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, θ,Kr,θ)–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′ = Ht − FV − FE is connected. When G′ is connected, a spanning tree of Kr,θ maps to exactly c − 1 omponentcrossing 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 Dr,θ ≤ 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.
Bader AlBader (Sun,) studied this question.
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