A min-cut approach to functional regionalization, with a case study of the Italian local labour market areas

In several economical, statistical and geographical applications, a territory must be subdivided into functional regions. Such regions are not fixed and politically delimited, but should be identified by analyzing the interactions among all its constituent localities. This is a very delicate and important task, that often turns out to be computationally difficult. In this work we propose an innovative approach to this problem based on the solution of minimum cut problems over an undirected graph called here transitions graph. The proposed procedure guarantees that the obtained regions satisfy all the statistical conditions required when considering this type of problems. Results on real-world instances show the effectiveness of the proposed approach

Modelling and Analysis Using GROOVE

In this paper we present case studies that describe how the graph transformation tool GROOVE has been used to model problems from a wide variety of domains. These case studies highlight the wide applicability of GROOVE in particular, and of graph transformation in general. They also give concrete templates for using GROOVE in practice. Furthermore, we use the case studies to analyse the main strong and weak points of GROOVE

k-Trails: Recognition, Complexity, and Approximations

The notion of degree-constrained spanning hierarchies, also called k-trails, was recently introduced in the context of network routing problems. They describe graphs that are homomorphic images of connected graphs of degree at most k. First results highlight several interesting advantages of k-trails compared to previous routing approaches. However, so far, only little is known regarding computational aspects of k-trails. In this work we aim to fill this gap by presenting how k-trails can be analyzed using techniques from algorithmic matroid theory. Exploiting this connection, we resolve several open questions about k-trails. In particular, we show that one can recognize efficiently whether a graph is a k-trail. Furthermore, we show that deciding whether a graph contains a k-trail is NP-complete; however, every graph that contains a k-trail is a (k+1)-trail. Moreover, further leveraging the connection to matroids, we consider the problem of finding a minimum weight k-trail contained in a graph G. We show that one can efficiently find a (2k-1)-trail contained in G whose weight is no more than the cheapest k-trail contained in G, even when allowing negative weights. The above results settle several open questions raised by Molnar, Newman, and Sebo

Induced minors and well-quasi-ordering

A graph $H$ is an induced minor of a graph $G$ if it can be obtained from an induced subgraph of $G$ by contracting edges. Otherwise, $G$ is said to be $H$-induced minor-free. Robin Thomas showed that $K_4$-induced minor-free graphs are well-quasi-ordered by induced minors [Graphs without $K_4$ and well-quasi-ordering, Journal of Combinatorial Theory, Series B, 38(3):240 -- 247, 1985]. We provide a dichotomy theorem for $H$-induced minor-free graphs and show that the class of $H$-induced minor-free graphs is well-quasi-ordered by the induced minor relation if and only if $H$ is an induced minor of the gem (the path on 4 vertices plus a dominating vertex) or of the graph obtained by adding a vertex of degree 2 to the complete graph on 4 vertices. To this end we proved two decomposition theorems which are of independent interest. Similar dichotomy results were previously given for subgraphs by Guoli Ding in [Subgraphs and well-quasi-ordering, Journal of Graph Theory, 16(5):489--502, 1992] and for induced subgraphs by Peter Damaschke in [Induced subgraphs and well-quasi-ordering, Journal of Graph Theory, 14(4):427--435, 1990]

Pseudograph associahedra

Given a simple graph G, the graph associahedron KG is a simple polytope whose face poset is based on the connected subgraphs of G. This paper defines and constructs graph associahedra in a general context, for pseudographs with loops and multiple edges, which are also allowed to be disconnected. We then consider deformations of pseudograph associahedra as their underlying graphs are altered by edge contractions and edge deletions.Comment: 25 pages, 22 figure