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Polyhedra of Small Order and Their Hamiltonian Properties
, 1995
"... ... The results of the enumeration were used to systematically search for certain smallest nonHamiltonian polyhedral graphs. In particular, the smallest nonHamiltonian planar graphs satisfying certain toughnesslike properties are presented here, as are the smallest nonHamiltonian, 3connected, D ..."
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Cited by 8 (1 self)
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... The results of the enumeration were used to systematically search for certain smallest nonHamiltonian polyhedral graphs. In particular, the smallest nonHamiltonian planar graphs satisfying certain toughnesslike properties are presented here, as are the smallest nonHamiltonian, 3connected, Delaunay tessellations and triangulations. Improved upper and lower bounds on the size of the smallest nonHamiltonian, inscribable polyhedra are also given.
Contractions, Removals and How to Certify 3Connectivity in Linear Time
"... One of the most noted construction methods of 3vertexconnected graphs is due to Tutte and based on the following fact: Any 3vertexconnected graph G = (V, E) on more than 4 vertices contains a contractible edge, i. e., an edge whose contraction generates a 3connected graph. This implies the exis ..."
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Cited by 1 (1 self)
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One of the most noted construction methods of 3vertexconnected graphs is due to Tutte and based on the following fact: Any 3vertexconnected graph G = (V, E) on more than 4 vertices contains a contractible edge, i. e., an edge whose contraction generates a 3connected graph. This implies the existence of a sequence of edge contractions from G to the complete graph K4, such that every intermediate graph is 3vertexconnected. A theorem of Barnette and Grünbaum gives a similar sequence using removals on edges instead of contractions. We show how to compute both sequences in optimal time, improving the previously best known running times of O(V  2) to O(E). This result has a number of consequences; an important one is a new lineartime test of 3connectivity that is certifying; finding such an algorithm has been a major open problem in the design of certifying algorithms in the last years. The test is conceptually different from wellknown lineartime 3connectivity tests and uses a certificate that is easy to verify in time O(E). We show how to extend the results to an optimal certifying test of 3edgeconnectivity. 1
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"... Recursive generation of simple planar quadrangulations with vertices of degree 3 and 4 ..."
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Recursive generation of simple planar quadrangulations with vertices of degree 3 and 4
Balanced Cayley graphs and balanced planar graphs
, 707
"... A balanced graph is a bipartite graph with no induced circuit of length 2 (mod 4). These graphs arise in linear programming. We focus on graphalgebraic properties of balanced graphs to prove a complete classification of balanced Cayley graphs on abelian groups. Moreover, in Section 5 of this paper, ..."
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A balanced graph is a bipartite graph with no induced circuit of length 2 (mod 4). These graphs arise in linear programming. We focus on graphalgebraic properties of balanced graphs to prove a complete classification of balanced Cayley graphs on abelian groups. Moreover, in Section 5 of this paper, we prove that there is no cubic balanced planar graph. Finally, some remarkable conjectures for balanced regular graphs are also presented. Key words: Cayley graph, balanced graph 1
Journal of Graph Algorithms and Applications
, 2010
"... Recursive generation of simple planar 5regular graphs and pentangulations ..."
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Recursive generation of simple planar 5regular graphs and pentangulations