正P-矩阵的逆零完成问题(英文)

本文基于不完备P-矩阵的1-通弦图和1-通弦块图的完成问题,获得了不完备正P-矩阵在一定条件下的k-通弦图和k-通弦块图的完成,同时解决了不完备正P-矩阵在k-通弦图和k-通弦块图下的逆零完成问题(k为大于1的整数).     

一类图设计的构造

设Kv是一个v点完全图,G是一个有限简单图,Kv上的一个图设计G-GD(v)是一个对子(X，B)，其中X是Kv的顶点集合，B是Kv的一些与G同构的子图(称为区组)的集合，使得Kv的任意一条边恰出现在B的一个区组中．文中讨论的简单图是C(r)10，即带有一条弦的10长圈(含有11条边)，其中r表示弦的两个端点之间的顶点个数，1≤r≤4．给出了C^（r）10-GD(v)的存在谱：v=0，1(mod11)且v≥11．     

由圈得到的一类优美图

本文证明了ｎ个顶点的圈增加若干条弦所得到的图优美．     

两类只含整数根的色多项式

研究了两类只含整数根的色多项式,给出其相应图G为弦图的必要条件,并完全刻画了G的色等价类[G].     

弦图的补图的最小填充

从图论观点讲,最小填充问题就是在一个图G中添加边集F,使得图G的母图G F是一个弦图而且所添边的边数| F|是最小的,其中最小值| F|称为图G的填充数,表示为f( G) .对一般图来说,最小填充问题是NP-困难的,但是对一些特殊图类来说,这个问题是在多项式时间内可解的.本文给出了弦图的补图-G的填充数f(-G) .     

带弦的圈是协调图

称具有ｅ条边的简单图Ｇ为协调图，若存在由Ｇ的顶点集到模ｅ的整数群Ｚｅ的一个单射ｈ，使得导出映射ｈ＾＊：ｈ＾＊（ｕｖ）≡ｈ（ｕ）＋ｈ（ｖ）（ｍｏｄ ｅ）是一个由Ｇ的边集到Ｚｅ的双射，带弦的圈Ｃ′ｎ是由含ｎ个顶点的圈Ｃｎ上添一条连结两个不相邻顶点的边而得到的图。本文中证明了，除了ｎ＝６且弦端点在Ｃｎ上的距离为２的情况外，所有带弦的圈都是协调图。     

两类广义控制问题的NP-完全性

研究两类广义控制问题的复杂性: k-步长控制问题和k-距离控制问题, 证明了k-步长控制问题在弦图和平面二部图上都是NP-完全的. 作为上述结果的推论, 给出了k-距离控制问题在弦图和二部图上NP-完全性的新的证明, 并进一步证明了k-距离控制问题在平面二部图上也是NP-完全的.     

平面可弦图的子式障碍

证明了平面可弦图的子式障碍恰由Ｋ５，Ｋ３，３，Ｋ２，２，２和Ｋ２×Ｃ５这四个图构成     

偶长圈加一条弦的分拆

设Kv是一个v点完全图.G是一个有限简单图.Kv上的一个图设计G-GD是一个对子（X,B）,其中X是Kv的顶点集合,B是Kv的一些与G同构的子图（称为区组）的集合,使得Kv的任意一条边恰出现在B的一个区组中.文中讨论的简单图是C^（r）2k,即带有一条弦的2k长圈,其中r表示弦的两个端点之间的顶点个数,1≤r≤k-1.文中给出了一个构作C^（r）m设计的统一方法,并得到关于v≡0,1（mod2k＋1）时C^（r）2k-GD（v）的一系列结果.     

“弦图”曼妙景 尽在横格中——2012年山东省滨州市中考数学一道压轴题的溯源与改进

我们知道,赵爽的"弦图(或勾股圆方图)"是由四个全等的直角三角形围成的,赵爽利用它巧妙地证明了勾股定理,其证法之优美、精巧,令人叹为观止,它是证明勾股定理最著名的证法之一,特别是"弦图"一图蕴含两种证法更是举世无双".弦图"是证明勾股定理的无字经典     

Characterizing directed path graphs by forbidden asteroids

An asteroidal triple is a stable set of three vertices such that each pair is connected by a path avoiding the neighborhood of the third vertex. Asteroidal triples play a central role in a classical characterization of interval graphs by Lekkerkerker and Boland. Their result says that a chordal graph is an interval graph if and only if it does not contain an asteroidal triple. In this paper, we prove an analogous theorem for directed path graphs which are the intersection graphs of directed paths in a directed tree. For this purpose, we introduce the notion of a special connection. Two non‐adjacent vertices are linked by a special connection if either they have a common neighbor or they are the endpoints of two vertex‐disjoint chordless paths satisfying certain conditions. A special asteroidal triple is an asteroidal triple such that each pair is linked by a special connection. We prove that a chordal graph is a directed path graph if and only if it does not contain a special asteroidal triple. © 2010 Wiley Periodicals, Inc. J Graph Theory 68:103‐112, 2011     

The clique-separator graph for chordal graphs

We present a new representation of a chordal graph called the clique-separator graph, whose nodes are the maximal cliques and minimal vertex separators of the graph. We present structural properties of the clique-separator graph and additional properties when the chordal graph is an interval graph, proper interval graph, or split graph. We also characterize proper interval graphs and split graphs in terms of the clique-separator graph. We present an algorithm that constructs the clique-separator graph of a chordal graph in O(n³) time and of an interval graph in O(n²) time, where n is the number of vertices in the graph.     

Cover-incomparability graphs and chordal graphs

The problem of recognizing cover-incomparability graphs (i.e. the graphs obtained from posets as the edge-union of their covering and incomparability graph) was shown to be NP-complete in general [J. Maxová, P. Pavlíkova, A. Turzík, On the complexity of cover-incomparability graphs of posets, Order 26 (2009) 229-236], while it is for instance clearly polynomial within trees. In this paper we concentrate on (classes of) chordal graphs, and show that any cover-incomparability graph that is a chordal graph is an interval graph. We characterize the posets whose cover-incomparability graph is a block graph, and a split graph, respectively, and also characterize the cover-incomparability graphs among block and split graphs, respectively. The latter characterizations yield linear time algorithms for the recognition of block and split graphs, respectively, that are cover-incomparability graphs.     

Polarity of chordal graphs

Polar graphs are a common generalization of bipartite, cobipartite, and split graphs. They are defined by the existence of a certain partition of vertices, which is NP-complete to decide for general graphs. It has been recently proved that for cographs, the existence of such a partition can be characterized by finitely many forbidden subgraphs, and hence tested in polynomial time. In this paper we address the question of polarity of chordal graphs, arguing that this is in essence a question of colourability, and hence chordal graphs are a natural restriction. We observe that there is no finite forbidden subgraph characterization of polarity in chordal graphs; nevertheless we present a polynomial time algorithm for polarity of chordal graphs. We focus on a special case of polarity (called monopolarity) which turns out to be the central concept for our algorithms. For the case of monopolar graphs, we illustrate the structure of all minimal obstructions; it turns out that they can all be described by a certain graph grammar, permitting our monopolarity algorithm to be cast as a certifying algorithm.     

Obstructions to chordal circular-arc graphs of small independence number

A blocking quadruple (BQ) is a quadruple of vertices of a graph such that any two vertices of the quadruple either miss (have no neighbours on) some path connecting the remaining two vertices of the quadruple, or are connected by some path missed by the remaining two vertices. This is akin to the notion of asteroidal triple used in the classical characterization of interval graphs by Lekkerkerker and Boland [Klee, V., What are the intersection graphs of arcs in a circle?, American Mathematical Monthly 76 (1976), pp. 810–813.].In this note, we first observe that blocking quadruples are obstructions for circular-arc graphs. We then focus on chordal graphs, and study the relationship between the structure of chordal graphs and the presence/absence of blocking quadruples.Our contribution is two-fold. Firstly, we provide a forbidden induced subgraph characterization of chordal graphs without blocking quadruples. In particular, we observe that all the forbidden subgraphs are variants of the subgraphs forbidden for interval graphs [Klee, V., What are the intersection graphs of arcs in a circle?, American Mathematical Monthly 76 (1976), pp. 810–813.]. Secondly, we show that the absence of blocking quadruples is sufficient to guarantee that a chordal graph with no independent set of size five is a circular-arc graph. In our proof we use a novel geometric approach, constructing a circular-arc representation by traversing around a carefully chosen clique tree.     

Introducing subclasses of basic chordal graphs

Basic chordal graphs arose when comparing clique trees of chordal graphs and compatible trees of dually chordal graphs. They were defined as those chordal graphs whose clique trees are exactly the compatible trees of its clique graph.In this work, we consider some subclasses of basic chordal graphs, like hereditary basic chordal graphs, basic DV and basic RDV graphs, we characterize them and we find some other properties they have, mostly involving clique graphs.     

Partitioning Perfect Graphs into Stars

The partition of graphs into “nice” subgraphs is a central algorithmic problem with strong ties to matching theory. We study the partitioning of undirected graphs into same‐size stars, a problem known to be NP‐complete even for the case of stars on three vertices. We perform a thorough computational complexity study of the problem on subclasses of perfect graphs and identify several polynomial‐time solvable cases, for example, on interval graphs and bipartite permutation graphs, and also NP‐complete cases, for example, on grid graphs and chordal graphs.     

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.     

Approximating the minimum clique cover and other hard problems in subtree filament graphs

Subtree filament graphs are the intersection graphs of subtree filaments in a tree. This class of graphs contains subtree overlap graphs, interval filament graphs, chordal graphs, circle graphs, circular-arc graphs, cocomparability graphs, and polygon-circle graphs. In this paper we show that, for circle graphs, the clique cover problem is NP-complete and the h-clique cover problem for fixed h is solvable in polynomial time. We then present a general scheme for developing approximation algorithms for subtree filament graphs, and give approximation algorithms developed from the scheme for the following problems which are NP-complete on circle graphs and therefore on subtree filament graphs: clique cover, vertex colouring, maximum k-colourable subgraph, and maximum h-coverable subgraph.