scholarly journals On short cycles through prescribed vertices of a polyhedral graph

2005 ◽  
Vol 25 (3) ◽  
pp. 419
Author(s):  
Erhard Hexel
Keyword(s):  
Polyhedral Graph

scholarly journals A Survey on Symmetry Group of Polyhedral Graphs

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 370 ◽  
Author(s):  
Modjtaba Ghorbani ◽  
Matthias Dehmer ◽  
Shaghayegh Rahmani ◽  
Mina Rajabi-Parsa
Keyword(s):  
Symmetry Group ◽  
Planar Graph ◽  
Polyhedral Graph

Every three-connected simple planar graph is a polyhedral graph and a cubic polyhedral graph with pentagonal and hexagonal faces is called as a classical fullerene. The aim of this paper is to survey some results about the symmetry group of cubic polyhedral graphs. We show that the order of symmetry group of such graphs divides 240.

scholarly journals Making a Graph Crossing-Critical by Multiplying its Edges

10.37236/2712 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Laurent Beaudou ◽  
César Hernández-Vélez ◽  
Gelasio Salazar
Keyword(s):  
Crossing Number ◽  
Polyhedral Graph ◽  
Nonplanar Graph

A graph is crossing-critical if the removal of any of its edges decreases its crossing number. This work is motivated by the following question: to what extent is crossing-criticality a property that is inherent to the structure of a graph, and to what extent can it be induced on a noncritical graph by multiplying (all or some of) its edges? It is shown that if a nonplanar graph $G$ is obtained by adding an edge to a cubic polyhedral graph, and $G$ is sufficiently connected, then $G$ can be made crossing-critical by a suitable multiplication of edges.

scholarly journals The number of corner polyhedra graphs

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Clement Dervieux ◽  
Dominique Poulalhon ◽  
Gilles Schaeffer
Keyword(s):  
Binary Trees ◽  
Recursive Decomposition ◽  
Cycle Cover ◽  
Rational Expression ◽  
Compositional Approach ◽  
Polyhedral Graph ◽  
International Audience ◽  
Simply Connected ◽  
Corner Polyhedra

International audience Corner polyhedra were introduced by Eppstein and Mumford (2014) as the set of simply connected 3D polyhedra such that all vertices have non negative integer coordinates, edges are parallel to the coordinate axes and all vertices but one can be seen from infinity in the direction (1, 1, 1). These authors gave a remarkable characterization of the set of corner polyhedra graphs, that is graphs that can be skeleton of a corner polyhedron: as planar maps, they are the duals of some particular bipartite triangulations, which we call hereafter corner triangulations.In this paper we count corner polyhedral graphs by determining the generating function of the corner triangulations with respect to the number of vertices: we obtain an explicit rational expression for it in terms of the Catalan gen- erating function. We first show that this result can be derived using Tutte's classical compositional approach. Then, in order to explain the occurrence of the Catalan series we give a direct algebraic decomposition of corner triangu- lations: in particular we exhibit a family of almond triangulations that admit a recursive decomposition structurally equivalent to the decomposition of binary trees. Finally we sketch a direct bijection between binary trees and almond triangulations. Our combinatorial analysis yields a simpler alternative to the algorithm of Eppstein and Mumford for endowing a corner polyhedral graph with the cycle cover structure needed to realize it as a polyhedral graph.

scholarly journals On Polyhedral Realization with Isosceles Triangles

2021 ◽  
Author(s):  
David Eppstein
Keyword(s):  
Convex Polyhedron ◽  
Infinite Family ◽  
Convex Polyhedra ◽  
Dominating Sets ◽  
Polyhedral Graph ◽  
Bounded Size ◽  
Scalene Triangle ◽  
Isosceles Triangles

AbstractAnswering a question posed by Joseph Malkevitch, we prove that there exists a polyhedral graph, with triangular faces, such that every realization of it as the graph of a convex polyhedron includes at least one face that is a scalene triangle. Our construction is based on Kleetopes, and shows that there exists an integer i such that all convex i-iterated Kleetopes have a scalene face. 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 answer another question of Malkevitch by observing that a spherical tiling of Dawson (Renaissance Banff, Bridges Conference, pp. 489–496, 2005) leads to a fourth infinite family of convex polyhedra in which all faces are congruent isosceles triangles, adding one to the three families previously known to Malkevitch. We prove that the graphs of convex polyhedra with congruent isosceles faces have bounded diameter and have dominating sets of bounded size.

On self duality of pathwidth in polyhedral graph embeddings

2007 ◽  
Vol 55 (1) ◽  
pp. 42-54 ◽  
Author(s):  
Fedor V. Fomin ◽  
Dimitrios M. Thilikos
Keyword(s):  
Graph Embeddings ◽  
Self Duality ◽  
Polyhedral Graph

scholarly journals A nonhamiltonian five-regular multitriangular polyhedral graph

1996 ◽  
Vol 150 (1-3) ◽  
pp. 387-392 ◽  
Author(s):  
Hansjoachim Walther
Keyword(s):  
Polyhedral Graph
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