• The KIPS Transactions:PartA
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• v.13A no.3 s.100
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• pp.253-260
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• 2006

In this paper, we analyze the Hamiltonian property of Pyramid graphs. We prove that it is always possible to construct a Hamiltonian cycle of length $(4^N-1)/3$ by applying the proposed algorithm to construct series of cycle expansion operations into two adjacent cycles in the Pyramid graph of height N.

• Proceedings of the Korea Information Processing Society Conference
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• 2005.11a
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• pp.867-870
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• 2005

본 논문에서는 피라미드 그래프를 대상으로 링을 임베딩하는 문제를 다룬다. 사이클 확장 연산을 이용하는 사이클의 크기를 확대시켜 나가는 일련의 과정을 통하여 최대 크기의 링을 의미하는 헤밀톤 사이클을 찾을 수 있는 알고리즘을 제시함으로써 임의의 높이 N인 피라미드 그래프 내에 길이 $4^N-1/3$인 링을 임베딩 할 수 있음을 증명한다.

• Journal of Korea Multimedia Society
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• v.13 no.8
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• pp.1183-1193
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• 2010

The pyramid graph is well known in parallel processing as a interconnection network topology based on regular square mesh and tree architectures. The enhanced pyramid graph is an alternative architecture by exchanging mesh into the corresponding torus on the base for upgrading performance than the pyramid. In this paper, we adopt a strategy of classification into two disjoint groups of edges in regular square torus as a basic sub-graph constituting of each layer in the enhanced pyramid graph. Edge set in the torus graph is considered as two disjoint sub-sets called NPC(represents candidate edge for neighbor-parent) and SPC(represents candidate edge for shared-parent) whether the parents vertices adjacent to two end vertices of the corresponding edge have a relation of neighbor or sharing in the upper layer of the enhanced pyramid graph. In addition, we also introduce a notion of shrink graph to focus only on the NPC-edges by hiding SPC-edges within the shrunk super-vertex on the resulting shrink graph. In this paper, we analyze that the lower and upper bounds on the number of NPC-edges in a Hamiltonian cycle constructed on $2^n{\times}2^n$ torus is $2^{2n-2}$ and $3{\cdot}2^{2n-2}$ respectively. By expanding this result into the enhanced pyramid graph, we also prove that the maximum number of NPC-edges containable in a Hamiltonian cycle is $4^{n-1}$-2n+1 in the n-dimensional enhanced pyramid.

• The Journal of the Korea Contents Association
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• v.9 no.12
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• pp.582-591
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• 2009

The pyramid graph is an interconnection network topology based on regular square mesh and tree structures. In this paper, we adopt a strategy of classification into two disjoint groups of edges in regular square mesh as a base sub-graph constituting of each layer in the pyramid graph. Edge set in the mesh can be divided into two disjoint sub-sets called as NPC(represents candidate edge for neighbor-parent) and SPC(represents candidate edge for shared-parent) whether the parents vertices adjacent to two end vertices of the corresponding edge have a relation of neighbor or shared in the upper layer of pyramid graph. In addition, we also introduce a notion of shrink graph to focus only on the NPC-edges by hiding SPC-edges in the original graph within the shrunk super-vertex on the resulting graph. In this paper, we analyze that the lower and upper bound on the number of NPC-edges in a Hamiltonian cycle constructed on $2^n\times2^n$ mesh is $2^{2n-2}$ and $3*(2^{2n-2}-2^{n-1})$ respectively. By expanding this result into the pyramid graph, we also prove that the maximum number of NPC-edges containable in a Hamiltonian cycle is $4^{n-1}-3*2^{n-1}$-2n+7 in the n-dimensional pyramid.

• Proceedings of the Korea Information Processing Society Conference
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• 2009.04a
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• pp.981-983
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• 2009

피라미드 그래프는 정방형 메쉬와 트리 구조를 기반으로 하는 상호연결망 토폴로지다. 정방형 메쉬 내에서 NPC-간선은 해당 메쉬를 피라미드의 부그래프 관점에서 해석할 때 NPC-간선의 양 끝 노드들의 부모 노드들이 상위 계층의 메쉬 부그래프 내에서 서로 인접하게 되는 간선으로써 사이클 확장이나 고장허용 특성의 관점에서 중요한 의미를 갖는 간선이다. 본 연구에서는 $2^n{\times}2^n$ 2-차원 정방형 메쉬 내에서 헤밀톤 사이클 구성 시 포함할 수 있는 NPC-간선 개수의 하한값이 $2^{2n-2}$임을 분석한다.