case of homomorphism–homogeneous graphs with loops allowed, showing that the corresponding decision problem is co–NP complete. Finally, we extend the list of considered morphism–types and show that the graphs for which monomorphisms can be extended to epimor‐phisms are complements of homomorphism–homogeneous graphs. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 253–261, 2010     

A New Method for Enumerating Independent Sets of a Fixed Size in General Graphs

We develop a new method for enumerating independent sets of a fixed size in general graphs, and we use this method to show that a conjecture of Engbers and Galvin [7] holds for all but finitely many graphs. We also use our method to prove special cases of a conjecture of Kahn [13]. In addition, we show that our method is particularly useful for computing the number of independent sets of small sizes in general regular graphs and Moore graphs, and we argue that it can be used in many other cases when dealing with graphs that have numerous structural restrictions.     

A note on vertex-transitive non-Cayley graphs from Cayley graphs generated by involutions

We show that the result of Watkins (1990) [19] on constructing vertex-transitive non-Cayley graphs from line graphs yields a simple method that produces infinite families of vertex-transitive non-Cayley graphs from Cayley graphs generated by involutions. We also prove that the graphs arising this way are hamiltonian provided that their valency is at least six.