Minimum Surgical Probing with Convexity Constraints

Abstract

We consider a tomographic problem on graphs, called Minimum Surgical Probing, introduced by Bar-Noy et al. [2]. Each vertex $$v \in V$$ of a graph $$G = (V,E)$$ is associated with an (unknown) label $$\ell _v$$ . The outcome of probing a vertex v is $$\mathcal{P}_v = \sum _{u \in N[v]} \ell _u$$ , where N[v] denotes the closed neighborhood of v. The goal is to uncover the labels given probes $$\mathcal{P}_v$$ for all $$v \in V$$ . 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 $$\ell _v$$ . In this paper, we introduce convexity constraints to Minimum Surgical Probing. For binary labels, convexity imposes constraints such as if $$\ell _u = \ell _v= 1$$ , then for all vertices w on a shortest path between u and v, we must have that $$\ell _w = 1$$ . 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 $$\left\lfloor |V|/2 \right\rfloor $$ 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.