Matheson–Tarjan conjecture on dominating sets in planar triangulations

Establish that every planar triangulation G on sufficiently large order n admits a dominating set D subset of V(G) with |D| ≤ n/4, i.e., determine whether for all sufficiently large n any maximal planar graph on n vertices has a dominating set of size at most n/4.

Background

The survey discusses plantri and related generators for planar graphs and highlights outstanding problems motivating such generation. Dominating sets in planar triangulations (maximal planar graphs) are a classical topic, and the Matheson–Tarjan conjecture proposes a universal upper bound of n/4 for sufficiently large n. The conjecture remains open and is a benchmark problem in computer-assisted explorations of planar structures.

References

An example of an interesting open conjecture for planar graphs is the Matheson-Tarjan conjecture stating that every planar triangulation $G$ (i.e., maximal planar graph) of sufficiently large order $n$ has a dominating set (i.e., a vertex set $D \subseteq V(G)$ such that every vertex in $V(G)\setminus D$ has a neighbor in $D$) of size at most $n/4$.

— Computer-assisted graph theory: a survey (2508.20825 - Jooken, 28 Aug 2025) in Section 2.1 (Generation algorithms and graph censuses), paragraph on planar graphs