And/or-convexity: a graph convexity based on processes and deadlock models

Deadlock prevention techniques are essential in the design of robust distributed systems. However, despite the large number of different algorithmic approaches to detect and solve deadlock situations, yet there remains quite a wide field to be explored in the study of deadlock-related combinatorial properties. In this work we consider a simplified AND-OR model, where the processes and their communication are given as a graph G. Each vertex of G is labelled AND or OR, in such a way that an AND-vertex (resp., OR-vertex) depends on the computation of all (resp., at least one) of its neighbors. We define a graph convexity based on this model, such that a set $$S \subseteq V(G)$$ S ⊆ V ( G ) is convex if and only if every AND-vertex (resp., OR-vertex) $$v \in V(G){\setminus }S$$ v ∈ V ( G ) \ S has at least one (resp., all) of its neighbors in $$V(G) {\setminus } S$$ V ( G ) \ S . We relate some classical convexity parameters to blocking sets that cause deadlock. In particular, we show that those parameters in a graph represent the sizes of minimum or maximum blocking sets, and also the computation time until system stability is reached. Finally, a study on the complexity of combinatorial problems related to such graph convexity is provided.