Abstract
This paper presents a generalization of the coupled-task sche-duling problem introduced by Shapiro \cite{Shapiro}, where considered tasks are subject to incompatibility constraints depicted by an undirected graph. The motivation of this problem comes from data acquisition and processing in a mono-processor torpedo used for underwater exploration. As we add the compatibility graph, we focus on complexity of the problem, and more precisely on the boundary between $\mathcal{P}$ and $\mathcal{NP}$-completeness when some other input parameters are restricted (e.g. the ratio between the durations of the two sub-tasks composing a task): we adapt the global visualization of the complexity of scheduling problems with coupled-task given by Orman and Potts \cite{op1997} to our model, determine new complexity results, and thus propose a new visualization including incompatibility constraints. In the end, we give a new polynomial-time approximation algorithm result which completes previous works.