Heavy paths and cycles in weighted graphs

A weighted graph is a graph in which each edge e is assigned a non-negative\ud number $w(e)$, called the weight of $e$. In this paper, some theorems on the\ud existence of long paths and cycles in unweighted graphs are generalized to heavy\ud paths and cycles in weighted graphs

Weighted degrees and heavy cycles in weighted graphs

AbstractA weighted graph is a graph provided with an edge-weighting function w from the edge set to nonnegative real numbers. Bondy and Fan [Annals of Discrete Math. 41 (1989), 53–69] began the study on the existence of heavy cycles in weighted graphs. Though several results with Dirac-type degree condition can be generalized to an Ore-type one in unweighted graphs, it is shown in Bondy et al. [Discuss. Math. Graph Theory 22 (2002), 7–15] that Bondy and Fan’s theorem, which uses Dirac-type condition, cannot be generalized analogously by using Ore-type condition.In this paper we investigate the property peculiar to weighted graphs, and prove a theorem on the existence of heavy cycles in weighted graphs under an Ore-type condition, which generalizes Bondy and Fan’s theorem. Moreover, we show the existence of heavy cycles passing through some specified vertices

Perfect Elimination Orderings for Symmetric Matrices

We introduce a new class of structured symmetric matrices by extending the notion of perfect elimination ordering from graphs to weighted graphs or matrices. This offers a common framework capturing common vertex elimination orderings of monotone families of chordal graphs, Robinsonian matrices and ultrametrics. We give a structural characterization for matrices that admit perfect elimination orderings in terms of forbidden substructures generalizing chordless cycles in graphs.Comment: 16 pages, 3 figure

Clustering in Complex Directed Networks

Many empirical networks display an inherent tendency to cluster, i.e. to form circles of connected nodes. This feature is typically measured by the clustering coefficient (CC). The CC, originally introduced for binary, undirected graphs, has been recently generalized to weighted, undirected networks. Here we extend the CC to the case of (binary and weighted) directed networks and we compute its expected value for random graphs. We distinguish between CCs that count all directed triangles in the graph (independently of the direction of their edges) and CCs that only consider particular types of directed triangles (e.g., cycles). The main concepts are illustrated by employing empirical data on world-trade flows

Cycles in weighted graphs and related topics

This thesis contains results on paths andcycles in graphs andon a more\ud or less relatedtopic, the vulnerability of graphs. In the first part of\ud the thesis, Chapters 2 through 5, we concentrate on paths andcycles\ud in weightedgraphs. A number of sufficient conditions are presentedfor\ud graphs to contain paths andcycles with certain properties, in particular\ud heavy (or long) paths andcycles. The secondpart of the thesis, Chapters\ud 6 and7, contains some basic results on paths andcycles in (edge-) colored\ud graphs. In the thirdpart of the thesis, Chapters 8 and9, we focus on\ud a graph parameter which can be usedto analyze the vulnerability of\ud graphs, i.e., the scattering number of graphs

Hitting Weighted Even Cycles in Planar Graphs

A classical branch of graph algorithms is graph transversals, where one seeks a minimum-weight subset of nodes in a node-weighted graph G which intersects all copies of subgraphs F from a fixed family F. Many such graph transversal problems have been shown to admit polynomial-time approximation schemes (PTAS) for planar input graphs G, using a variety of techniques like the shifting technique (Baker, J. ACM 1994), bidimensionality (Fomin et al., SODA 2011), or connectivity domination (Cohen-Addad et al., STOC 2016). These techniques do not seem to apply to graph transversals with parity constraints, which have recently received significant attention, but for which no PTASs are known. In the even-cycle transversal (ECT) problem, the goal is to find a minimum-weight hitting set for the set of even cycles in an undirected graph. For ECT, Fiorini et al. (IPCO 2010) showed that the integrality gap of the standard covering LP relaxation is ?(log n), and that adding sparsity inequalities reduces the integrality gap to 10. Our main result is a primal-dual algorithm that yields a 47/7 ? 6.71-approximation for ECT on node-weighted planar graphs, and an integrality gap of the same value for the standard LP relaxation on node-weighted planar graphs

Symbolic powers of monomial ideals and Cohen-Macaulay vertex-weighted digraphs

In this paper we study irreducible representations and symbolic Rees algebras of monomial ideals. Then we examine edge ideals associated to vertex-weighted oriented graphs. These are digraphs having no oriented cycles of length two with weights on the vertices. For a monomial ideal with no embedded primes we classify the normality of its symbolic Rees algebra in terms of its primary components. If the primary components of a monomial ideal are normal, we present a simple procedure to compute its symbolic Rees algebra using Hilbert bases, and give necessary and sufficient conditions for the equality between its ordinary and symbolic powers. We give an effective characterization of the Cohen--Macaulay vertex-weighted oriented forests. For edge ideals of transitive weighted oriented graphs we show that Alexander duality holds. It is shown that edge ideals of weighted acyclic tournaments are Cohen--Macaulay and satisfy Alexander dualityComment: Special volume dedicated to Professor Antonio Campillo, Springer, to appea