Dominating plane triangulations

Dominating Sets in Triangulations

In 1996, Matheson and Tarjan conjectured that any n-vertex triangulation with n sufficiently large has a dominating set of size at most n/4. We prove this for graphs of maximum degree 6.

Good Triangulations in Plane

Dominating Sets in Triangulations on Surfaces

A dominating set D ⊆ V (G) of a graph G is a set such that each vertex v ∈ V (G) is either in the set or adjacent to a vertex in the set. Matheson and Tarjan (1996) proved that any n-vertex plane triangulation has a dominating set of size at most n/3, and conjectured a bound of n/4 for n sufficiently large. King and Pelsmajer recently proved this for graphs with maximum degree at most 6. Plumme...

h-perfect plane triangulations

We characterise t-perfect plane triangulations by forbidden induced subgraphs. As a consequence, we obtain that a plane triangulation is h-perfect if and only if it is perfect.

Plane triangulations are 6-partitionable

Given a graph G = (V; E) and k positive integers n1; n2; : : : ; nk such that ∑k i=1 ni = |V |, we wish to 2nd a partition P1; P2; : : : ; Pk of the vertex set V such that |Pi| = ni and Pi induces a connected subgraph of G for all i; 16 i6 k. Such a partition is called a k-partition of G. A graph G with n vertices is said to be k-partitionable if there exists a k-partition of G for any partitio...