Re-Rooting-Assisted Edge-Minimum Runtime Repair for Node and Link Failures in Dense Eisenstein--Jacobi Broadcast Networks

One-to-all broadcasting in dense Eisenstein–Jacobi (EJ) networks relies on diameter-level spanning trees that fragment when nodes or links fail. This paper introduces the selected triple $(r,\theta ,{\text{Kcomp}}_{r,\theta })$–a chosen root, a chosen EJ coordinate-reduction orientation, and the healthy component graph induced by that choice–as the fundamental unit of analysis for joint node/link fault recovery. The central result is a necessary and sufficient condition: hybrid repair succeeds if and only if the healthy EJ graph $G^{\mathrm{\prime }}=\text{Ht}-\text{Fv}-\text{Fe}$ is connected. When $G^{\mathrm{\prime }}$ is connected, a spanning tree of ${\text{Kcomp}}_{r,\theta }$ maps to exactly $c-1$ component-crossing repair edges, which is minimum for the selected pruned tree. Deterministic guarantees include: one/two faulty nodes are always placed on the distance-$t$ boundary by re-rooting; a single failed link is either avoided or repaired by exactly one crossing edge; and the repaired depth satisfies $D_{r,\theta }\le 2t+1$ under shallowest-layer entry selection. A 260,000-trial validation campaign confirms 100% recovery and substantial repair-edge reduction over fixed-source repair across five network scales up to $N=120601$ nodes, while global-BFS, near-miss, and cap-sensitivity audits clarify the tradeoff between reachability, forwarding-state changes, and ranked root selection.