A Generalization of Distance Domination
Alicia Muth, E. Dov Neimand
TL;DR
Problem: generalize distance domination via distance-$\ell$ domination and $k$-component neighbor connectivity, and minimize the size of a failure set $F$ where every component of $V \setminus N_\ell[F]$ has order $< k$, i.e., minimize $|F|$ for $\lambda_{\ell}^{(k)}(G)$. The authors show NP-hardness on general graphs and, focusing on trees, present a polynomial-time algorithm achieving $O(n^2)$ time by defining a component-order function $\mathrm{comporder}$ and a recursive selection process. Correctness is supported by a mapping argument $M$ that relates any failure set to the constructed one, establishing optimality for trees. Practically, the approach informs efficient service-location designs under relaxed coverage constraints and offers a foundation for extensions to weighted or directed variants.
Abstract
Expanding on the graph theoretic ideas of k-component order connectivity and distance-l domination, we present a quadratic-complexity algorithm that finds a tree's minimum failure-set cardinality, i.e., the minimum cardinality any subset of the tree's vertices must have so that all clusters of vertices further away than some l do not exceed a cardinality threshold. Applications of solutions to the expanded problems include choosing service center locations so that no large neighborhoods are excluded from service, while reducing the redundancy inherent in distance domination problems.