Quotient tree partitioning of undirected graphs

The partitioning of the vertices of an undirected graph, in a way that makes its quotient graph a tree, mirrors a way of permuting a square symmetric matrix to allow its factoring with little fill-in. We analyze the complexity of finding the best partitioning and show that it is NP-complete. We also give a new and simpler implementation of an algorithm that finds a maximal quotient tree.     

Partitions and homomorphisms in directed and undirected graphs

Boyle has given a condition for defining a homomorphism in terms of minimal paths for undirected graphs. The purpose of such homomorphisms is to provide a simpler graph which will reflect the structure of the more complex graph, and thereby enable the researcher to make observations which may have been shrouded by a preponderance of nodes and edges. This paper develops Boyle's ideas and introduces further homomorphisms for directed as well as undirected graphs. The relationships between the various homomorphisms are also examined.     

The lattice of quasivarieties of undirected graphs

In this note, it is shown that the quasivariety of undirected graphs is Q-universal. Received March 3, 2000; accepted in final form January 31, 2001.     

SUPER EDGE-CONNECTIVITY OF DE BRUIJN AND KAUTZ UNDIRECTED GRAPHS

The super edge-connectivity of a graph is an important parameter to measure fault-tolerance of interconnection networks. This note shows that the Kautz undirected graph is super edge-connected,and provides a short proof of Lue and Zhang‘s result on super edge-connectivity of the de Bruijn undirected graph.     

Paths and edge-connectivity in graphs

Mader proved that for every k-edge-connected graph G (k ≥ 4), there exists a path joining two given vertices such that the subgraph obtained from G by deleting the edges of the path is (k - 2)-edge-connected. A generalization of this and a sufficient condition for existance of 3, 4, or 5 terminus k edge-disjoint paths in graphs are given.     

On tree congestion of graphs

This paper investigates the problem of embedding a graph into a tree with the same vertex set (a spanning tree in particular), such that the maximum congestion of the edges is minimized. We calculate exact formulas for the tree congestion and spanning tree congestion for various families of graphs, including grids and complete bipartite graphs.     

On P-quasigroups and decompositions of complete undirected graphs

In [2], A. Kotzig has introduced the concepts of P-groupoid and P-quasigroup and has shown how these concepts are related to the decomposition of a complete undirected graph into disjoint closed paths. To each closed path of the graph associated with a given P-quasigroup Q there corresponds a cyclic partial transversal in the Latin square L which is defined by the multiplication table of Q. In this paper, it is shown that cyclic transversals are connected with Hamiltonian decompositions of complete undirected graphs having an even number of vertices and a connection between the order of a particular type of P-quasigroup and the length of its cyclic partial transversals is established. An indirect connection with the work of Yap [4] is established via the concept of isotopy.     

An algorithm for generating minimal cutsets of undirected graphs

In this paper, we propose an algorithm for generating minimal cutsets of undirected graphs. The algorithm is based on a blocking mechanism for generating every minimal cutset exactly once. The algorithm has an advantage of not requiring any preliminary steps to find minimal cutsets. The algorithm generates minimal cutsets atO(e · n) {wheree,n = number of (edges, vertices) in the graph} computational effort per cutset. Formal proofs of the algorithm are presented.     

On spanning tree congestion of graphs

Let G be a connected graph and T be a spanning tree of G. For e∈E(T), the congestion of e is the number of edges in G connecting two components of T−e. The edge congestion ofGinT is the maximum congestion over all edges in T. The spanning tree congestion ofG is the minimum congestion of G in its spanning trees. In this paper, we show the spanning tree congestion for the complete k-partite graphs and the two-dimensional tori. We also address lower bounds of spanning tree congestion for the multi-dimensional grids and the hypercubes.     

On Riesz and Wishart distributions associated with decomposable undirected graphs

Classical Wishart distributions on the open convex cone of positive definite matrices and their fundamental features are extended to generalized Riesz and Wishart distributions associated with decomposable undirected graphs using the basic theory of exponential families. The families of these distributions are parameterized by their expectations/natural parameter and multivariate shape parameter and have a non-trivial overlap with the generalized Wishart distributions defined in Andersson and Wojnar (2004) [4] and [8]. This work also extends the Wishart distributions of type I in Letac and Massam (2007) [7] and, more importantly, presents an alternative point of view on the latter paper.     

On 3-restricted edge connectivity of undirected binary Kautz graphs

An edge cut of a connected graph is m-restricted if its removal leaves every component having order at least m. The size of minimum m-restricted edge cuts of a graph G is called its m-restricted edge connectivity. It is known that when m≤4, networks with maximal m-restricted edge connectivity are most locally reliable. The undirected binary Kautz graph UK(2,n) is proved to be maximal 2- and 3-restricted edge connected when n≥3 in this work. Furthermore, every minimum 2-restricted edge cut disconnects this graph into two components, one of which being an isolated edge.     

On degree sequences of undirected,directed, and bidirected graphs

Bidirected graphs generalize directed and undirected graphs in that edges are oriented locally at every node. The natural notion of the degree of a node that takes into account (local) orientations is that of net-degree. In this paper, we extend the following four topics from (un)directed graphs to bidirected graphs:

• –Erdős–Gallai-type results: characterization of net-degree sequences,
• –Havel–Hakimi-type results: complete sets of degree-preserving operations,
• –Extremal degree sequences: characterization of uniquely realizable sequences, and
• –Enumerative aspects: counting formulas for net-degree sequences.

To underline the similarities and differences to their (un)directed counterparts, we briefly survey the undirected setting and we give a thorough account for digraphs with an emphasis on the discrete geometry of degree sequences. In particular, we determine the tight and uniquely realizable degree sequences for directed graphs.     

On the sizes of graphs and their powers: The undirected case

Let G be an undirected graph and G^r be its r-th power. We study different issues dealing with the number of edges in G and G^r. In particular, we answer the following question: given an integer r≥2 and all connected graphs G of order n such that G^r≠K_n, what is the minimum number of edges that are added when going from G to G^r, and which are the graphs achieving this bound?     

Branch and cut algorithms for detecting critical nodes in undirected graphs

In this paper we deal with the critical node problem, where a given number of nodes has to be removed from an undirected graph in order to maximize the disconnections between the node pairs of the graph. We propose an integer linear programming model with a non-polynomial number of constraints but whose linear relaxation can be solved in polynomial time. We derive different valid inequalities and some theoretical results about them. We also propose an alternative model based on a quadratic reformulation of the problem. Finally, we perform many computational experiments and analyze the corresponding results.     

Paths and cycles in matroid base graphs

LetG be the base graph of any simple matroid. It is proved thatG is Hamilton-connected, edge-pancyclic, and if for any two vertices ofG there are paths of lengthsm andn joining them,m < n, then there is a path of lengthk joining them for all integersk satisfyingm < k < n. This research was partially supported by the Natural Sciences and Engineering Council of Canada under Grant A-4792.This research was done while the author was a Visiting Scholar at Simon Fraser University.     

Paths of homomorphisms from stable Kneser graphs

We denote by SG _n,k the stable Kneser graph (Schrijver graph) of stable n-subsets of a set of cardinality 2n+k. For k≡3 (mod 4) and n≥2 we show that there is a component of the χ-colouring graph of SG _n,k which is invariant under the action of the automorphism group of SG _n,k . We derive that there is a graph G with χ(G)=χ(SG _n,k ) such that the complex Hom(SG _n,k ,G) is non-empty and connected. In particular, for k≡3 (mod 4) and n≥2 the graph SG _n,k is not a test graph.     

Efficient algorithms for finding minimum spanning trees in undirected and directed graphs

Recently, Fredman and Tarjan invented a new, especially efficient form of heap (priority queue). Their data structure, theFibonacci heap (or F-heap) supports arbitrary deletion inO(logn) amortized time and other heap operations inO(1) amortized time. In this paper we use F-heaps to obtain fast algorithms for finding minimum spanning trees in undirected and directed graphs. For an undirected graph containingn vertices andm edges, our minimum spanning tree algorithm runs inO(m logβ (m, n)) time, improved fromO(mβ(m, n)) time, whereβ(m, n)=min {i|log⁽ⁱ⁾ n ≦m/n}. Our minimum spanning tree algorithm for directed graphs runs inO(n logn + m) time, improved fromO(n log n +m log log log_(m/n+2) n). Both algorithms can be extended to allow a degree constraint at one vertex. Research supported in part by National Science Foundation Grant MCS-8302648. Research supported in part by National Science Foundation Grant MCS-8303139. Research supported in part by National Science Foundation Grant MCS-8300984 and a United States Army Research Office Program Fellowship, DAAG29-83-GO020.