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Confirming a conjecture of Ne\v{s}et\v{r}il, we show that up to isomorphism there is only a finite number of finite minimal asymmetric undirected graphs. In fact, there are exactly 18 such graphs. We also show that these graphs are exactly the finite minimal involution-free graphs., Comment: 11 pages, 3 figures

The ADTool is free, open source software assisting graphical modeling and quantitative analysis of security, using attack-defense trees. The main features of the ADTool are easy creation, efficient editing, and automated bottom-up evaluation of security-relevant measures. The tool also supports the usage of attack trees, protection trees and defense trees, which are all particular instances of attack-defense trees., Comment: This is an extended version of the tool demonstration paper accepted for publication at the 10th International Conference on Quantitative Evaluation of SysTems (QEST 2013)

This paper presents the current state of the art on attack and defense modeling approaches that are based on directed acyclic graphs (DAGs). DAGs allow for a hierarchical decomposition of complex scenarios into simple, easily understandable and quantifiable actions. Methods based on threat trees and Bayesian networks are two well-known approaches to security modeling. However there exist more than 30 DAG-based methodologies, each having different features and goals. The objective of this survey is to present a complete overview of graphical attack and defense modeling techniques based on DAGs. This consists of summarizing the existing methodologies, comparing their features and proposing a taxonomy of the described formalisms. This article also supports the selection of an adequate modeling technique depending on user requirements.

Attack-defense trees are a novel methodology for graphical security modeling and assessment. The methodology includes visual, intuitive tree models whose analysis is supported by a rigorous mathematical formalism. Both, the intuitive and the formal components of the approach can be used for quantitative analysis of attack-defense scenarios. In practice, we use intuitive questions to ask about aspects of scenarios we are interested in. Formally, a computational procedure, defined with the help of attribute domains and a bottom-up algorithm, is applied to derive the corresponding numerical values. This paper bridges the gap between the intuitive and the formal way of quantitatively assessing attack-defense scenarios. We discuss how to properly specify a question, so that it can be answered unambiguously. Given a well specified question, we then show how to derive an appropriate attribute domain which constitutes the corresponding formal model. Since any attack tree is in particular an attack-defense tree, our analysis is also an advancement of the attack tree methodology., Comment: technical report including formal pruning and additional figures

The competition graph of a directed acyclic graph D is the undirected graph on the same vertex set as D in which two distinct vertices are adjacent if they have a common out-neighbor in D. The competition number of an undirected graph G is the least number of isolated vertices that have to be added to G to make it the competition graph of a directed acyclic graph. We resolve two conjectures concerning competition graphs. First we prove a conjecture of Opsut by showing that the competition number of every quasi-line graph is at most 2. Recall that a quasi-line graph, also called a locally co-bipartite graph, is a graph for which the neighborhood of every vertex can partitioned into at most two cliques. To prove this conjecture we devise an alternative characterization of quasi-line graphs to the one by Chudnovsky and Seymour. Second, we prove a conjecture of Kim by showing that the competition number of any graph is at most one greater than the number of holes in the graph. Our methods also allow us to prove a strengthened form of this conjecture recently proposed by Kim, Lee, Park and Sano, showing that the competition number of any graph is at most one greater than the dimension of the subspace of the cycle space spanned by the holes., Comment: 16 pages, 2 figures. v3 contains minor typographic changes. As accepted for SIAM J Disc Math

Attack--defense trees are used to describe security weaknesses of a system and possible countermeasures. In this paper, the connection between attack--defense trees and game theory is made explicit. We show that attack--defense trees and binary zero-sum two-player extensive form games have equivalent expressive power when considering satisfiability, in the sense that they can be converted into each other while preserving their outcome and their internal structure., Comment: Added link to springerlink; Proceedings of GameSec 2010