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.     

Note on a conjecture of toft

A conjecture of Toft [17] asserts that any 4-critical graph (or equivalently, every 4-chromatic graph) contains a fully odd subdivision ofK ₄. We show that if a graphG has a degree three nodev such thatG-v is 3-colourable, then eitherG is 3-colourable or it contains a fully oddK ₄. This resolves Toft's conjecture in the special case where a 4-critical graph has a degree three node, which is in turn used to prove the conjecture for line-graphs. The proof is constructive and yields a polynomial algorithm which given a 3-degenerate graph either finds a 3-colouring or exhibits a subgraph that is a fully odd subdivision ofK ₄. (A graph is 3-degenerate if every subgraph has some node of degree at most three.)     

Second order evolution equations with time-dependent subdifferentials

We study an abstract second order nonlinear evolution equation in a real Hilbert space. We consider time-dependent convex functions and their subdifferentials operating on the first derivative of the unknown function. Introducing appropriate assumptions on the convex functions and other data, we prove the existence and uniqueness of a strong solution, and give some applications of the abstract theorem to hyperbolic variational inequalities with time-dependent constraints.      

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.     

Approximating the single source unsplittable min-cost flow problem

In the single source unsplittable min-cost flow problem, commodities must be routed simultaneously from a common source vertex to certain destination vertices in a given graph with edge capacities and costs; the demand of each commodity must be routed along a single path so that the total flow through any edge is at most its capacity. Moreover, the total cost must not exceed a given budget. This problem has been introduced by Kleinberg [7] and generalizes several NP-complete problems from various areas in combinatorial optimization such as packing, partitioning, scheduling, load balancing, and virtual-circuit routing. Kolliopoulos and Stein [9] and Dinitz, Garg, and Goemans [4] developed algorithms improving the first approximation results of Kleinberg for the problem of minimizing the violation of edge capacities and for other variants. However, known techniques do not seem to be capable of providing solutions without also violating the cost constraint. We give the first approximation results with hard cost constraints. Moreover, all our results dominate the best known bicriteria approximations. Finally, we provide results on the hardness of approximation for several variants of the problem. Received: August 23, 2000 / Accepted: April 20, 2001?Published online October 2, 2001     

Extremal positive and self-adjoint extensions of suboperators

This work is supported in part by the fund OTKA (No. 5-134).     

A proof of Parrott's theorem on quotient norms

We prove, as an application of our positive extension argument, a theorem of Parrott concerning the quotient norm with respect to spaces of Hilbert space operators.     

The dual of a finitely generated multiplication module

This paper is based on the M. Sc. thesis written by the third author under the supervision of the first two authors. It was submitted to the University of Baghdad in 1986.     

Nuclear Effects in Structure Functions xFa（x, Q^2） from Charge Current Neutrino Deep Inelastic Scattering

By taking advantage of the model-independent nuclear parton distributions, the structure functions xF3（x, Q^2） are calculated, in comparison with the experimental data from CCFR neutrino-nuclei charge current deep inelastic scattering. It is shown that shadowing and anti-shadowing effects occur in valence quark distributions for small and medium x regions, respectively. It is suggested that the neutrino experimental data should be employed in the future for pinning down the nuclear patton distributions.     

On the 2-rainbow domination in graphs

The concept of 2-rainbow domination of a graph G coincides with the ordinary domination of the prism G□K₂. In this paper, we show that the problem of deciding if a graph has a 2-rainbow dominating function of a given weight is NP-complete even when restricted to bipartite graphs or chordal graphs. Exact values of 2-rainbow domination numbers of several classes of graphs are found, and it is shown that for the generalized Petersen graphs GP(n,k) this number is between ⌈4n/5⌉ and n with both bounds being sharp.