Locating domination upper-bound conjecture

Prove that every isolate-free and twin-free graph G of order n has locating domination number at most n/2.

Background

The paper situates its partitioning result within the longstanding upper-bound problem for locating dominating sets. For an isolate-free, twin-free graph of order n, the conjectured bound is that a locating dominating set of size at most n/2 exists. The paper notes that the best general bound known at the time is ceil(5n/8), while the conjecture is known for several special graph classes.

A stronger formulation asks whether the vertex set of every isolate-free, twin-free graph can be partitioned into two locating sets, which would provide a route toward the conjectured upper bound.

References

A notable conjecture in the study of locating dominating sets is to show that the locating domination number of an isolate-free and twin-free graph $G$ of order $n$ is at most $\frac{1}{2}n$.

— A note on partitioning the vertex set of a graph into a dominating set and a locating dominating set  (2610.03131 - Chakraborty et al., 2 Oct 2026) in Abstract; Conjecture 1, Section 1, subsection “Motivation”