We study the problem of scheduling jobs on a single machine with a rejection possibility, concurrently minimizing the total tardiness of the scheduled jobs and the total cost of the rejected ones. The model we consider is fully bi-objective, i.e. its aim is to enumerate the Pareto front. We tackle the problem both with and without the presence of hard deadlines. For the case without deadlines, we provide a pseudo-polynomial time algorithm, based on the dynamic program of [25], thereby proving that the problem is weakly NP-hard. For the case with deadlines, we propose a branch-and-bound algorithm and prove its efficiency by comparing it to an µ-constrained approach on benchmark instances based on those proposed in the literature on similar problems.