From Constraints to Coordinates: A Computer-Science Playbook for Graph Layout Algorithm

This paper codifies a computer-science approach to graph layout by unifying constraint-driven tree positioning with force-based placement into reproducible, implementation-ready methods. It casts tree drawing as a linear optimization with level, separation, and centering requirements that implicitly ensure non-intersection, establishing planarity proofs and analyzing the asymptotic growth of constraints. For general graphs, it develops force-based layouts and Laplacian equilibrium methods with explicit convergence criteria, alongside studies of initialization sensitivity and resolution limits as graph size increases. Reference C++ implementations detail data structures, indexing passes, and solver choices leveraging open numerical libraries to realize these models efficiently in practice. Case studies on canonical families reveal symmetry preservation and degradation patterns, offering guidance on when to favor constraints versus forces in systems engineering and visualization pipelines.