Diagonal flips in outer-Klein-bottle triangulations

Diagonal flips in outer-triangulations on closed surfaces

“…It is easy to show that any two outer-triangulations on the sphere with the same number of vertices are equivalent to each other, up to isotopy. Moreover, the same fact has been shown for the projective plane [8], the torus [5] and Klein bottle [6]. (The cases for the torus and the Klein bottle have been solved, under the condition up to homeomorphism".)…”

“…An outer-triangulation G with no contractible outer edge is said to be irreducible. In order to prove Lemma 6, we need the following lemma shown in [6]. Let G be an outer-triangulation on a closed surface F 2 and let xyz be a face of G such that xy is an outer edge.…”

Irreducible triangulations are small

“…By Lemma 6, v has at least three neighbours in H i , implying e(v, N) e(N, Z ) + e( A, Z ). (11) Since [4,10] i|…”

“…Summing (3), 6 × (11) and 6 × (12) gives 3| A| + 6 j∈ [4,10] j| A j | + 60| Z | 12e( A) + 7e(N, A) + 12e( A, Z ) + 12e(Z ) + 6e(N, Z ). (19) Since |A| = i∈ [4,10] |A i |, summing 3 × (15) with (18) and (19) gives…”

On local transformations in plane geometric graphs embedded on small grids

“…For example, an edge contraction 1 is a local transformation that affects the neighborhood of two vertices. Once a local transformation has been defined, properties and applications of the local transformation with respect to a given class of graphs are studied [3][4][5][6][9][10][11][12]14,[16][17][18][19][20]23]. A natural question with respect to local transformations is: Does performing a local transformation of a graph in a given class keep the graph in the same class?…”