While graph-based dynamic programming (DP) is a cornerstone of genomics and network analytics, its efficiency is hampered by fundamentally conflicting computational patterns. Matrix-centric DP drives regular, compute-bound network analytics, while topology-centric DP handles irregular, memory-bound genomic traversals. These two categories of DP have substantially different computation patterns and dataflows, which makes it difficult for a single homogeneous processing-in-memory (PIM) architecture to efficiently support both. This work presents GEN-Graph, a novel heterogeneous PIM chiplet that integrates two types of specialized compute tiles within a 2.5D package: Matrix-tile, a processing-using-memory (PUM) tile optimized for matrix-centric workloads, such as all-pairs shortest path (APSP); and traversal-tile, a processing-near-memory (PNM) tile optimized for traversal-centric DP workloads, such as DNA sequence alignment. Our hardware-software co-design employs recursive partitioning and reconfigurable windowed bit-parallel logic to ensure exact computation. Results show the matrix tile achieves 42.8x speedup and 392x energy efficiency over the NVIDIA H100 GPU for APSP. For sequence-to-graph alignment, the traversal tile sustains 2.56 million reads/s (short-reads) and 39.3 thousand reads/s (long-reads), outperforming state-of-the-art accelerators by up to 2.56x in throughput. GEN-Graph provides the first scalable, exact solution for general DP dataflows by matching hardware specialization to algorithmic structure.