In this work, for a given oriented graph D, we study its interval and hull numbers, denoted by ⃗in (D) and ⃗hn (D), respectively, in the oriented geodetic, ⃗P3 and ⃗P*3 convexities. This last one, we believe to be formally defined and first studied in this paper, although its undirected version is well-known in the literature. Concerning bounds, for a strongly oriented graph D and the oriented geodetic convexity, we prove that ⃗hn g(D) ≤ m(D)-n(D) + 2 and that there is at least one such that ⃗hn g(D) = m(D) - n(D). We also determine exact values for the hull numbers in these three convexities for tournaments, which imply polynomial-time algorithms to compute them. These results allow us to deduce polynomial-time algorithms to compute ⃗hn P3 (D) when the underlying graph of D is split or cobipartite. Moreover, we provide a meta-theorem by proving that if deciding whether ⃗ing(D) ≤ k or ⃗hn g(D) ≤ k is NP-hard or W[i]-hard parameterized by k, for some i ϵ Z*+, then the same holds even if the underlying graph of D is bipartite. Next, we prove that deciding whether ⃗hn P3 (D) ≤ k or ⃗hn P3* (D) ≤ k is W[2]-hard parameterized by k, even if the underlying graph of D is bipartite; that deciding whether ⃗in P3(D) ≤ k or whether ⃗in P3*(D) ≤ k is NP-complete, and the same for ⃗hn P3*(D) ≤ k even if D has no directed cycles and the underlying graph of D is a chordal bipartite graph; and that deciding whether ⃗in P3(D) ≤ k or whether ⃗in P3*(D) ≤ k is W[2]-hard parameterized by k, even if the underlying graph of D is split. Finally, we also argue that the interval and hull numbers in the ⃗P3 and ⃗P*3 convexities can be computed in polynomial time for directed graphs with underlying graph of bounded tree-width by using Courcelle's theorem.

Kanté and Nourine [ SIAM J. Discrete Math., 30 (2016), pp. 311--326] present a polynomial time algorithm for the computation of the hull number of chordal graphs. We point out a gap in the correctness proof of their algorithm for chordal graphs and show that computing the hull number of a chordal graph is NP-hard, which most likely rules out the existence of a polynomial time algorithm.

We study perfect matchings in graphs that have the two properties of being robust as well as recoverable; where robust means that the failure of a set of not too many edges of can be compensated, and recoverable means that this compensation can be done in an efficient way, that is, has a perfect matching for which the symmetric difference of and is small. We establish the hardness of several related algorithmic problems and identify some tractable cases. Among others we show the hardness of the well known matching preclusion number of a graph. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 66(3), 210–213 2015

We study the graphs $G$ for which the hull number $h(G)$ and the geodetic number $g(G)$ with respect to $P_3$-convexity coincide. These two parameters correspond to the minimum cardinality of a set $U$ of vertices of $G$ such that the simple expansion process which iteratively adds to $U$ all vertices outside of $U$ having two neighbors in $U$ produces the whole vertex set of $G$ either eventually or after one iteration, respectively. We establish numerous structural properties of the graphs $G$ with $h(G)=g(G)$, allowing for the constructive characterization as well as the efficient recognition of all such graphs that are triangle-free. Furthermore, we characterize---in terms of forbidden induced subgraphs---the graphs $G$ that satisfy $h(G')=g(G')$ for every induced subgraph $G'$ of $G$.

We present a modular redesign of TrustedPals, a smart card-based security framework for solving Secure Multiparty Computation (SMC). Originally, TrustedPals assumed a synchronous network setting and allowed to reduce SMC to the problem of fault-tolerant consensus among smart cards. We explore how to make TrustedPals applicable in environments with less synchrony and show how it can be used to solve asynchronous SMC. Within the redesign we investigate the problem of solving consensus in a general omission failure model augmented with failure detectors. To this end, we give novel definitions of both consensus and the class \diamond {\cal P} of failure detectors in the omission model, which we call \diamond {\cal P}({ om}), and show how to implement \diamond {\cal P}({ om}) and have consensus in such a system with very weak synchrony assumptions. The integration of failure detection and consensus into the TrustedPals framework uses tools from privacy enhancing techniques such as message padding and dummy traffic.

For \documentclass{article} \usepackage{amsmath,amsfonts,amssymb}\pagestyle{empty}\begin{document} $n\in \mathbb{N}$ \end{document} and \documentclass{article} \usepackage{amsmath,amsfonts,amssymb}\pagestyle{empty}\begin{document} $D\subseteq \mathbb{N}$ \end{document}, the distance graph P has vertex set {0,1,…,n − 1} and edge set {ij | 0 ≤ i,j ≤ n − 1,|j − i| ∈ D}. The class of distance graphs generalizes the important and very well-studied class of circulant graphs, which have been proposed for numerous network applications. In view of fault tolerance and delay issues in these applications, the connectivity and diameter of circulant graphs have been studied in great detail. Our contributions are hardness results concerning computational problems related to the connectivity and the diameter of distance graphs and a characterization of the connected distance graphs P for |D| = 2. © 2010 Wiley Periodicals, Inc. NETWORKS, Vol. 57(4), 310-315 2011

No abstract available.