<SYSTEM_TASK:>
Solve the following problem using Python, implementing the functions described below, one line at a time
<END_TASK>
<USER_TASK:>
Description:
 def _prune(self, filename: str, df: pd.DataFrame) -> pd.DataFrame: """Depth-first search through the dependency graph and prune dependent DataFrames along the way. """

        dependencies = []
        for _, depf, data in self._config.out_edges(filename, data=True):
            deps = data.get("dependencies")
            if deps is None:
                msg = f"Edge missing `dependencies` attribute: {filename}->{depf}"
                raise ValueError(msg)
            dependencies.append((depf, deps))

        if not dependencies:
            return df

        for depfile, column_pairs in dependencies:
            # Read the filtered, cached file dependency
            depdf = self.get(depfile)
            for deps in column_pairs:
                col = deps[filename]
                depcol = deps[depfile]
                # If applicable, prune this dataframe by the other
                if col in df.columns and depcol in depdf.columns:
                    df = df[df[col].isin(depdf[depcol])]

        return df