\subsection{Algorithms}
\label{app:complexity}
In this section, we present the complete algorithmic details of our method. 
We first outline the overall training procedure, followed by the core modules: 
topology-aware graph regularizer, GNN-based appearance propagation module, and semantic-rarity and boundary-aware modulator. Each algorithm is provided in pseudocode form for clarity 
and reproducibility.

\paragraph{Training Loop}

We summarize the end-to-end training procedure of our method. 
The schedule integrates photometric reconstruction, topology-aware graph regularizer, 
GNN-based appearance propagation module, and semantic-rarity and boundary-aware modulator.

\input{algs/training}

\paragraph{Topology-Aware Graph Regularizer}

We detail the topology-aware graph regularizer, which prunes small components, 
detects geometric voids, and repairs holes via bridging or filling strategies.

\input{algs/topo}

\paragraph{GNN-Based Appearance Propagation Module}

We describe the appearance refinement module, where a graph neural network 
propagates information across Gaussians with visibility-adaptive residual blending.

\input{algs/gnn}

\paragraph{Semantic-rarity and Boundary-aware
Modulator}

We present the semantic rarity \& boundary-aware modulator, which 
emphasizes rare categories and preserves fine structures near semantic boundaries.

\input{algs/srb}
