Existence of graphs not partitionable into two locating dominating sets

Determine whether there exists an isolate-free and twin-free graph whose vertex set cannot be partitioned into two locating dominating sets.

Background

The paper discusses the stronger partition problem in which an isolate-free, twin-free graph must be divided into two locating dominating sets. This property is known for several graph classes, including bipartite graphs and block graphs, but the general situation is unresolved.

The authors explicitly note that even the existence of a counterexample is unknown. Finding such a graph would not by itself disprove the locating-domination upper-bound conjecture, because the graph could still possess a minimum locating dominating set of size at most n/2.

References

However, as a preliminary investigation, it is not yet known to the authors if there exist examples of isolate-free and twin-free graphs whose vertex sets cannot be partitioned into two locating dominating sets (notice that finding such an example does not, however, disprove Conjecture~\ref{conj:Ldom-bound} as long as a minimum locating dominating set of the example has cardinality 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 Section 1, subsection “Motivation,” paragraph following Problem 2