• Journal of the Korean Society for Precision Engineering
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• v.22 no.9 s.174
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• pp.202-211
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• 2005

This paper introduces and illustrates the results of a new method fer offsetting triangular mesh by moving all vertices along the multiple normal vectors of a vertex. The multiple normal vectors of a vertex are set the same as the normal vectors of the faces surrounding the vertex, while the two vectors with the smallest difference are joined repeatedly until the difference is smaller than allowance. Offsetting with the multiple normal vectors of a vertex does not create a gap or overlap at the smooth edges, thereby making the mesh size uniform and the computation time short. In addition, this offsetting method is accurate at the sharp edges because the vertices are moved to the normal directions of faces and joined by the blend surface. The method is also useful for rapid prototyping and tool path generation if the triangular mesh is tessellated part of the solid models with curved surfaces and sharp edges. The suggested method and previous methods are implemented on a PC using C++ and illustrated using an OpenGL library.

• Bulletin of the Korean Mathematical Society
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• v.58 no.1
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• pp.31-46
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• 2021

Let a and b be positive integers, and let V (G), ��(G), and ��2(G) be the vertex set of a graph G, the minimum degree of G, and the minimum degree sum of two non-adjacent vertices in V (G), respectively. An even [a, b]-factor of a graph G is a spanning subgraph H such that for every vertex v ∈ V (G), dH(v) is even and a ≤ dH(v) ≤ b, where dH(v) is the degree of v in H. Matsuda conjectured that if G is an n-vertex 2-edge-connected graph such that $n{\geq}2a+b+{\frac{a^2-3a}{b}}-2$, ��(G) ≥ a, and ${\sigma}_2(G){\geq}{\frac{2an}{a+b}}$, then G has an even [a, b]-factor. In this paper, we provide counterexamples, which are highly connected. Furthermore, we give sharp sufficient conditions for a graph to have an even [a, b]-factor. For even an, we conjecture a lower bound for the largest eigenvalue in an n-vertex graph to have an [a, b]-factor.

• Journal of the Korean Mathematical Society
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• v.59 no.4
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• pp.757-774
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• 2022

Let a and b be positive even integers. An even [a, b]-factor of a graph G is a spanning subgraph H such that for every vertex v ∈ V (G), d_H(v) is even and a ≤ dH(v) ≤ b. Let κ(G) be the minimum size of a vertex set S such that G - S is disconnected or one vertex, and let σ₂(G) = min_uv∉E(G) (d(u)+d(v)). In 2005, Matsuda proved an Ore-type condition for an n-vertex graph satisfying certain properties to guarantee the existence of an even [2, b]-factor. In this paper, we prove that for an even positive integer b with b ≥ 6, if G is an n-vertex graph such that n ≥ b + 5, κ(G) ≥ 4, and σ₂(G) ≥ ${\frac{8n}{b+4}}$, then G contains an even [4, b]-factor; each condition on n, κ(G), and σ₂(G) is sharp.

• The KIPS Transactions:PartA
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• v.14A no.7
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• pp.435-442
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• 2007

This paper is concerned with the mesh segmentation problem that can be applied to diverse applications such as texture mapping, simplification, morphing, compression, and shape matching for 3D mesh models. The mesh segmentation is the process of dividing a given mesh into the disjoint set of sub-meshes. We propose a method for segmenting meshes by simultaneously reflecting global and local geometric characteristics of the meshes. First, we extract sharp vertices over mesh vertices by interpreting the curvatures and convexity of a given mesh, which are respectively contained in the local and global geometric characteristics of the mesh. Next, we partition the sharp vertices into the $\kappa$ number of clusters by adopting the $\kappa$-means clustering method [29] based on the Euclidean distances between all pairs of the sharp vertices. Other vertices excluding the sharp vertices are merged into the nearest clusters by Euclidean distances. Also we implement the proposed method and visualize its experimental results on several 3D mesh models.

• International Journal of Contents
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• v.5 no.4
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• pp.55-61
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• 2009

In this paper, we propose a hierarchical method for segmenting a given 3D mesh, which hierarchically clusters sharp vertices of the mesh using the metric of geodesic distance among them. Sharp vertices are extracted from the mesh by analyzing convexity that reflects global geometry. As well as speeding up the computing time, the sharp vertices of this kind avoid the problem of local optima that may occur when feature points are extracted by analyzing the convexity that reflects local geometry. For obtaining more effective results, the sharp vertices are categorized according to the priority from the viewpoint of cognitive science, and the reasonable number of clusters is automatically determined by analyzing the geometric features of the mesh.

• Journal of the Korean Mathematical Society
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• v.60 no.5
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• pp.959-986
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• 2023

Zhai and Lin recently proved that if G is an n-vertex connected 𝜃(1, 2, r + 1)-free graph, then for odd r and n ⩾ 10r, or for even r and n ⩾ 7r, one has ${\rho}(G){\leq}{\sqrt{{\lfloor}{\frac{n^2}{4}}{\rfloor}}}$, and equality holds if and only if G is $K_{{\lceil}{\frac{n}{2}}{\rceil},{\lfloor}{\frac{n}{2}}{\rfloor}}$. In this paper, for large enough n, we prove a sharp upper bound for the spectral radius in an n-vertex H-free non-bipartite graph, where H is 𝜃(1, 2, 3) or 𝜃(1, 2, 4), and we characterize all the extremal graphs. Furthermore, for n ⩾ 137, we determine the maximum number of edges in an n-vertex 𝜃(1, 2, 4)-free non-bipartite graph and characterize the unique extremal graph.

• Bulletin of the Korean Mathematical Society
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• v.58 no.1
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• pp.171-181
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• 2021

The fixing number of a graph G is the smallest order of a subset S of its vertex set V (G) such that the stabilizer of S in G, ��S(G) is trivial. Let G1 and G2 be the disjoint copies of a graph G, and let g : V (G1) → V (G2) be a function. A functigraph FG consists of the vertex set V (G1) ∪ V (G2) and the edge set E(G1) ∪ E(G2) ∪ {uv : v = g(u)}. In this paper, we study the behavior of fixing number in passing from G to FG and find its sharp lower and upper bounds. We also study the fixing number of functigraphs of some well known families of graphs like complete graphs, trees and join graphs.

• The Journal of the Korea Contents Association
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• v.8 no.5
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• pp.94-103
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• 2008

In this paper, we adapt the $\kappa$-means clustering technique to segmenting a given 3D mesh. In order to avoid the locally minimal convergence and speed up the computing time, first we extract sharp vertices from the mesh by analysing its curvature and convexity that respectively reflect the local and global geometric characteristics from the viewpoint of cognitive science. Next the sharp vertices are partitioned into $\kappa$ clusters by iterated converging with the $\kappa$-means clustering method based on the geodesic distance instead of the Euclidean distance between each pair of the sharp vertices. For obtaining the effective result of $\kappa$-means clustering method, it is crucial to assign an initial value to $\kappa$ appropriately. Hence, we automatically compute a reasonable number of clusters as an initial value of $\kappa$. Finally the mesh segmentation is completed by merging other vertices except the sharp vertices into the nearest cluster by geodesic distance.

• ETRI Journal
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• v.33 no.1
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• pp.89-99
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• 2011

We construct surfaces with darts, creases, and corners by blending different types of local geometries. We also render these surfaces efficiently using programmable graphics hardware. Points on the blending surface are evaluated using simplified computation which can easily be performed on a graphics processing unit. Results show an eighteen-fold to twenty-fold increase in rendering speed over a CPU version. We also demonstrate how these surfaces can be trimmed using textures.

• East Asian mathematical journal
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• v.30 no.3
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• pp.355-360
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• 2014

Let D be a finite digraph with the vertex set V (D) and the arc set A(D). A function f : $V(D){\rightarrow}\{-1,\;1\}$ defined on the vertices of a digraph D is called a bad function if $f(N^-(v)){\leq}1$ for every v in D. The weight of a bad function is $f(V(D))=\sum\limits_{v{\in}V(D)}f(v)$. The maximum weight of a bad function of D is the the negative decision number ${\beta}_D(D)$ of D. Wang [4] studied several sharp upper bounds of this number for an undirected graph. In this paper, we study sharp upper bounds of the negative decision number ${\beta}_D(D)$ of for a digraph D.