Safety-critical systems use redundant input units to improve their reliability and fault tolerance. A voting logic is then used to select a reliable input from the redundant sources. A fault detection and isolation rules help in selecting input units that can participate in voting. This work deals with the formal requirement formulation, design, verification and synthesis of a generic voting unit for an N-modular redundant measurement system used for control applications in avionics systems. The work follows a correct-by-construction approach, using the Rocq theorem prover.

Arc-Disjoint Cycle Packing is a classical NP -complete problem and we study it from two perspectives: (1) by restricting the cycles in the packing to be of a fixed length, and (2) by restricting the inputs to bipartite tournaments. Focusing first on Arc-Disjoint r -Cycle Packing (where the cycles in the packing are required to be of length r ), we show NP -completeness in oriented graphs with girth r for each r ≥ 3 and study the parameterized complexity of the problem with respect to two parameterizations (solution size and vertex cover size) for r = 4 in oriented graphs. Moving on to Arc-Disjoint Cycle Packing in bipartite tournaments, we show that every bipartite tournament either contains k arc-disjoint cycles or has a feedback arc set of size at most 7 ( k − 1 ) . This result adds to the set of Erdös-Pósa-type results known in the combinatorics literature for packing and covering problems.

A D-disc around a vertex v of a graph G=(V,E) is the subgraph induced by all vertices of distance at most D from v. We show that the structure of an outerplanar graph on n vertices is determined, up to modification (insertion or deletion) of at most epsilon n edges, by a set of D-discs around the vertices, for D=D(epsilon) that is independent of the size of the graph. Such a result was already known for planar graphs (and any hyperfinite graph class), in the limited case of bounded degree graphs (that is, their maximum degree is bounded by some fixed constant, independent of |V|). We prove this result with no assumption on the degree of the graph. A pure combinatorial consequence of this result is that two outerplanar graphs that share the same local views are close to be isomorphic. We also obtain the following property testing results in the sparse graph model: * graph isomorphism is testable for outerplanar graphs by poly(log n) queries. * every graph property is testable for outerplanar graphs by poly(log n) queries. We note that we can replace outerplanar graphs by a slightly more general family of k-edge-outerplanar graphs. The only previous general testing results, as above, where known for forests (Kusumoto and Yoshida), and for some power-law graphs that are extremely close to be bounded degree hyperfinite (by Ito).