• Korean Journal of Mathematics
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• v.31 no.4
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• pp.505-519
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• 2023

Let G = (V, E) be a graph. A subset S of V is called a dominating set of G if every vertex not in S is adjacent to some vertex in S. The domination number γ(G) of G is the minimum cardinality taken over all dominating sets of G. A dominating set S is called a connected dominating set if the subgraph induced by S is connected. The minimum cardinality taken over all connected dominating sets of G is called the connected domination number of G, and is denoted by γ_c(G). In this paper, we investigate the Nordhaus-Gaddum type results for the connected domination number and its derived graphs like line graph, subdivision graph, power graph, block graph and total graph, and characterize the extremal graphs.

• 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.28 no.3_4
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• pp.893-908
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• 2010

In this paper we prove the existence of spanning 3-trees in a 3-connected infinite locally finite VAP-free plane graph. Together with the results of Barnette and the author, this yields that every finite or infinite 3-connected locally finite VAP-free plane graph contains a spanning 3-tree.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.55 no.6
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• pp.1667-1690
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• 2018

We consider the minimum order of a graph G with a given lazy cop number $c_L(G)$. Sullivan, Townsend and Werzanski [7] showed that the minimum order of a connected graph with lazy cop number 3 is 9 and $k_3{\square}k_3$ is the unique graph on nine vertices which requires three lazy cops. They conjectured that for a graph G on n vertices with ${\Delta}(G){\geq}n-k^2$, $c_L(G){\leq}k$. We proved that the conjecture is true for k = 4. Furthermore, we showed that the Petersen graph is the unique connected graph G on 10 vertices with ${\Delta}(G){\leq}3$ having lazy cop number 3 and the minimum order of a connected graph with lazy cop number 4 is 16.

• Communications of the Korean Mathematical Society
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• v.33 no.1
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• pp.361-369
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• 2018

For a given graph G = (V, E), a set $D{\subseteq}V(G)$ is said to be an outer-connected vertex edge dominating set if D is a vertex edge dominating set and the graph $G{\backslash}D$ is connected. The outer-connected vertex edge domination number of a graph G, denoted by ${\gamma}^{oc}_{ve}(G)$, is the cardinality of a minimum outer connected vertex edge dominating set of G. We characterize trees T of order n with l leaves, s support vertices, for which ${\gamma}^{oc}_{ve}(T)=(n-l+s+1)/3$ and also characterize trees with equal domination number and outer-connected vertex edge domination number.

• Journal of KIISE:Computer Systems and Theory
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• v.34 no.10
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• pp.539-554
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• 2007

In this Paper, we investigate conditions for proper interval graphs to have k-disjoint path covers of three types each: one-to-one, one-to-many, and many-to-many. It was proved that for $k{\geq}2$, a proper interval graph is one-to-one k-disjoint path coverable if and only if the graph is k-connected, and is one-to-many k-disjoint path coverable if and only if the graph is k+1-connected. For $k{\geq}3$, a Proper interval graph is (paired) many-to-many k-disjoint path coverable if and only if the graph is 2k-1-connected.

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.6 no.4
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• pp.1188-1202
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• 2012

In this paper, we propose a new multi-scale connected coherence tree algorithm (MCCTA) by improving the connected coherence tree algorithm (CCTA). In contrast to many multi-scale image processing algorithms, MCCTA works on multiple scales space of an image and can adaptively change the parameters to capture the coarse and fine level details. Furthermore, we design a Multi-scale Connected Coherence Tree algorithm plus Spectral graph partitioning (MCCTSGP) by combining MCCTA and Spectral graph partitioning in to a new framework. Specifically, the graph nodes are the regions produced by CCTA and the image pixels, and the weights are the affinities between nodes. Then we run a spectral graph partitioning algorithm to partition on the graph which can consider the information both from pixels and regions to improve the quality of segments for providing image segmentation. The experimental results on Berkeley image database demonstrate the accuracy of our algorithm as compared to existing popular methods.

• Bulletin of the Korean Mathematical Society
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• v.52 no.4
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• pp.1169-1183
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• 2015

We describe generalized Cayley graphs of rectangular groups, so that we obtain (1) an equivalent condition for two Cayley graphs of a rectangular group to be isomorphic to each other, (2) a necessary and sufficient condition for a generalized Cayley graph of a rectangular group to be (strong) connected, (3) a necessary and sufficient condition for the colour-preserving automorphism group of such a graph to be vertex-transitive, and (4) a sufficient condition for the automorphism group of such a graph to be vertex-transitive.

• The Journal of the Korea Contents Association
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• v.22 no.6
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• pp.56-68
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• 2022

Recently, as the demand for real-time processing increases, studies on a dynamic graph that changes over time has been actively done. There is a connected components processing algorithm as one of the algorithms for analyzing dynamic graphs. GPUs are suitable for large-scale graph calculations due to their high memory bandwidth and computational performance. However, when computing the connected components of a dynamic graph using the GPU, frequent data exchange occurs between the CPU and the GPU during real graph processing due to the limited memory of the GPU. The proposed scheme utilizes the Weighted-Quick-Union algorithm to process large-scale graphs on the GPU. It supports fast connected components computation by applying the size to the connected component label. It computes the connected component by determining the parts to be recalculated and minimizing the data to be transmitted to the GPU. In addition, we propose a processing structure in which the GPU and the CPU execute asynchronously to reduce the data transfer time between GPU and CPU. We show the excellence of the proposed scheme through performance evaluation using real dataset.