Abstract: Although Large Language Models (LLMs) are successful in abstractive summarization of short dialogues, summarization of long dialogues remains challenging. To address this challenge, we propose a novel algorithm that processes complete dialogues comprising thousands of tokens into topic-segment-level Abstract Meaning Representation (AMR) graphs, which explicitly capture the dialogue structure, highlight salient semantics, and preserve high-level information. We also develop a new text-graph attention to leverage both graph semantics and a pretrained LLM that exploits the text. Finally, we propose an AMR node selection loss used jointly with conventional cross-entropy loss, to create additional training signals that facilitate graph feature encoding and content selection. Experiments show that our system outperforms the state-of-the-art models on multiple long dialogue summarization datasets, especially in low-resource settings, and generalizes well to out-of-domain data.