• Journal of applied mathematics & informatics
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• v.41 no.4
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• pp.839-848
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• 2023

Let G be a simple undirected graph. A planar graph known as a Halin graph(HG) is characterised by having three connected and pendent vertices of a tree that are connected by an outer cycle. A subset S of V is said to be a dominating set of the graph G if each vertex u that is part of V is dominated by at least one element v that is a part of S. The domination number of a graph is denoted by the γ(G), and it corresponds to the minimum size of a dominating set. A dominating set S is called a secure dominating set if for each v ∈ V＼S there exists u ∈ S such that v is adjacent to u and S₁ = (S＼{v}) ∪ {u} is a dominating set. The minimum cardinality of a secure dominating set of G is equal to the secure domination number γ_s(G). In this article we found the secure domination number of Halin graph(HG) with perfet k-ary tree and also we determined secure domination of rooted product of special trees.

• Journal of applied mathematics & informatics
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• v.30 no.5_6
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• pp.871-882
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• 2012

For a connected graph G of order $n$, an ordered set $S=\{u_1,u_2,{\cdots},u_k\}$ of vertices in G is a linear edge geodetic set of G if for each edge $e=xy$ in G, there exists an index $i$, $1{\leq}i$ < $k$ such that e lie on a $u_i-u_{i+1}$ geodesic in G, and a linear edge geodetic set of minimum cardinality is the linear edge geodetic number $leg(G)$ of G. A graph G is called a linear edge geodetic graph if it has a linear edge geodetic set. The linear edge geodetic numbers of certain standard graphs are obtained. Let $g_l(G)$ and $eg(G)$ denote the linear geodetic number and the edge geodetic number, respectively of a graph G. For positive integers $r$, $d$ and $k{\geq}2$ with $r$ < $d{\leq}2r$, there exists a connected linear edge geodetic graph with rad $G=r$, diam $G=d$, and $g_l(G)=leg(G)=k$. It is shown that for each pair $a$, $b$ of integers with $3{\leq}a{\leq}b$, there is a connected linear edge geodetic graph G with $eg(G)=a$ and $leg(G)=b$.

• Journal of applied mathematics & informatics
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• v.14 no.1_2
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• pp.185-191
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• 2004

Given a connected graph G, we say that a set EC\;{\subseteq}\;V(G)$ is convex in G if, for every pair of vertices x, $y\;{\in}\;C$, the vertex set of every x - y geodesic in G is contained in C. The convexity number of G is the cardinality of a maximal proper convex set in G. In this paper, we show that every pair k, n of integers with $2\;{\leq}k\;{\leq}\;n\;-\;1$ is realizable as the convexity number and order, respectively, of some connected triangle-free graph, and give a lower bound for the convexity number of k-regular graphs of order n with n > k+1.

• Kyungpook Mathematical Journal
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• v.16 no.2
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• pp.243-246
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• 1976

• Kyungpook Mathematical Journal
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• v.11 no.2
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• pp.169-172
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• 1971

• Journal of the Chungcheong Mathematical Society
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• v.20 no.3
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• pp.343-348
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• 2007

In this paper, we show that if a homeomorphism has the pseudo-orbit-tracing-property and its nonwandering set is locally connected, then the limit sets of wandering points whose stable sets have nonempty interior consist of single periodic orbit.

• Journal of the Chungcheong Mathematical Society
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• v.23 no.4
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• pp.871-878
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• 2010

In this paper, we show that if a homeomorphism has the pseudo-orbit-tracing-property and its nonwandering set is locally connected, then the points whose stable sets have nonempty interior are periodic points.

• Journal of the Korean Institute of Intelligent Systems
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• v.16 no.6
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• pp.683-690
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• 2006

Reducing power consumption to extend network lifetime is one of the most important challenges in designing wireless sensor networks. One promising approach to conserving system energy is to keep only a minimal number of sensors active and put others into low-powered sleep mode, while the active sensors can maintain a connected covet set for the target area. The problem of computing such minimum working sensor set is NP-hard. In this paper, a centralized Voronoi tessellation (CVT) based approximate algorithm is proposed to construct the near optimal cover set. When sensor's communication radius is at least twice of its sensing radius, the covet set is connected at the same time; In case of sensor's communication radius is smaller than twice of its sensing radius, a connection scheme is proposed to calculate the assistant nodes needed for constructing the connectivity of the cover set. Finally, the performance of the proposed algorithm is evaluated through theoretical analysis and extensive numerical experiments. Experimental results show that the proposed algorithm outperforms the greedy algorithm in terms of the runtime and the size of the constructed connected cover set.

• Proceedings of the IEEK Conference
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• 1998.10a
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• pp.731-734
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• 1998

A scalable parallel algorithm is proposed for efficient image component labeling with local operatos on a mesh connected SIMD computer. In contrast to the conventional parallel labeling algorithms, where a single pixel is assigned to each PE, the algorithm presented here is scalable and can assign m$\times$m pixel set to each PE according to the input image size. The assigned pixel set is converted to a single pixel that has representative value, and the amount of the required memory and processing time can be highly reduced. For N$\times$N image, if m$\times$m pixel set is assigned to each PE of P$\times$P mesh, where P=N/m, the time complexity due to the communication of each PE and the computation complexity are reduced to O(PlogP) bit operations and O(P) bit operations, respectively, which is 1/m of each of the conventional method. This method also diminishes the amount of memory in each PE to O(P), and can decrease the number of PE to O(P2) =Θ(N2/m2) as compared to O(N2) of conventional method. Because the proposed parallel labeling algorithm is scalable, we can adapt to the increase of image size without the hardware change of the given mesh connected SIMD computer.

• Journal of the Chungcheong Mathematical Society
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• v.28 no.1
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• pp.139-143
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• 2015

In this paper, we prove that (r, s)-semi-irresolute image of the (r, s)-semi-${\theta}$-connected set is also (r, s)-semi-${\theta}$-connected. Moreover, we prove that an (r, s)-semi-irresolute mapping preserves (r; s)-$S^*$-closedness.