MINIMUM SURGICAL PROBING with convexity constraints

We consider a tomographic problem on graphs, called MINIMUM SURGICAL PROBING, introduced by Bar-Noy et al. [4]. Each vertex v ∈ V of a graph G=(V,E) is associated with an (unknown) label ℓ_v. The outcome of probing a vertex v is P_v=∑_u∈N[v]ℓ_u, where N[v] denotes the closed neighborhood of v. The goal is to uncover the labels given probes P_v for all v ∈ V. For some graphs, the labels cannot be determined (uniquely), and the use of surgical probes is permitted but must be minimized. A surgical probe at vertex v returns ℓ_v. In this paper, we introduce convexity constraints to MINIMUM SURGICAL PROBING. For binary labels, convexity imposes constraints such as if ℓ_u=ℓ_v=1, then for all vertices w on a shortest path between u and v, we must have that ℓ_w=1. We show that convexity constraints reduce the number of required surgical probes for several graph families. Specifically, they allow us to recover the labels without using surgical probes for trees and bipartite graphs where otherwise ⌊|V|/2⌋ surgical probes might be needed. Our analysis is based on restricting the size of cliques in a graph using the concept of K_h-free graphs (forbidden induced subgraphs). Utilizing this approach, we analyze grid graphs, the King's graph, and (maximal-) outerplanar graphs.