Query containment is a fundamental problem of databases. Given two queries q1 and q2, it asks whether the set of answers to q1 is included in the set of answers to q2 for any database. In this paper, we investigate this problem for conjunctive queries with negated subgoals. We use graph homomorphism as the core notion, which leads us to extend the results presented in [Ull97] and [WL03]. First, we exhibit sufficient (but not necessary) conditions for query containment based on special subgraphs of q2, which generalize that proposed in [WL03]. As a corollary, we obtain a case where the time complexity of the problem decreases. From a practical viewpoint, these properties can be exploited in algorithms, as shown in the paper. Second, we propose an algorithm based on the exploration of a space of graphs, which improves existing algorithms.