Minimum edge ranking spanning trees of split graphs

Given a graph G, the minimum edge ranking spanning tree problem (MERST) is to find a spanning tree of G whose edge ranking is minimum. However, this problem is known to be NP-hard for general graphs. In this paper, we show that the problem MERST has a polynomial time algorithm for split graphs, which have useful applications in practice. The result is also significant in the sense that this is a first non-trivial graph class for which the problem MERST is found to be polynomially solvable. We also show that the problem MERST for threshold graphs can be solved in linear time, where threshold graphs are known to be split.     

On encodings of spanning trees

Deo and Micikevicius recently gave a new bijection for spanning trees of complete bipartite graphs. In this paper we devise a generalization of Deo and Micikevicius's method, which is also a modification of Olah's method for encoding the spanning trees of any complete multipartite graph K(n₁,…,n_r). We also give a bijection between the spanning trees of a planar graph and those of any of its planar duals. Finally we discuss the possibility of bijections for spanning trees of DeBriujn graphs, cubes, and regular graphs such as the Petersen graph that have integer eigenvalues.     

Eliminating graphs by means of parallel knock-out schemes

In 1997 Lampert and Slater introduced parallel knock-out schemes, an iterative process on graphs that goes through several rounds. In each round of this process, every vertex eliminates exactly one of its neighbors. The parallel knock-out number of a graph is the minimum number of rounds after which all vertices have been eliminated (if possible). The parallel knock-out number is related to well-known concepts like perfect matchings, hamiltonian cycles, and 2-factors.We derive a number of combinatorial and algorithmic results on parallel knock-out numbers: for families of sparse graphs (like planar graphs or graphs of bounded tree-width), the parallel knock-out number grows at most logarithmically with the number n of vertices; this bound is basically tight for trees. Furthermore, there is a family of bipartite graphs for which the parallel knock-out number grows proportionally to the square root of n. We characterize trees with parallel knock-out number at most 2, and we show that the parallel knock-out number for trees can be computed in polynomial time via a dynamic programming approach (whereas in general graphs this problem is known to be NP-hard). Finally, we prove that the parallel knock-out number of a claw-free graph is either infinite or less than or equal to 2.     

Paired domination on interval and circular-arc graphs

We study the paired-domination problem on interval graphs and circular-arc graphs. Given an interval model with endpoints sorted, we give an O(m+n) time algorithm to solve the paired-domination problem on interval graphs. The result is extended to solve the paired-domination problem on circular-arc graphs in O(m(m+n)) time.     

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.     

On the maximum cardinality search lower bound for treewidth

The Maximum Cardinality Search (MCS) algorithm visits the vertices of a graph in some order, such that at each step, an unvisited vertex that has the largest number of visited neighbours becomes visited. A maximum cardinality search ordering (MCS-ordering) of a graph is an ordering of the vertices that can be generated by the MCS algorithm. The visited degree of a vertex v in an MCS-ordering is the number of neighbours of v that are before v in the ordering. The visited degree of an MCS-ordering ψ of G is the maximum visited degree over all vertices v in ψ. The maximum visited degree over all MCS-orderings of graph G is called its maximum visited degree. Lucena [A new lower bound for tree-width using maximum cardinality search, SIAM J. Discrete Math. 16 (2003) 345-353] showed that the treewidth of a graph G is at least its maximum visited degree.We show that the maximum visited degree is of size O(logn) for planar graphs, and give examples of planar graphs G with maximum visited degree k with O(k!) vertices, for all k∈N. Given a graph G, it is NP-complete to determine if its maximum visited degree is at least k, for any fixed k?7. Also, this problem does not have a polynomial time approximation algorithm with constant ratio, unless P=NP. Variants of the problem are also shown to be NP-complete.In this paper, we also propose some heuristics for the problem, and report on an experimental analysis of them. Several tiebreakers for the MCS algorithm are proposed and evaluated. We also give heuristics that give upper bounds on the value of the maximum visited degree of a graph, which appear to give results close to optimal on many graphs from real life applications.     

Topological sweep of the complete graph

We present a novel, simple and easily implementable algorithm to report all intersections in an embedding of a complete graph. For graphs with N vertices and complexity K measured as the number of segments of the embedding, the running time of the algorithm is Θ(K+NM), where M is the maximum number of edges cut by any vertical line. Our algorithm handles degeneracies, such as vertical edges or multiply intersecting edges, without requiring numerical perturbations to achieve general position.The algorithm is based on the sweep line technique, one of the most fundamental techniques in computational geometry, where an imaginary line passes through a given set of geometric objects, usually from left to right. The algorithm sweeps the graph using a topological line, borrowing the concept of horizon trees from the topological sweep method [H. Edelsbrunner, L.J. Guibas, Topologically sweeping an arrangement, J. Comput. Syst. Sci. 38 (1989) 165-194; J. Comput. Syst. Sci. 42 (1991) 249-251 (corrigendum)].The novelty in our approach is to control the topological line through the use of the moving wall that separates at any time the graph into two regions: the region of known structure, in front of the moving wall, and the region that may contain intersections generated by edges-that have not yet been registered in the sweep process-behind the wall.Our method has applications to graph drawing and for depth-based statistical analysis, for computing the simplicial depth median for a set of N data points [G. Aloupis, S. Langerman, M. Soss, G. Toussaint, Algorithms for bivariate medians and a Fermat-Torricelli problem for lines, Comp. Geom. Theory Appl. 26 (1) (2003) 69-79].We present the algorithm, its analysis, experimental results and extension of the method to general graphs.     

Feedback numbers of Kautz digraphs

A subset of vertices (resp. arcs) of a graph G is called a feedback vertex (resp. arc) set of G if its removal results in an acyclic subgraph. Let f(d,n) (f_a(d,n)) denote the minimum cardinality over all feedback vertex (resp. arc) sets of the Kautz digraph K(d,n). This paper proves that for any integers d?2 and n?1

     

Dense sets and embedding binary trees into hypercubes

The purpose of this paper is to describe a method for embedding binary trees into hypercubes based on an iterative embedding into their subgraphs induced by dense sets. As a particular application, we present a class of binary trees, defined in terms of size of their subtrees, whose members allow a dilation two embedding into their optimal hypercubes. This provides a partial evidence in favor of a long-standing conjecture of Bhatt and Ipsen which claims that such an embedding exists for an arbitrary binary tree.     

On the average hyperoscillations of planted plane trees

Assume that the leaves of a planted plane tree are enumerated from left to right by 1, 2, .... Thej-ths-turn of the tree is defined to be the root of the (unique) subtree of minimal height with leavesj, j+1, ...,j+s−1. If all trees withn nodes are regarded equally likely, the average level number of thej-ths-turn tends to a finite limitα _s (j), which is of orderj ^1/2. Thej-th ”s-hyperoscillation”α ₁(j)−α _s+1(j) is given by 1/2α ₁(s)+O(j ^−1/2) and therefore tends (forj → ∞) to a constant behaving like √8/π·s ^1/2 fors → ∞. These results are obtained by setting up appropriate generating functions, which are expanded about their (algebraic) singularities nearest to the origin, so that the asymptotic formulas are consequences of the so-called Darboux-Pólyamethod.     

Constructing disjoint paths on expander graphs

In a typical parallel or distributed computation model processors are connected by a spars interconnection network. To establish open-line communication between pairs of processors that wish to communicate interactively, a set of disjoint paths has to be constructed on the network. Since communication needs vary in time, paths have to be dynamically constructed and destroyed.We study the complexity of constructing disjoint paths between given pairs of vertices on expander interconnection graphs. These graphs have been shown before to possess desirable properties for other communication tasks.We present a sufficient condition for the existence ofKn ^Q edge-disjoint paths connecting any set ofK pairs of vertices on an expander graph, wheren is the number of vertices and<1 is some constant. We then show that the computational problem of constructing these paths lies in the classes Deterministic-P and Random-P C.Furthermore, we show that the set of paths can be constructed in probabilistic polylog time in the parallel-distributed model of computation, in which then participating processors reside in the nodes of the communication graph and all communication is done through edges of the graph. Thus, the disjoint paths are constructed in the very computation model that uses them.Finally, we show how to apply variants of our parallel algorithms to find sets ofvertex-disjoint paths when certain conditions are satisfied.Supported in part by a Weizmann fellowship and by contract ONR N00014-85-C-0731.     

Some graphs with small second eigenvalue

For any prime,p, we construct a Cayley graph on the group,G, of affine linear transformations ofℤ/pℤ of degree 2(p−1) and second eigenvalue with the following special property: the adjacency matrix of the graph is supported on the “blocks” associated to the trivial representation and the irreducible representation of sizep−1. SinceG is of orderp(p−1), the correspondingt-uniform Cayley hypergraph has essentially optimal second eigenvalue for this degree and size of the graph (see [2] for definitions). En route we give, for any integerk>1, a simple Cayley graph onp ^k nodes of degreep of second eigenvalue . The author wishes to acknowledge the National Science Foundation for supporting this research in part under Grant CCR-8858788, and the Office of Naval Research under Grant N00014-87-K-0467.     

Linearity of grid minors in treewidth with applications through bidimensionality

We prove that any H-minor-free graph, for a fixed graph H, of treewidth w has an Ω(w) × Ω(w) grid graph as a minor. Thus grid minors suffice to certify that H-minorfree graphs have large treewidth, up to constant factors. This strong relationship was previously known for the special cases of planar graphs and bounded-genus graphs, and is known not to hold for general graphs. The approach of this paper can be viewed more generally as a framework for extending combinatorial results on planar graphs to hold on H-minor-free graphs for any fixed H. Our result has many combinatorial consequences on bidimensionality theory, parameter-treewidth bounds, separator theorems, and bounded local treewidth; each of these combinatorial results has several algorithmic consequences including subexponential fixed-parameter algorithms and approximation algorithms. A preliminary version of this paper appeared in the ACM-SIAM Symposium on Discrete Algorithms (SODA 2005) [16].     

Maximum andk-th maximal spanning trees of a weighted graph

LetA be a maximum spanning tree andP be an arbitrary spanning tree of a connected weighted graphG. Then we prove that there exists a bijectionψ fromA/P intoP/A such that for any edgea∈A/P, (P/ψ(a)) ∪a is a spanning tree ofG whose weight is greater than or equal to that ofP. We apply this theorem to some problems concerning spanning trees of a weighted graph.     

赋权图的w密度和w平衡性

The notion of w-density for the graphs with positive weights on vertices and nonnegative weights on edges is introduced. A weighted graph is called w-balanced if its w-density is no less than the w-density of any subgraph of it. In this paper,a good characterization of w-balanced weighted graphs is given. Applying this characterization ,many large w-balanced weighted graphs are formed by combining smaller ones. In the case where a graph is not w-balanced,a polynomial-time algorithm to find a subgraph of maximum w-density is proposed. It is shown that the w-density theory is closely related to the study of SEW(G,w) games.     

On the complexity of the disjoint paths problem

In this paper we consider the disjoint paths problem. Given a graphG and a subsetS of the edge-set ofG the problem is to decide whether there exists a family of disjoint circuits inG each containing exactly one edge ofS such that every edge inS belongs to a circuit inC. By a well-known theorem of P. Seymour the edge-disjoint paths problem is polynomially solvable for Eulerian planar graphsG. We show that (assumingPNP) one can drop neither planarity nor the Eulerian condition onG without losing polynomial time solvability. We prove theNP-completeness of the planar edge-disjoint paths problem by showing theNP-completeness of the vertex disjoint paths problem for planar graphs with maximum vertex-degree three. This disproves (assumingPNP) a conjecture of A. Schrijver concerning the existence of a polynomial time algorithm for the planar vertex-disjoint paths problem. Furthermore we present a counterexample to a conjecture of A. Frank. This conjecture would have implied a polynomial algorithm for the planar edge-disjoint paths problem. Moreover we derive a complete characterization of all minorclosed classes of graphs for which the disjoint paths problem is polynomially solvable. Finally we show theNP-completeness of the half-integral relaxation of the edge-disjoint paths problem. This implies an answer to the long-standing question whether the edge-disjoint paths problem is polynomially solvable for Eulerian graphs.Supported by Sonderforschungsbereich 303 (DFG)     

Edge disjoint spanning trees in random graphs

We show that almost everyG _m-out containsm edge disjoint spanning trees.     

On the power of white pebbles

We construct a family (G _p |p) of directed acyclic graphs such that any black pebble strategy forG _p requiresp ² pebbles whereas 3p+1 pebbles are sufficient when white pebbles are allowed.Supported by the National Science Foundation under contract no. DCR-8407256 and by the office of Naval Research under contract no. N0014-80-0517.