Defensive alliances in graphs: a survey

A set $S$ of vertices of a graph $G$ is a defensive $k$-alliance in $G$ if every vertex of $S$ has at least $k$ more neighbors inside of $S$ than outside. This is primarily an expository article surveying the principal known results on defensive alliances in graph. Its seven sections are: Introduction, Computational complexity and realizability, Defensive $k$-alliance number, Boundary defensive $k$-alliances, Defensive alliances in Cartesian product graphs, Partitioning a graph into defensive $k$-alliances, and Defensive $k$-alliance free sets.Comment: 25 page

On defensive alliances and line graphs

Let $\Gamma$ be a simple graph of size $m$ and degree sequence $\delta_1\ge \delta_2\ge ... \ge \delta_n$. Let ${\cal L}(\Gamma)$ denotes the line graph of $\Gamma$. The aim of this paper is to study mathematical properties of the alliance number, ${a}({\cal L}(\Gamma)$, and the global alliance number, $\gamma_{a}({\cal L}(\Gamma))$, of the line graph of a simple graph. We show that $\lceil\frac{\delta_{n}+\delta_{n-1}-1}{2}\rceil \le {a}({\cal L}(\Gamma))\le \delta_1.$ In particular, if $\Gamma$ is a $\delta$-regular graph ($\delta>0$), then $a({\cal L}(\Gamma))=\delta$, and if $\Gamma$ is a $(\delta_1,\delta_2)$-semiregular bipartite graph, then $a({\cal L}(\Gamma))=\lceil \frac{\delta_1+\delta_2-1}{2} \rceil$. As a consequence of the study we compare $a({\cal L}(\Gamma))$ and ${a}(\Gamma)$, and we characterize the graphs having $a({\cal L}(\Gamma))<4$. Moreover, we show that the global-connected alliance number of ${\cal L}(\Gamma)$ is bounded by $\gamma_{ca}({\cal L}(\Gamma)) \ge \lceil\sqrt{D(\Gamma)+m-1}-1\rceil,$ where $D(\Gamma)$ denotes the diameter of $\Gamma$, and we show that the global alliance number of ${\cal L}(\Gamma)$ is bounded by $\gamma_{a}({\cal L}(\Gamma))\geq \lceil\frac{2m}{\delta_{1}+\delta_{2}+1}\rceil$. The case of strong alliances is studied by analogy

The Community Structure of R&D Cooperation in Europe. Evidence from a social network perspective

The focus of this paper is on pre-competitive R&D cooperation across Europe, as captured by R&D joint ventures funded by the European Commission in the time period 1998-2002, within the 5th Framework Program. The cooperations in this Framework Program give rise to a bipartite network with 72,745 network edges between 25,839 actors (representing organizations that include firms, universities, research organizations and public agencies) and 9,490 R&D projects. With this construction, participating actors are linked only through joint projects. In this paper we describe the community identification problem based on the concept of modularity, and use the recently introduced label-propagation algorithm to identify communities in the network, and differentiate the identified communities by developing community-specific profiles using social network analysis and geographic visualization techniques. We expect the results to enrich our picture of the European Research Area by providing new insights into the global and local structures of R&D cooperation across Europe

Relevance of Negative Links in Graph Partitioning: A Case Study Using Votes From the European Parliament

In this paper, we want to study the informative value of negative links in signed complex networks. For this purpose, we extract and analyze a collection of signed networks representing voting sessions of the European Parliament (EP). We first process some data collected by the VoteWatch Europe Website for the whole 7 th term (2009-2014), by considering voting similarities between Members of the EP to define weighted signed links. We then apply a selection of community detection algorithms, designed to process only positive links, to these data. We also apply Parallel Iterative Local Search (Parallel ILS), an algorithm recently proposed to identify balanced partitions in signed networks. Our results show that, contrary to the conclusions of a previous study focusing on other data, the partitions detected by ignoring or considering the negative links are indeed remarkably different for these networks. The relevance of negative links for graph partitioning therefore is an open question which should be further explored.Comment: in 2nd European Network Intelligence Conference (ENIC), Sep 2015, Karlskrona, Swede