On the size of identifying codes in triangle-free graphs

In an undirected graph $G$, a subset $C\subseteq V(G)$ such that $C$ is a dominating set of $G$, and each vertex in $V(G)$ is dominated by a distinct subset of vertices from $C$, is called an identifying code of $G$. The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. For a given identifiable graph $G$, let \M(G) be the minimum cardinality of an identifying code in $G$. In this paper, we show that for any connected identifiable triangle-free graph $G$ on $n$ vertices having maximum degree $\Delta\geq 3$, \M(G)\le n-\tfrac{n}{\Delta+o(\Delta)}. This bound is asymptotically tight up to constants due to various classes of graphs including $(\Delta-1)$-ary trees, which are known to have their minimum identifying code of size $n-\tfrac{n}{\Delta-1+o(1)}$. We also provide improved bounds for restricted subfamilies of triangle-free graphs, and conjecture that there exists some constant $c$ such that the bound \M(G)\le n-\tfrac{n}{\Delta}+c holds for any nontrivial connected identifiable graph $G$

Subgraph covers -- An information theoretic approach to motif analysis in networks

Many real world networks contain a statistically surprising number of certain subgraphs, called network motifs. In the prevalent approach to motif analysis, network motifs are detected by comparing subgraph frequencies in the original network with a statistical null model. In this paper we propose an alternative approach to motif analysis where network motifs are defined to be connectivity patterns that occur in a subgraph cover that represents the network using minimal total information. A subgraph cover is defined to be a set of subgraphs such that every edge of the graph is contained in at least one of the subgraphs in the cover. Some recently introduced random graph models that can incorporate significant densities of motifs have natural formulations in terms of subgraph covers and the presented approach can be used to match networks with such models. To prove the practical value of our approach we also present a heuristic for the resulting NP-hard optimization problem and give results for several real world networks.Comment: 10 pages, 7 tables, 1 Figur

Animating the evolution of software

The use and development of open source software has increased significantly in the last decade. The high frequency of changes and releases across a distributed environment requires good project management tools in order to control the process adequately. However, even with these tools in place, the nature of the development and the fact that developers will often work on many other projects simultaneously, means that the developers are unlikely to have a clear picture of the current state of the project at any time. Furthermore, the poor documentation associated with many projects has a detrimental effect when encouraging new developers to contribute to the software. A typical version control repository contains a mine of information that is not always obvious and not easy to comprehend in its raw form. However, presenting this historical data in a suitable format by using software visualisation techniques allows the evolution of the software over a number of releases to be shown. This allows the changes that have been made to the software to be identified clearly, thus ensuring that the effect of those changes will also be emphasised. This then enables both managers and developers to gain a more detailed view of the current state of the project. The visualisation of evolving software introduces a number of new issues. This thesis investigates some of these issues in detail, and recommends a number of solutions in order to alleviate the problems that may otherwise arise. The solutions are then demonstrated in the definition of two new visualisations. These use historical data contained within version control repositories to show the evolution of the software at a number of levels of granularity. Additionally, animation is used as an integral part of both visualisations - not only to show the evolution by representing the progression of time, but also to highlight the changes that have occurred. Previously, the use of animation within software visualisation has been primarily restricted to small-scale, hand generated visualisations. However, this thesis shows the viability of using animation within software visualisation with automated visualisations on a large scale. In addition, evaluation of the visualisations has shown that they are suitable for showing the changes that have occurred in the software over a period of time, and subsequently how the software has evolved. These visualisations are therefore suitable for use by developers and managers involved with open source software. In addition, they also provide a basis for future research in evolutionary visualisations, software evolution and open source development