arXiv:2610.36106v1 Announce Type: new Abstract: We study the simultaneous recovery of an unknown initial signal and unknown vertex wave speeds for the wave equation on a finite graph from time-dependent measurements on a prescribed set of vertices. With the graph geometry known, we first show that a single experiment determines both quantities under a new two-witness zero forcing condition and generic nondegeneracy assumptions. We then develop a more general frontier-propagation framework in which the redundancy required for simultaneous recovery is distributed across successive graph contours rather than localized at individual forcing vertices. We derive Lipschitz stability estimates for the frontier propagation problem. We also show the recovery and stability results extend to positively weighted graphs.
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