Locally triangular graphs and normal quotients of the <Emphasis Type="Italic">n</Emphasis>-cube

For an integer \(n\ge 2\), the triangular graph has vertex set the 2-subsets of \(\{1,\ldots ,n\}\) and edge set the pairs of 2-subsets intersecting at one point. Such graphs are known to be halved graphs of bipartite rectagraphs, which are connected triangle-free graphs in which every 2-path lies in a unique quadrangle. We refine this result and provide a characterisation of connected locally triangular graphs as halved graphs of normal quotients of n-cubes. To do so, we study a parameter that generalises the concept of minimum distance for a binary linear code to arbitrary automorphism groups of the n-cube.     

Set graphs. I. Hereditarily finite sets and extensional acyclic orientations

A graph $G$ is said to be a set graph if it admits an acyclic orientation which is also extensional, in the sense that the out-neighborhoods of its vertices are pairwise distinct. Equivalently, a set graph is the underlying graph of the digraph representation of a hereditarily finite set.In this paper, we initiate the study of set graphs. On the one hand, we identify several necessary conditions that every set graph must satisfy. On the other hand, we show that set graphs form a rich class of graphs containing all connected claw-free graphs and all graphs with a Hamiltonian path. In the case of claw-free graphs, we provide a polynomial-time algorithm for finding an extensional acyclic orientation. Inspired by manipulations of hereditarily finite sets, we give simple proofs of two well-known results about claw-free graphs. We give a complete characterization of unicyclic set graphs, and point out two NP-complete problems closely related to the problem of recognizing set graphs. Finally, we argue that these three problems are solvable in linear time on graphs of bounded treewidth.     

The signless Laplacian spectral radius of bicyclic graphs with prescribed degree sequences

In this paper, we characterize all extremal connected bicyclic graphs with the largest signless Laplacian spectral radius in the set of all connected bicyclic graphs with prescribed degree sequences. Moreover, the signless Laplacian majorization theorem is proved to be true for connected bicyclic graphs. As corollaries, all extremal connected bicyclic graphs having the largest signless Laplacian spectral radius are obtained in the set of all connected bicyclic graphs of order n (resp. all connected bicyclic graphs with a specified number of pendant vertices, and all connected bicyclic graphs with given maximum degree).     

一类二层多目标规划的若干性质

本文对于下层为线性多目标规划的二层规划问题,在约束域非空有界的条件下证明了可行集的弱拟凸性、连通性,为算法设计提供了理论依据．     

Compatible connectedness in graphs and topological spaces

A topology on the vertex set of a graphG iscompatible with the graph if every induced subgraph ofG is connected if and only if its vertex set is topologically connected. In the case of locally finite graphs with a finite number of components, it was shown in [11] that a compatible topology exists if and only if the graph is a comparability graph and that all such topologies are Alexandroff. The main results of Section 1 extend these results to a much wider class of graphs. In Section 2, we obtain sufficient conditions on a graph under which all the compatible topologies are Alexandroff and in the case of bipartite graphs we show that this condition is also necessary.     

On connected domination in unit ball graphs

Given a simple undirected graph, the minimum connected dominating set problem is to find a minimum cardinality subset of vertices D inducing a connected subgraph such that each vertex outside D has at least one neighbor in D. Approximations of minimum connected dominating sets are often used to represent a virtual routing backbone in wireless networks. This paper first proposes a constant-ratio approximation algorithm for the minimum connected dominating set problem in unit ball graphs and then introduces and studies the edge-weighted bottleneck connected dominating set problem, which seeks a minimum edge weight in the graph such that the corresponding bottleneck subgraph has a connected dominating set of size k. In wireless network applications this problem can be used to determine an optimal transmission range for a network with a predefined size of the virtual backbone. We show that the problem is hard to approximate within a factor better than 2 in graphs whose edge weights satisfy the triangle inequality and provide a 3-approximation algorithm for such graphs. We also show that for fixed k the problem is polynomially solvable in unit disk and unit ball graphs.     

Feasibility of motion planning on acyclic and strongly connected directed graphs

Motion planning is a fundamental problem of robotics with applications in many areas of computer science and beyond. Its restriction to graphs has been investigated in the literature, for it allows one to concentrate on the combinatorial problem abstracting from geometric considerations. In this paper, we consider motion planning over directed graphs, which are of interest for asymmetric communication networks. Directed graphs generalize undirected graphs, while introducing a new source of complexity to the motion planning problem: moves are not reversible. We first consider the class of acyclic directed graphs and show that the feasibility can be solved in time linear in the product of the number of vertices and the number of arcs. We then turn to strongly connected directed graphs. We first prove a structural theorem for decomposing strongly connected directed graphs into strongly biconnected components. Based on the structural decomposition, we show that the feasibility of motion planning on strongly connected directed graphs can be decided in linear time.     

Brambles and independent packings in chordal graphs

An independent packing of triangles is a set of pairwise disjoint triangles, no two of which are joined by an edge. A triangle bramble is a set of triangles, every pair of which intersect or are joined by an edge. More generally, I consider independent packings and brambles of any specified connected graphs, not just triangles. I give a min-max theorem for the maximum number of graphs in an independent packing of any family of connected graphs in a chordal graph, and a dual min-max theorem for the maximum number of graphs in a bramble in a chordal graph.     

Geometry of the vacant set left by random walk on random graphs,Wright's constants,and critical random graphs with prescribed degrees

We provide an explicit algorithm for sampling a uniform simple connected random graph with a given degree sequence. By products of this central result include: (1) continuum scaling limits of uniform simple connected graphs with given degree sequence and asymptotics for the number of simple connected graphs with given degree sequence under some regularity conditions, and (2) scaling limits for the metric space structure of the maximal components in the critical regime of both the configuration model and the uniform simple random graph model with prescribed degree sequence under finite third moment assumption on the degree sequence. As a substantive application we answer a question raised by ?erný and Teixeira study by obtaining the metric space scaling limit of maximal components in the vacant set left by random walks on random regular graphs.     

On an approach to the modelling of problems connected with conflicting economic interests

The aim of the present paper is to show a possible way to use the recent results obtained in linear and linear-fractional programming in applications connected with the coordination of conflicting economic interests. We consider the relationship between linear and linear-fractional programming problems with the same feasible set and show how it is possible to reorientate objective functions such that all objective functions considered lead to the same optimal solution. We show also that in certain cases the dual estimates of linear and linear-fractional programming are closely connected. The economic interpretation of the results obtained is outlined and a numerical example is presented.     

覆盖数不超过3的图上渡河问题

1000多年前,英国著名学者Alcuin曾提出一个古老的渡河问题,即狼、羊和卷心菜的渡河问题。2006年,Prisner把该问题推广到任意的冲突图上,考虑了一类情况更一般的渡河运输问题。所谓冲突图是指一个图G=(V,E),这里V代表某些物品的集合,V中的两个点有边连结当且仅当这两个点是冲突的,即在无人监管的情况下不允许留在一起的点。图G=(V,E)的一个可行运输方案是指在保证不发生任何冲突的前提下,把V的点所代表的物品全部摆渡到河对岸的一个运输方案。图G的Alcuin数定义为它存在可行运输方案时所需船的最小容量。本文讨论了覆盖数不超过3的连通图的Alcuin数,给出了该类图Alcuin数的完全刻画。     

Convex domination in the composition and cartesian product of graphs

In this paper we characterize the convex dominating sets in the composition and Cartesian product of two connected graphs. The concepts of clique dominating set and clique domination number of a graph are defined. It is shown that the convex domination number of a composition G[H] of two non-complete connected graphs G and H is equal to the clique domination number of G. The convex domination number of the Cartesian product of two connected graphs is related to the convex domination numbers of the graphs involved.     

On bounds for the index of double nested graphs

The index of a simple graph is the largest eigenvalue of its adjacency matrix. It is well-known that in the set of all connected graphs with fixed order and size the graphs with maximal index are nested split graphs. It was recently observed that double nested graphs assume the same role if we restrict ourselves to bipartite graphs. In this paper we provide some bounds (lower and upper) for the index of double nested graphs. Some computational results are also included.     

Crown reductions for the Minimum Weighted Vertex Cover problem

The paper studies crown reductions for the Minimum Weighted Vertex Cover problem introduced recently in the unweighted case by Fellows et al. [Blow-Ups, Win/Win's and crown rules: some new directions in FPT, in: Proceedings of the 29th International Workshop on Graph Theoretic Concepts in Computer Science (WG’03), Lecture notes in computer science, vol. 2880, 2003, pp. 1-12, Kernelization algorithms for the vertex cover problem: theory and experiments, in: Proceedings of the Workshop on Algorithm Engineering and Experiments (ALENEX), New Orleans, Louisiana, January 2004, pp. 62-69]. We describe in detail a close relation of crown reductions to Nemhauser and Trotter reductions that are based on the linear programming relaxation of the problem. We introduce and study the so-called strong crown reductions, suitable for finding (or counting) all minimum vertex covers, or finding a minimum vertex cover under some additional constraints. It is described how crown decompositions and strong crown decompositions suitable for such problems can be computed in polynomial time. For weighted König-Egerváry graphs (G,w) we observe that the set of vertices belonging to all minimum vertex covers, and the set of vertices belonging to no minimum vertex covers, can be efficiently computed.Further, for some specific classes of graphs, simple algorithms for the MIN-VC problem with a constant approximation factor r<2 are provided. On the other hand, we conclude that for the regular graphs, or for the Hamiltonian connected graphs, the problem is as hard to approximate as for general graphs.It is demonstrated how the results about strong crown reductions can be used to achieve a linear size problem kernel for some related vertex cover problems.     

Coloring planar perfect graphs by decomposition

This paper describes a decomposition scheme for coloring perfect graphs. Based on this scheme, one need only concentrate on coloring highly connected (at least 3-connected) perfect graphs. This idea is illustrated on planar perfect graphs, which yields a straightforward coloring algorithm. We suspect that, under appropriate definition, highly connected perfect graphs might possess certain regular properties that are amenable to coloring algorithms. This research has been supported in part by National Science Foundation under grant ECS—8105989 to Northwestern University.     

On Homotopy of connected graphs having the same degree function

Let denote the set of simple graphs having the degree function . It is known that any two graphs from are homotopic in the sense that one can be obtained from the other by a series of simple two-edge switches so that every intermediate graph is in . Let denote the set of simple graphs having at most k components. A natural question arises whether it is possible to extend the above homotopy result on the class , and in particular, whether it is possible to obtain a connected simple graph from another connected simple graph of the same degree function by a series of two edge switches preserving connectivity. In this paper we give an answer to this question. We also consider two-edge switches of a special type and single out some classes of graphs for which these special operations provide the corresponding homotopy between its members.     

Degenerate Linear Multicriteria Sensitivity

Sensitivity to perturbations of the vector objective function and/or the constraints is studied for the efficient (or weakly) set associated to a linear multicriteria program whose the feasible polyhedron is not necessarily assumed to be nondegenerate. Namely, the efficient set in the linear case being a connected union of some faces of the polyhedron, then we establish firstly second order necessary and sufficient conditions for the feasibility of a degenerate vertex under small perturbations, and finally, we give necessary and sufficient conditions of Simplex type, for a such vertex, or incident face to it, to be efficient and mostly to preserve this quality after the perturbations.     

Facet defining inequalities among graph invariants: The system GraPHedron

We present a new computer system, called GraPHedron, which uses a polyhedral approach to help the user to discover optimal conjectures in graph theory. We define what should be optimal conjectures and propose a formal framework allowing to identify them. Here, graphs with n nodes are viewed as points in the Euclidian space, whose coordinates are the values of a set of graph invariants. To the convex hull of these points corresponds a finite set of linear inequalities. These inequalities computed for a few values of n can be possibly generalized automatically or interactively. They serve as conjectures which can be considered as optimal by geometrical arguments.We describe how the system works, and all optimal relations between the diameter and the number of edges of connected graphs are given, as an illustration. Other applications and results are mentioned, and the forms of the conjectures that can be currently obtained with GraPHedron are characterized.     

Dominating and large induced trees in regular graphs

Recently Chen et al. [Tree domination in graphs, Ars Combin. 73 (2004) 193-203] asked for characterizations of the class of graphs and the class of regular graphs that have an induced dominating tree, i.e. for which there exists a dominating set that induces a tree.We give a somewhat negative answer to their question by proving that the corresponding decision problems are NP-complete. Furthermore, we prove essentially best-possible lower bounds on the maximum order of induced trees in connected cacti of maximum degree 3 and connected cubic graphs.Finally, we give a forbidden induced subgraph condition for the existence of induced dominating trees.