On the Metric Dimension of Infinite Graphs

A set of vertices S resolves a connected graph G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of a graph G is the minimum cardinality of a resolving set. In this paper we undertake the metric dimension of infinite locally finite graphs, i.e., those infinite graphs such that all its vertices have finite degree. We give some necessary conditions for an infinite graph to have finite metric dimension and characterize infinite trees with finite metric dimension. We also establish some general results about the metric dimension of the Cartesian product of finite and infinite graphs, and obtain the metric dimension of the Cartesian product of several families of graphs.     

On the Metric Dimension of Imprimitive Distance-Regular Graphs

A resolving set for a graph \({\Gamma}\) is a collection of vertices S, chosen so that for each vertex v, the list of distances from v to the members of S uniquely specifies v. The metric dimension of \({\Gamma}\) is the smallest size of a resolving set for \({\Gamma}\). Much attention has been paid to the metric dimension of distance-regular graphs. Work of Babai from the early 1980s yields general bounds on the metric dimension of primitive distance-regular graphs in terms of their parameters. We show how the metric dimension of an imprimitive distance-regular graph can be related to that of its halved and folded graphs. We also consider infinite families (including Taylor graphs and the incidence graphs of certain symmetric designs) where more precise results are possible.     

The Local Metric Dimension of Strong Product Graphs

A vertex \(v\in V(G)\) is said to distinguish two vertices \(x,y\in V(G)\) of a nontrivial connected graph G if the distance from v to x is different from the distance from v to y. A set \(S\subset V(G)\) is a local metric generator for G if every two adjacent vertices of G are distinguished by some vertex of S. A local metric generator with the minimum cardinality is called a local metric basis for G and its cardinality, the local metric dimension of G. It is known that the problem of computing the local metric dimension of a graph is NP-Complete. In this paper we study the problem of finding exact values or bounds for the local metric dimension of strong product of graphs.     

Connected Resolvability of Graphs

For an ordered set W = {w ₁, w ₂,..., w _k} of vertices and a vertex v in a connected graph G, the representation of v with respect to W is the k-vector r(v|W) = (d(v, w ₁), d(v, w ₂),... d(v, w _k)), where d(x, y) represents the distance between the vertices x and y. The set W is a resolving set for G if distinct vertices of G have distinct representations with respect to W. A resolving set for G containing a minimum number of vertices is a basis for G. The dimension dim(G) is the number of vertices in a basis for G. A resolving set W of G is connected if the subgraph 〈W〉 induced by W is a nontrivial connected subgraph of G. The minimum cardinality of a connected resolving set in a graph G is its connected resolving number cr(G). Thus 1 ≤ dim(G) ≤ cr(G) ≤ n?1 for every connected graph G of order n ≥ 3. The connected resolving numbers of some well-known graphs are determined. It is shown that if G is a connected graph of order n ≥ 3, then cr(G) = n?1 if and only if G = K _n or G = K _1,n?1. It is also shown that for positive integers a, b with a ≤ b, there exists a connected graph G with dim(G) = a and cr(G) = b if and only if $\left( {a,b} \right) \notin \left\{ {\left( {1,k} \right):k = 1\;{\text{or}}\;k \geqslant 3} \right\}$ Several other realization results are present. The connected resolving numbers of the Cartesian products G × K ₂ for connected graphs G are studied.     

图的连通因子

连通因子问题与Hamilton问题和信息网络有着密切的联系．1993年，首先由M.Kano就该问题在[6]中提出了许多问题和猜想，接下来他又在[16]中提出了一些猜想，本文就连通因子问题在过去十年的主要进展进行了回顾并提出了若干可进一步研究的问题和猜想．     

Hamiltonian Connected Claw-Free Graphs

A well-known result by O. Ore is that every graph of order n with d(u)+d(v)n+1 for any pair of nonadjacent vertices u and v is hamiltonian connected (i.e., for every pair of vertices, there is a hamiltonian path joining them). In this paper, we show that every 3-connected claw-free graph G on at most 5–8 vertices is hamiltonian connected, where denotes the minimum degree in G. This result generalizes several previous results.Acknowledgments. The author would like to thank the referee for his important suggestions and careful corrections.Final version received: March 12, 2003Supported by the project of Nature Science Funds of China     

泛连通图的Faudree-Schelp定理和哈密尔顿连通图的Ore定理的注记

In this note more short proofs are given for Faudree-Schelp theorem and Ore theorem.     

Weakly Connected Domination in Graphs

In the last 50 years, Graph theory has seen an explosive growth due to interaction with areas like computer science, electrical and communication engineering, Operations Research etc. Perhaps the fastest growing area within graph theory is the study of domination, the reason being its many and varied applications in such fields as social sciences, communication networks, algorithm designs, computational complexity etc. Henda C. Swart has rightly commented that the theory of domination in graphs is like a ‘growth industry’. There are several types of domination depending upon the nature of domination and the nature of the dominating set. In the following, we present weakly connected domination in connected graphs.     

连通图的度序列及连通平面图的低度点个数

美国数学家Bondy给出了一个非负整数序列为简单图的度序列的充要条件.本文对此进行了发展,证明了一个正整数序列为连通简单图的度序列的充要条件;然后在此基础上又探讨了平面图的低度点个数问题并定义了描述连通平面图的低度点个数的一个概念φ(n,m),并对某些低阶平面图求出了φ(n,m)的值.最后给出了φ(n,m)的上下界.     

Circumference and Pathwidth of Highly Connected Graphs

Birmele [J Graph Theory 2003] proved that every graph with circumference t has treewidth at most . Under the additional assumption of 2‐connectivity, such graphs have bounded pathwidth, which is a qualitatively stronger conclusion. Birmele's theorem was extended by Birmele et al. [Combinatorica 2007] who showed that every graph without k disjoint cycles of length at least t has treewidth . Our main result states that, under the additional assumption of ‐connectivity, such graphs have bounded pathwidth. In fact, they have pathwidth . Moreover, examples show that ‐connectivity is required for bounded pathwidth to hold. These results suggest the following general question: for which values of k and graphs H does every k‐connected H‐minor‐free graph have bounded pathwidth? We discuss this question and provide a few observations.     

连通的顶点可迁图的色唯一性

本文给出从一个已知的顶点可迁的非色唯一图出发,构造无穷多个顶点可迁的非色唯一图的一种方法,据此给出若干类无穷多个连通的顶点可迁,但不是色唯一的图簇,从而进一步否定地回答了Chia在[1]中提出的问题.     

Metric Uniformization and Spectral Bounds for Graphs

We present a method for proving upper bounds on the eigenvalues of the graph Laplacian. A main step involves choosing an appropriate “Riemannian” metric to uniformize the geometry of the graph. In many interesting cases, the existence of such a metric is shown by examining the combinatorics of special types of flows. This involves proving new inequalities on the crossing number of graphs.     

The Decomposition Dimension of Graphs

 For an ordered k-decomposition ? = {G ₁, G ₂,…,G _k } of a connected graph G and an edge e of G, the ?-representation of e is the k-tuple r(e|?) = (d(e, G ₁), d(e, G ₂),…,d(e, G _k )), where d(e, G _i ) is the distance from e to G _i . A decomposition ? is resolving if every two distinct edges of G have distinct representations. The minimum k for which G has a resolving k-decomposition is its decomposition dimension dec(G). It is shown that for every two positive integers k and n≥ 2, there exists a tree T of order n with dec(T) = k. It is also shown that dec(G) ≤n for every graph G of order n≥ 3 and that dec(K _n ) ≤⌊(2n + 5)/3⌋ for n≥ 3. Received: June 17, 1998 Final version received: August 10, 1999     

Accessibility and Reliability for Connected Bipartite Graphs

In the recently published atlas of graphs [9] the general listing of graphs with diagrams went up to V=7 vertices but the special listing for connected bipartite graphs carried further up to V=8. In this paper we wish to study the random accessibility of these connected bipartite graphs by means of random walks on the graphs using the degree of the gratis starting point as a weighting factor. The accessibility is then related to the concept of reliability of the graphs with only edge failures. Exact expectation results for accessibility are given for any complete connected bipartite graph N₁ cbp N₂ (where cbp denotes connected bipartite) for several values of J (the number of new vertices searched for). The main conjecture in this paper is that for any complete connected bipartite graph N₁ cbp N₂: if |N₁–N₂| 1, then the graph is both uniformly optimal in reliability and optimal in random accessibility within its family. Numerical results are provided to support the conjecture.     

Connected Factors and Spanning Trees in Graphs

 Let G be a graph, and g, f, f′ be positive integer-valued functions defined on V(G). If an f′-factor of G is a spanning tree, we say that it is f′-tree. In this paper, it is shown that G contains a connected (g, f+f′−1)-factor if G has a (g, f)-factor and an f′-tree. Received: October 30, 2000 Final version received: August 20, 2002     

Grid Intersection Graphs and Order Dimension

We study subclasses of grid intersection graphs from the perspective of order dimension. We show that partial orders of height two whose comparability graph is a grid intersection graph have order dimension at most four. Starting from this observation we provide a comprehensive study of classes of graphs between grid intersection graphs and bipartite permutation graphs and the containment relation on these classes. Order dimension plays a role in many arguments.     

Connected Domatic Number in Planar Graphs

A dominating set in a graph G is a connected dominating set of G if it induces a connected subgraph of G. The connected domatic number of G is the maximum number of pairwise disjoint, connected dominating sets in V(G). We establish a sharp lower bound on the number of edges in a connected graph with a given order and given connected domatic number. We also show that a planar graph has connected domatic number at most 4 and give a characterization of planar graphs having connected domatic number 3.     

Ehrhart Series for Connected Simple Graphs

The Ehrhart ring of the edge polytope ${\mathcal{P}_G}$ for a connected simple graph G is known to coincide with the edge ring of the same graph if G satisfies the odd cycle condition. This paper gives for a graph which does not satisfy the condition, a generating set of the defining ideal of the Ehrhart ring of the edge polytope, described by combinatorial information of the graph. From this result, two factoring properties of the Ehrhart series are obtained; the first one factors out bipartite biconnected components, and the second one factors out a even cycle which shares only one edge with other part of the graph. As an application of the factoring properties, the root distribution of Ehrhart polynomials for bipartite polygon trees is determined.