Abstract
An {\em annulus triangulation} $G$ is a 2-connected plane graph with two disjoint faces $f_1$ and $f_2$ such that every face other than $f_1$ and $f_2$ are triangular, and that every vertex of $G$ is contained in the boundary cycle of $f_1$ or $f_2$. In this paper, we prove that every annulus triangulation $G$ with $t$ vertices of degree 2 has a dominating set with cardinality at most $\lfloor \frac{|V(G)|+t+1}{4} \rfloor$ if $G$ is not isomorphic to the octahedron. In particular, this bound is best possible.
Highlights
In this paper, all graphs are undirected and simple
For S, T ⊂ V (G), we say that S dominates T if T ⊂ S ∪ N (S)
⌋, and the estimation in Theorem is best possible. The examples they constructed are maximal outerplanar graphs, (i.e., a 2-connected plane graph such that there is a single face f containing all vertices on the boundary cycle, and that every face other than f is triangular), and so they asked what happens if every face is triangular: Conjecture 2 (Matheson and Tarjan [3]) Let G be a planar triangulation with n vertices
Summary
All graphs are undirected and simple. For a graph G, let V (G) and E(G) denote the vertex set and the edge set of G, respectively. Matheson and Tarjan proved the following theorem by an elegant coloring method: Theorem 1 (Matheson and Tarjan [3]) Let G be a disk triangulation with n vertices They constructed a disk triangulation with n vertices in which any dominating sets have least. ⌋, and the estimation in Theorem is best possible The examples they constructed are maximal outerplanar graphs, (i.e., a 2-connected plane graph such that there is a single face f containing all vertices on the boundary cycle, and that every face other than f is triangular), and so they asked what happens if every face is triangular: Conjecture 2 (Matheson and Tarjan [3]) Let G be a planar triangulation with n vertices. By Theorem 1, every maximal outerplanar graph vertices has domination number at most. Campos and Wakabayashi [1] pointed out that maximal outerplanar graphs with a large domination number have many vertices of degree 2, and they (and Tokunaga independently)proved the following theorem.
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