We present a general approach for analyzing structural parameters of a relational event graph within arbitrary query time intervals using colored range query data structures. Relational event graphs generally represent social network datasets, where each graph edge carries a timestamp. We provide data structures based on colored range searching to efficiently compute several graph parameters (e...

Variable Neighborhood Search (VNS) is a recent metaheuristic, or framework for building heuristics, which exploits systematically the idea of neighborhood change, both in the descent to local minima and in the escape from the valleys which contain them. In this tutorial we first present the ingredients of VNS, i.e., Variable Neighborhood Descent (VND) and Reduced VNS (RVNS) followed by the basi...

Let G = (V,E) be a simple undirected graph. The neighborhood hypergraph N (G) = (V, EN ) of G has the edge set EN = {e ⊆ V | |e| ≥ 1 ∧ ∃x ∈ V : e = NG(x)}. In a certain sense, this is a generalization of the well-known notion of the neighborhood graph N(G) = (V,EN ). For several products G1 ◦G2 of simple undirected graphs G1 and G2, we investigate the question how N(G1 ◦G2) / N (G1 ◦G2) can be ...

We present a simple method for meshing a 3D point cloud to a manifold genus-0 mesh. Our approach is based on recent methods for spherical embedding of planar graphs, where we use instead a k-nearest neighborhood graph of the point cloud. Our approach proceeds in two steps: We first embed the neighborhood graph on a sphere using an iterative procedure, minimizing the tangential Laplacian. Then w...

Let G be a simple undirected graph of order n. For an independent set S C V(G) of k vertices, we define the k neighborhood intersections S, = { u E V(G)\SIIN(u) n Sl = 11, 1 5 ir k, with s; = ISJ. Using the concept of insertible vertices and the concept of neighborhood intersections, we prove the following theorem. Theorem. Let G be a graph of order n and connectivity K 2 2. Then G is hamiltoni...

The topic of this thesis is the complexity of some hard problems on graphs when parametrized by neighborhood diversity, a recently introduced graph parameter capturing some structural graph properties, and possibly by other related parameters. The student will survey relevant interesting problems and results (e.g., problems that are hard when parametrized by treewidth or which fall into a cathe...

An odd neighborhood transversal of a graph is a set of its vertices that intersects the set of neighbors of each of its vertices in an odd number of elements. In the case of grid graphs this odd number will be either one or three. We characterize those grid graphs that have odd neighborhood transversals.

A graph G is chromatically k–connected if every vertex cutset induces a subgraph with chromatic number at least k. Thus, in particular each neighborhood has to induce a k–chromatic subgraph. In [3], Godsil, Nowakowski and Nešetřil asked whether there exists a k–chromatically connected graph such that every minimal cutset induces a subgraph with no triangles. We show that the answer is positive ...

This paper presents a very large scale neighborhood (VLSN) search algorithm for the capacitated facility location problem with single-source constraints. The neighborhood structures are induced by customer multi-exchanges and by facility moves. We consider both single-customer multi-exchanges, detected on a suitably de ned customer improvement graph, and multi-customer multi-exchanges, detected...

Chemical graphs, as other ones, are regular if all their vertices have the same degree. Otherwise, they are irregular, and it is of interest to measure their irregularity both for descriptive purposes and for QSAR/QSPR studies. Three indices have been proposed in the literature for that purpose: those of Collatz-Sinogowitz, of Albertson, and of Bell's variance of degrees. We study their propert...