• Journal of the Korea Society of Computer and Information
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• v.18 no.11
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• pp.159-165
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• 2013

This paper proposes a O(E) polynomial-time algorithm that has been devised to simultaneously solve edge-coloring problem and graph classification problem both of which remain NP-complete. The proposed algorithm selects an edge connecting maximum and minimum degree vertices so as to determine the number of edge coloring ${\chi}^{\prime}(G)$. Determined ${\chi}^{\prime}(G)$ is in turn either ${\Delta}(G)$ or ${\Delta}(G)+1$. Eventually, the result could be classified as class 1 if ${\chi}^{\prime}(G)={\Delta}(G)$ and as category 2 if ${\chi}^{\prime}(G)={\Delta}(G)+1$. This paper also proves Vizing's planar graph conjecture, which states that 'all simple, planar graphs with maximum degree six or seven are of class one, closing the remaining possible case', which has known to be NP-complete.

• Journal of KIISE:Software and Applications
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• v.26 no.5
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• pp.692-701
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• 1999

수퍼스칼라(superscalar)나 VLIW 와 같은 명령어 수준 병렬화(ILP) 프로세서의 성능을 극대화하는 과감한 명령어 스케쥴링은 소프트웨어 파이프라이닝과같은 스케쥴링 과정을 거치면서 일반적인 복사 명령어 제거 기법으로 없앨 수 없는 서로 간섭하는 복사 명령을 많이 만들어내는데 루프 내부에 생성된 이러한 복사명령은 적절한 루프 펼침을 수행하여 간섭관계를 없앰으로서 제거할 수 있다. 본 논문에서는 이와 같이 루프 펼침이 수행된 루프 내부의 복사명령을 제거하는 기법으로 그래프 컬러링 상에 구현한 낙관적 융합기법을 제안한다. 그래프 컬러링에서의 융합기법은 간선의 개수가 많은 노드를 만들어 낼수 있으므로 채색성에 부정적인 영향을 주는 것으로 알려져 왔으나 본 기법에서는 융합되는 노드에 동시에 간섭하는 노드의 간선의 수가 줄어드는 긍정적인 영향을 최대한 이용하여 채색성을 높이고 융합된 노드에 대한 실제 버림(spill)이 일어나는 경우 유효 범위 분절(live range splitting)을 통하여 버림의 부담을 최대한 줄이도록 하였으며 이를 VLIW 스케쥴링 된 SPEC 정수벤치마크 루프내부의 복사 명령 제거에 적용한 결과 제거 가능한 복사 명령의 99%를 제거하면서도 버림명령은 다른 융합 기법과 비교하여 가장 적게 발생하는 우수한 결과를 얻을수 있었다.

• The Journal of the Institute of Internet, Broadcasting and Communication
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• v.14 no.5
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• pp.263-269
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• 2014

In this paper, I disprove Hadwiger conjecture of the vertex coloring problem, which asserts that "All $K_k$-minor free graphs can be colored with k-1 number of colors, i.e., ${\chi}(G)=k$ given $K_k$-minor." Pursuant to Hadwiger conjecture, one shall obtain an NP-complete k-minor to determine ${\chi}(G)=k$, and solve another NP-complete vertex coloring problem as a means to color vertices. In order to disprove Hadwiger conjecture in this paper, I propose an algorithm of linear time complexity O(V) that yields the exact solution to the vertex coloring problem. The proposed algorithm assigns vertex with the minimum degree to the Maximum Independent Set (MIS) and repeats this process on a simplified graph derived by deleting adjacent edges to the MIS vertex so as to finally obtain an MIS with a single color. Next, it repeats the process on a simplified graph derived by deleting edges of the MIS vertex to obtain an MIS whose number of vertex color corresponds to ${\chi}(G)=k$. Also presented in this paper using the proposed algorithm is an additional algorithm that searches solution of ${\chi}^{{\prime}{\prime}}(G)$, the total chromatic number, which also remains NP-complete. When applied to a $K_4$-minor graph, the proposed algorithm has obtained ${\chi}(G)=3$ instead of ${\chi}(G)=4$, proving that the Hadwiger conjecture is not universally applicable to all the graphs. The proposed algorithm, however, is a simple algorithm that directly obtains an independent set minor of ${\chi}(G)=k$ to assign an equal color to the vertices of each independent set without having to determine minors in the first place.

• Journal of the Korea Society of Computer and Information
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• v.16 no.6
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• pp.179-186
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• 2011

This paper suggests the minimum number of vertex guards algorithm. Given n rooms, the exact number of minimum vertex guards is proposed. However, only approximation algorithms are presented about the maximum number of vertex guards for polygon and orthogonal polygon without or with holes. Fisk suggests the maximum number of vertex guards for polygon with n vertices as follows. Firstly, you can triangulate with n-2 triangles. Secondly, 3-chromatic vertex coloring of every triangulation of a polygon. Thirdly, place guards at the vertices which have the minority color. This paper presents the minimum number of vertex guards using dominating set. Firstly, you can obtain the visibility graph which is connected all edges if two vertices can be visible each other. Secondly, you can obtain dominating set from visibility graph or visibility matrix. This algorithm applies various art galley problems. As a results, the proposed algorithm is simple and can be obtain the minimum number of vertex guards.

• Journal of the Korea Society of Computer and Information
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• v.16 no.7
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• pp.85-93
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• 2011

The Vertex Coloring Problem hasn't been solved in polynomial time, so this problem has been known as NP-complete. This paper suggests linear time algorithm for Vertex Coloring Problem (VCP). The proposed algorithm is based on assumption that we can't know a priori the minimum chromatic number ${\chi}(G)$=k for graph G=(V,E) This algorithm divides Vertices V of graph into two parts as independent sets $\overline{C}$ and cover set C, then assigns the color to $\overline{C}$. The element of independent sets $\overline{C}$ is a vertex ${\upsilon}$ that has minimum degree ${\delta}(G)$ and the elements of cover set C are the vertices ${\upsilon}$ that is adjacent to ${\upsilon}$. The reduced graph is divided into independent sets $\overline{C}$ and cover set C again until no edge is in a cover set C. As a result of experiments, this algorithm finds the ${\chi}(G)$=k perfectly for 26 Graphs that shows the number of selecting ${\upsilon}$ is less than the number of vertices n.