Problem-solving strategies that physics undergraduates learn should prepare them for real-world contexts as they transition from novices to experts. Yet, graduate students in physics-intensive research face problems that go beyond "problem sets" they experienced as undergraduates and are solved by different strategies than are typically emphasized in undergraduate coursework. We conducted semi-structured interviews with ten graduate students to determine problem-solving strategies they found useful in their research. We coded these interviews using emergent and grounded theory approaches. Our findings explore problem-solving strategies (e.g., planning ahead, breaking down problems, evaluating options), contexts (e.g., designing software and troubleshooting equipment), and characteristics of successful problem-solvers (e.g., initiative, persistence, and motivation). Graduate students also relied on problem representations such as test cases, approximations, and simulations in their problem-solving process. Understanding problem-solving strategies, contexts, and characteristics has implications for how we approach problem-solving in undergraduate physics and physics education research.