scholarly journals Locally Hamiltonian Graphs and Minimal Size of Maximal Graphs on a Surface

10.37236/9286 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
James Davies ◽  
Carsten Thomassen
Keyword(s):  
Complete Graph ◽  
Hamiltonian Graph ◽  
Minimal Size ◽  
Hamiltonian Graphs ◽  
The Face ◽  
Multiple Edges ◽  
Dimensional Surface

We prove that every locally Hamiltonian graph with $n$ vertices and possibly with multiple edges has at least $3n-6$ edges with equality if and only if it triangulates the sphere. As a consequence, every edge-maximal embedding of a graph $G$ on some 2-dimensional surface $\Sigma$ (not necessarily compact) has at least $3n-6$ edges with equality if and only if $G$ also triangulates the sphere. If, in addition, $G$ is simple, then for each vertex $v$, the cyclic ordering of the edges around $v$ on $\Sigma$ is the same as the clockwise or anti-clockwise orientation around $v$ on the sphere. If $G$ contains no complete graph on 4 vertices, then the face-boundaries are the same in the two embeddings.

Recognising Facial Surfaces

Perception ◽  
1991 ◽  
Vol 20 (6) ◽  
pp. 755-769 ◽  
Author(s):  
Vicki Bruce ◽  
Patrick Healey ◽  
Mike Burton ◽  
Tony Doyle ◽  
Anne Coombes ◽  
...  
Keyword(s):  
Computer Aided Design ◽  
Three Dimensional ◽  
Head Model ◽  
Matching Accuracy ◽  
The People ◽  
Computer Aided ◽  
Closed Head ◽  
The Face ◽  
Aided Design ◽  
Dimensional Surface

The extent to which faces depicted as surfaces devoid of pigmentation and with minimal texture cues (‘head models’) could be matched with photographs (when unfamiliar) and identified (when familiar) was examined in three experiments. The head models were obtained by scanning the three-dimensional surface of the face with a laser, and by displaying the surface measured in this way by using standard computer-aided design techniques. Performance in all tasks was above chance but far from ceiling. Experiment 1 showed that matching of unfamiliar head models with photographs was affected by the resolution with which the surface was displayed, suggesting that subjects based their decisions, at least in part, on three-dimensional surface structure. Matching accuracy was also affected by other factors to do with the viewpoints shown in the head models and test photographs, and the type of lighting used to portray the head model. In experiment 2 further evidence for the importance of the nature of the illumination used was obtained, and it was found that the addition of a hairstyle (not that of the target face) did not facilitate matching. In experiment 3 identification of the head models by colleagues of the people shown was compared with identification of photographs where the hair was concealed and eyes were closed. Head models were identified less well than these photographs, suggesting that the difficulties in their recognition are not solely due to the lack of hair. Women's heads were disproportionately difficult to recognise from the head models. The results are discussed in terms of their implications for the use of such three-dimensional head models in forensic and surgical applications.

6. Graphs

2016 ◽  
pp. 86-108
Author(s):  
Robin Wilson
Keyword(s):  
Graph Theory ◽  
Bipartite Graph ◽  
Complete Graph ◽  
Planar Graphs ◽  
Complete Bipartite Graph ◽  
Travelling Salesman Problem ◽  
Chemical Bonds ◽  
Hamiltonian Graphs ◽  
Road Map ◽  
Cycle Graph

Graph theory is about collections of points that are joined in pairs, such as a road map with towns connected by roads or a molecule with atoms joined by chemical bonds. ‘Graphs’ revisits the Königsberg bridges problem, the knight’s tour problem, the Gas–Water–Electricity problem, the map-colour problem, the minimum connector problem, and the travelling salesman problem and explains how they can all be considered as problems in graph theory. It begins with an explanation of a graph and describes the complete graph, the complete bipartite graph, and the cycle graph, which are all simple graphs. It goes on to describe trees in graph theory, Eulerian and Hamiltonian graphs, and planar graphs.

scholarly journals The Ramsey number for stripes

1975 ◽  
Vol 19 (2) ◽  
pp. 252-256 ◽  
Author(s):  
E. J. Cockayne ◽  
P. J. Lorimer
Keyword(s):  
Complete Graph ◽  
Ramsey Number ◽  
Multiple Edges

If G1,…,Gc are graphs without loops or multiple edges there is a smallest integer r(G1,…,Gc) such that if the edges of a complete graph Kn, with n ≧ r(G1,…,Gc), are painted arbitrarily with c colours the ith coloured subgraph contains Gi as a subgraph for at least one i. r(G1,…Gc) is called the Ramsey number of the graphs G1,…,Gc.

scholarly journals Reliability of 3‐dimensional surface imaging of the face using a whole‐body surface scanner

2021 ◽  
Author(s):  
Ya Xu ◽  
Konstantin Frank ◽  
Lukas Kohler ◽  
Denis Ehrl ◽  
Michael Alfertshofer ◽  
...  
Keyword(s):  
Body Surface ◽  
Whole Body ◽  
Surface Imaging ◽  
3 Dimensional ◽  
The Face ◽  
Dimensional Surface

Hamiltonian Paths and Cycles

2013 ◽  
pp. 96-105
Author(s):  
Mahtab Hosseininia ◽  
Faraz Dadgostari
Keyword(s):  
Hamiltonian Cycle ◽  
Travelling Salesman Problem ◽  
Sufficient Conditions ◽  
Hamiltonian Cycles ◽  
Hamiltonian Graph ◽  
Hamiltonian Graphs ◽  
Hamiltonian Paths ◽  
Graph Problems ◽  
History Of ◽  
Knight’S Tour

In this chapter, the concepts of Hamiltonian paths and Hamiltonian cycles are discussed. In the first section, the history of Hamiltonian graphs is described, and then some concepts such as Hamiltonian paths, Hamiltonian cycles, traceable graphs, and Hamiltonian graphs are defined. Also some most known Hamiltonian graph problems such as travelling salesman problem (TSP), Kirkman’s cell of a bee, Icosian game, and knight’s tour problem are presented. In addition, necessary and (or) sufficient conditions for existence of a Hamiltonian cycle are investigated. Furthermore, in order to solve Hamiltonian cycle problems, some algorithms are introduced in the last section.

ON n-HAMILTONIAN GRAPHS OF MINIMAL SIZE

1981 ◽  
Vol 14 (3) ◽  
Author(s):  
Tomasz Traczyk ◽  
Mirosław Truszczyński
Keyword(s):  
Minimal Size ◽  
Hamiltonian Graphs

On the Number of Hamiltonian Cycles in Bipartite Graphs

1996 ◽  
Vol 5 (4) ◽  
pp. 437-442 ◽  
Author(s):  
Carsten Thomassen
Keyword(s):  
Hamiltonian Cycle ◽  
Reduction Method ◽  
Minimum Degree ◽  
Bipartite Graphs ◽  
Hamiltonian Cycles ◽  
Hamiltonian Graph ◽  
Hamiltonian Graphs ◽  
Color Class ◽  
General Reduction

We prove that a bipartite uniquely Hamiltonian graph has a vertex of degree 2 in each color class. As consequences, every bipartite Hamiltonian graph of minimum degree d has at least 21−dd! Hamiltonian cycles, and every bipartite Hamiltonian graph of minimum degree at least 4 and girth g has at least (3/2)g/8 Hamiltonian cycles. We indicate how the existence of more than one Hamiltonian cycle may lead to a general reduction method for Hamiltonian graphs.

On the Ramsey Number r(F, Km) Where F is a Forest

1975 ◽  
Vol 27 (3) ◽  
pp. 585-589 ◽  
Author(s):  
Saul Stahl
Keyword(s):  
Complete Graph ◽  
Ramsey Number ◽  
Multiple Edges

The graphs considered here are finite and have no loops or multiple edges. In particular, Km denotes the complete graph on m vertices. For any graph G,V(G) and E(G) denote, respectively, the vertex and edge sets of G. A forest is a graph which has no cycles and a tree is a connected forest. The reader is referred to [1] or [4] for the meaning of terms not defined in this paper.

scholarly journals The number of Hamiltonian circuits in large, heavily edged graphs

1977 ◽  
Vol 18 (1) ◽  
pp. 35-37 ◽  
Author(s):  
J. Sheehan ◽  
E. M. Wright
Keyword(s):  
Complete Graph ◽  
Maximum Degree ◽  
Hamiltonian Circuits ◽  
Multiple Edges ◽  
Image Position

G is a graph on n nodes with q edges, without loops or multiple edges. We write α = q/n and β for the maximum degree of any node of G. We writeand H for the number of Hamiltonian circuits (H.c.) in the complement of G, or, what is the same thing, the number of those H.c. in the complete graph Kn which have no edge in common with G. Our object here is to prove the following theorem.

Precision in 3-Dimensional Surface Imaging of the Face: A Handheld Scanner Comparison Performed in a Cadaveric Model

2018 ◽  
Vol 39 (4) ◽  
pp. NP36-NP44 ◽  
Author(s):  
Konstantin C Koban ◽  
Sebastian Cotofana ◽  
Konstantin Frank ◽  
Jeremy B Green ◽  
Lucas Etzel ◽  
...  
Keyword(s):  
Surface Imaging ◽  
3 Dimensional ◽  
The Face ◽  
Handheld Scanner ◽  
Cadaveric Model ◽  
Dimensional Surface
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