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### Total domination in plane triangulations [article]

M. Claverol, A. García, G. Hernández, C. Hernando, M. Maureso, M. Mora, J. Tejel
2020 arXiv   pre-print
A total dominating set of a graph G=(V,E) is a subset D of V such that every vertex in V is adjacent to at least one vertex in D.  ...  The total domination number of G, denoted by γ _t (G), is the minimum cardinality of a total dominating set of G.  ...  The upper bound 2n 5 on the total domination number in near-triangulations is proved in Section 4.  ...

### Enumeration of Enumeration Algorithms [article]

Kunihiro Wasa
2016 arXiv   pre-print
In this paper, we enumerate enumeration problems and algorithms. This survey is under construction. If you know some results not in this survey or there is anything wrong, please let me know.  ...  Comment A based plane triangulation is a plane triangulation with one designated edge on the outer face.  ...  Comment A based plane triangulation is a plane triangulation with one designated edge on the outer face.  ...

### Dominating Plane Triangulations [article]

Michael D. Plummer, Dong Ye, Xiaoya Zha
2014 arXiv   pre-print
In 1996, Tarjan and Matheson proved that if G is a plane triangulated disc with n vertices, γ (G)< n/3, where γ (G) denotes the domination number of G.  ...  In the present paper, it is proved that if G is a hamiltonian plane triangulation with |V(G)|=n vertices and minimum degree at least 4, then γ (G)<{ 2n/7, 5n/16}.  ...  Preferred Hamilton cycles in plane triangulations Let G be a plane triangulation with δ(G) ≥ 4 and suppose G contains a Hamilton cycle H.  ...

### Dominating maximal outerplane graphs and Hamiltonian plane triangulations [article]

Michael D. Plummer, Dong Ye, Xiaoya Zha
2019 arXiv   pre-print
Let $G$ be a graph and $\gamma (G)$ denote the domination number of $G$, i.e. the cardinality of a smallest set of vertices $S$ such that every vertex of $G$ is either in $S$ or adjacent to a vertex in  ...  by distance at least 3 on the boundary of $G$; and (2) a Hamiltonian plane triangulation $G$ with $n \ge 23$ vertices has $\gamma (G)\le 5n/16$.  ...  Introduction A plane triangulation is a plane graph in which every face is bounded by a triangle.  ...

### Combinatorial bounds on connectivity for dominating sets in maximal outerplanar graphs

Santiago Canales, Irene Castro, Gregorio Hernández, Mafalda Martins
2016 Electronic Notes in Discrete Mathematics
In this article we study some variants of the domination concept attending to the connectivity of the subgraph generated by the dominant set. This study is restricted to maximal outerplanar graphs.  ...  We establish tight combinatorial bounds for connected domination, semitotal domination, independent domination and weakly connected domination for any n-vertex maximal outerplaner graph.  ...  A maximal outerplanar graph embedded in the plane corresponds to a triangulation of a polygon.  ...

### Efficient plane sweeping in parallel

M J Atallah, M T Goodrich
1986 Proceedings of the second annual symposium on Computational geometry - SCG '86
We present techniques which result in improved parallel algorithms for a number of problems whose efficient sequential algorithms use the plane-sweeping paradigm.  ...  The problems for which we give improved algorithms include intersection detection, trapezoidal decomposition, triangulation, and planar point location.  ...  (v)) are totally ordered.  ...

### Distance domination, guarding and vertex cover for maximal outerplanar graph [article]

Santiago Canales, Gregorio Hernández, Mafalda Martins, Inês Matos
2013 arXiv   pre-print
This paper discusses a distance guarding concept on triangulation graphs, which can be associated with distance domination and distance vertex cover.  ...  We show how these subjects are interconnected and provide tight bounds for any n-vertex maximal outerplanar graph: the 2d-guarding number, g_2d(n) = n/5; the 2d-distance domination number, gamma_2d(n)  ...  Such graph is called triangulation graph (triangulation, for short), because is the graph of a triangulation of a set of points in the plane (see Figures 1 and 2) .  ...

### On Polyhedral Realization with Isosceles Triangles [article]

David Eppstein
2020 arXiv   pre-print
However, we also show that all Kleetopes of triangulated polyhedral graphs have non-convex non-self-crossing realizations in which all faces are isosceles.  ...  We prove that the graphs of convex polyhedra with congruent isosceles faces have bounded diameter and have dominating sets of bounded size.  ...  angle, so the total angle is ≤ kπ + O(1), but in order to achieve total angular deficit at most 4π the total angle must be ≥ (2k − 4)π.  ...

### Dominating Sets in Planar Graphs

Lesley R. Matheson, Robert E. Tarjan
1996 European journal of combinatorics (Print)
Motivated by an application to unstructured multigrid calculations, we consider the problem of asymptotically minimizing the size of dominating sets in triangulated planar graphs.  ...  For triangulated discs we obtain a tight bound of = 1 3 . The upper bound proof yields a linear-time algorithm for nding an n=3 -size dominating set.  ...  Small Dominating Sets in Triangulated Discs Our method for nding small dominating sets in triangulated discs is based on a stronger result.  ...

### On Polyhedral Realization with Isosceles Triangles

David Eppstein
2021 Graphs and Combinatorics
However, we also show that all Kleetopes of triangulated polyhedral graphs have non-convex non-self-crossing realizations in which all faces are isosceles.  ...  We prove that the graphs of convex polyhedra with congruent isosceles faces have bounded diameter and have dominating sets of bounded size.  ...  angle, so the total angle is kp þ Oð1Þ, but in order to achieve total angular deficit at most 4p the total angle must be !  ...

### TRANSLATION QUERIES FOR SETS OF POLYGONS

MARK DE BERG, HAZEL EVERETT, HUBERT WAGENER
1995 International journal of computational geometry and applications
Let S be a set of m polygons in the plane with a total of n vertices.  ...  A translation order for S in direction d is an order on the polygons such that no collisions occur if the polygons are moved one by one to infinity in direction d according to this order.  ...  plane.  ...

### Thoroughly Distributed Colorings [article]

Wayne Goddard, Michael A. Henning
2016 arXiv   pre-print
Equivalently, every color is a total dominating set. We define (G) as the maximum number of colors in such a coloring and (G) as the fractional version thereof.  ...  We also consider the related concepts in hypergraphs.  ...  In order to totally dominate the new vertices added in each face of the base graph G, all total dominating sets of T (G) contain at least two vertices of G.  ...

### Dominating sets in triangulations on surfaces

Hong Liu, Michael J. Pelsmajer
2011 Ars Mathematica Contemporanea
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.  ...  We would also like to thank the referees for their careful work, which led to critical improvements in the paper.  ...

### Triangulations and soliton graphs for totally positive Grassmannian [article]

Rachel Karpman, Yuji Kodama
2018 arXiv   pre-print
For the positive Grassmannian Gr(2,M)_>0, Kodama and Williams showed that soliton graphs are in bijection with triangulations of the M-gon.  ...  It is well known that regular soliton solutions of the KP equation may be constructed from points in the totally nonnegative Grassmannian Gr(N,M)_≥ 0.  ...  That is, f M(A) (x, y, t) represents a dominant plane z = Θ I (x, y, t) in R 3 for fixed t.  ...

### Dominating Sets in Plane Triangulations [article]

Erika L.C. King, Michael J. Pelsmajer
2010 arXiv   pre-print
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.  ...  For any closed walk W in the plane dual of G ′ , let h(W ) be the total number of such vertices for all cycles in W .  ...  A triangulation is a plane graph in which every face is a 3-face. A triangle is a subgraph isomorphic to K 3 .  ...
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