Recent advancements in text-attributed graphs (TAGs) have significantly improved the quality of node features by using the textual modeling capabilities of language models. Despite this success, utilizing text attributes to enhance the predefined graph structure remains largely unexplored. Our extensive analysis reveals that conventional edges on TAGs, treated as a single relation (e.g., hyperlinks) in previous literature, actually encompass mixed semantics (e.g., "advised by" and "participates in"). This simplification hinders the representation learning process of Graph Neural Networks (GNNs) on downstream tasks, even when integrated with advanced node features. In contrast, we discover that decomposing these edges into distinct semantic relations significantly enhances the performance of GNNs. Despite this, manually identifying and labeling of edges to corresponding semantic relations is labor-intensive, often requiring domain expertise. To this end, we introduce RoSE (Relation-oriented Semantic Edge-decomposition), a novel framework that leverages the capability of Large Language Models (LLMs) to decompose the graph structure by analyzing raw text attributes - in a fully automated manner. RoSE operates in two stages: (1) identifying meaningful relations using an LLM-based generator and discriminator, and (2) categorizing each edge into corresponding relations by analyzing textual contents associated with connected nodes via an LLM-based decomposer. Extensive experiments demonstrate that our model-agnostic framework significantly enhances node classification performance across various datasets, with improvements of up to 16% on the Wisconsin dataset.