.We study two fundamental operations on free graphic subarrangements of the braid arrangement : theextension(addition of a hyperplane) and thedeletion(removal of a hyperplane). For extensions, we prove that is free if and only if is chordal. For deletions, we prove the stronger result that isalwaysfree for every , with no combinatorial condition required. Together these results characterize the simplicial complex of free graphic subarrangements, identify the minimal non-free extensions as exactly those creating an induced 4-cycle, and provide a complete projective-dimension formula. Detailed proofs, worked examples, explicit pseudocode, and complexity estimates for the associated algorithms are included throughout.