• Journal of applied mathematics & informatics
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• v.39 no.1_2
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• pp.155-163
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• 2021

For a connected graph G of order n, a set S of vertices is called a double geodetic set of G if for each pair of vertices x, y in G there exist vertices u, v ∈ S such that x, y ∈ I[u, v]. The double geodetic number dg(G) is the minimum cardinality of a double geodetic set. Any double godetic set of cardinality dg(G) is called a dg-set of G. A connected double geodetic set of G is a double geodetic set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected double geodetic set of G is the connected double geodetic number of G and is denoted by dgc(G). A connected double geodetic set of cardinality dgc(G) is called a dgc-set of G. Connected graphs of order n with connected double geodetic number 2 or n are characterized. For integers n, a and b with 2 ≤ a < b ≤ n, there exists a connected graph G of order n such that dg(G) = a and dgc(G) = b. It is shown that for positive integers r, d and k ≥ 5 with r < d ≤ 2r and k - d - 3 ≥ 0, there exists a connected graph G of radius r, diameter d and connected double geodetic number k.

• Honam Mathematical Journal
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• v.24 no.1
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• pp.75-86
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• 2002

A Nielsen number $\bar{N}(f:X-A)$ is a homotopy invariant lower bound for the number of fixed points on X-A where X is a compact connected polyhedron and A is a connected subpolyhedron of X. This number is extended to Nielsen type numbers $\bar{NP_{n}}(f:X-A)$ of least period n and $\bar{N{\phi}_{n}}(f:X-A)$ of the nth iterate on X-A where the subpolyhedron A of a compact connected polyhedron X is no longer path connected.

• Annals of Fuzzy Mathematics and Informatics
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• v.16 no.3
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• pp.301-316
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• 2018

In this paper, the concept of connected geodesic number, $gn_c(G)$, of a fuzzy graph G is introduced and its limiting bounds are identified. It is proved that all extreme nodes of G and all cut-nodes of the underlying crisp graph $G^*$ belong to every connected geodesic cover of G. The connected geodesic number of complete fuzzy graphs, fuzzy cycles, fuzzy trees and of complete bipartite fuzzy graphs are obtained. It is proved that for any pair k, n of integers with $3{\leq}k{\leq}n$, there exists a connected fuzzy graph G : (V, ${\sigma}$, ${\mu}$) on n nodes such that $gn_c(G)=k$. Also, for any positive integers $2{\leq}a<b{\leq}c$, it is proved that there exists a connected fuzzy graph G : (V, ${\sigma}$, ${\mu}$) such that the geodesic number gn(G) = a and the connected geodesic number $gn_c(G)=b$.

• Journal of applied mathematics & informatics
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• v.10 no.1_2
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• pp.27-40
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• 2002

For two integers m, n with m $\leq$ n, an [m,n]-factor F in a graph G is a spanning subgraph of G with m $\leq$ d$\_$F/(v) $\leq$ n for all v ∈ V(F). In 1996, H. Enomoto et al. proved that every 3-connected Planar graph G with d$\_$G/(v) $\geq$ 4 for all v ∈ V(G) contains a [2,3]-factor. In this paper. we extend their result to all 3-connected locally finite infinite planar graphs containing no unbounded faces.

• IE interfaces
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• v.21 no.3
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• pp.254-261
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• 2008

Connected-(r,s)-out of-(m,n) : F system is an important topic in redundancy design of the complex system reliability and it's maintenance policy. Previous studies applied Monte Carlo simulation and genetic, simulated annealing algorithms to tackle the difficulty of maintenance policy problem. These algorithms suggested most suitable maintenance cycle to optimize maintenance pattern of connected-(r,s)-out of-(m,n) : F system. However, genetic algorithm is required long execution time relatively and simulated annealing has improved computational time but rather poor solutions. In this paper, we propose the ant colony optimization approach for connected-(r,s)-out of-(m,n) : F system that determines maintenance cycle and minimum unit cost. Computational results prove that ant colony optimization algorithm is superior to genetic algorithm, simulated annealing and tabu search in both execution time and quality of solution.

• Journal of the Korea Institute of Information and Communication Engineering
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• v.22 no.11
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• pp.1538-1543
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• 2018

For n pairs of points (a,b) on a circle, the line segment to connect two points is called a chord. These chords define a new graph G. Each chord corresponds to a vertex of G, and if two chords intersect, the two vertices corresponding to them are connected by an edge. This makes a graph, called by a circle graph. In this paper, we deal with the problem to find the connected components of a circle graph. The connected component of a graph G is a maximal subgraph H such that any two vertices in H can be connected by a path. When the adjacent matrix of G is given, the problem to find them can be solved by either the depth-first search or the breadth-first search. But when only the information for the chords is given as an input, it takes ${\Omega}(n^2)$ time to obtain the adjacent matrix. In this paper, we do not make the adjacent matrix and develop an $O(n{\log}^2n)$ algorithm for the problem.

• Communications of the Korean Mathematical Society
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• v.27 no.1
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• pp.57-68
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• 2012

The commuting graph of an arbitrary ring R, denoted by ${\Gamma}(R)$, is a graph whose vertices are all non-central elements of R, and two distinct vertices a and b are adjacent if and only if ab = ba. In this paper, we investigate the connectivity, the diameter, the maximum degree and the minimum degree of the commuting graph of group ring $Z_nQ_8$. The main result is that $\Gamma(Z_nQ_8)$ is connected if and only if n is not a prime. If $\Gamma(Z_nQ_8)$ is connected, then diam($Z_nQ_8$)= 3, while $\Gamma(Z_nQ_8)$ is disconnected then every connected component of $\Gamma(Z_nQ_8)$ must be a complete graph with a same size. Further, we obtain the degree of every vertex in $\Gamma(Z_nQ_8)$, the maximum degree and the minimum degree of $\Gamma(Z_nQ_8)$.

• Journal of Korean Institute of Industrial Engineers
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• v.30 no.3
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• pp.250-260
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• 2004

This study considers a linear connected-(r,s)-out-of-(m,n):F lattice system whose components are ordered like the elements of a linear (m,n )-matrix. We assume that all components are in the state 1 (operating) or 0 (failed) and identical and s-independent. The system fails whenever at least one connected (r,s)-submatrix of failed components occurs. The purpose of this paper is to present an optimization scheme that aims at minimizing the expected cost per unit time. To find the optimal threshold of maintenance intervention, we use a genetic algorithm for the cost optimization procedure. The expected cost per unit time is obtained by Monte Carlo simulation. The sensitivity analysis to the different cost parameters has also been made.

• International Journal of Reliability and Applications
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• v.14 no.2
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• pp.71-78
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• 2013

The problem of designing an experiment to estimate the reliability of a system that has N subsystems connected in series where each subsystem n has n $T_n$ components connected in parallel is investigated both theoretically and by simulation. An accelerated sampling sheme is introduced. It is shown that the accelerated sampling scheme is asymptotically optimal as the total number of units goes to infinity. Numerical comparisons for a system that has two subsystems connected in series where each subsystem has two components connected in parallel are also given. They indicate that the accelerated sampling scheme performs better than the batch sequential sampling scheme and is nearly optimal.

• Journal of Communications and Networks
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• v.16 no.3
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• pp.335-343
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• 2014

This paper considers the crucial problem of deploying wireless relays for achieving a connected wireless sensor network in indoor environments, an important aspect related to the management of the sensor network. Several algorithms have been proposed for ensuring full sensing coverage and network connectivity. These algorithms are not applicable to indoor environments because of the complexity of indoor environments, in which a radio signal can be dramatically degraded by obstacles such as walls. We first prove theoretically that the indoor relay placement problem is NP-hard. We then predict the radio coverage of a given relay deployment in indoor environments. We consider two practical scenarios; wire-connected relays and radio-connected relays. For the network with wire-connected relays, we propose an efficient greedy algorithm to compute the deployment locations of relays for achieving the required coverage percentage. This algorithm is proved to provide a $H_n$ factor approximation to the theoretical optimum, where $H_n=1+{\frac{1}{2}}+{\cdots}+{\frac{1}{n}}={\ln}(n)+1$, and n is the number of all grid points. In the network with radio-connected relays, relays have to be connected in an ad hoc mode. We then propose an algorithm based on the previous algorithm for ensuring the connectivity of relays. Experimental results demonstrate that the proposed algorithms achieve better performance than baseline algorithms.