• Korean Journal of Mathematics
• /
• 제31권4호
• /
• pp.505-519
• /
• 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
• /
• 제16권3호
• /
• pp.301-316
• /
• 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
• /
• 제28권3_4호
• /
• pp.893-908
• /
• 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
• /
• 제39권1_2호
• /
• pp.155-163
• /
• 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.

• 대한수학회보
• /
• 제55권6호
• /
• pp.1667-1690
• /
• 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.

• 대한수학회논문집
• /
• 제33권1호
• /
• pp.361-369
• /
• 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.

• 한국정보과학회논문지:시스템및이론
• /
• 제34권10호
• /
• pp.539-554
• /
• 2007

이 논문에서는 진구간 그래프(proper interval graph)가 각각 일대일, 일대다. 다대다 k-서로소인 경로 커버를 가질 조건을 고찰한다. 진구간 그래프는 $k{\geq}2$인 경우, k-연결되어 있는 경우에만 일대일 k-서로소인 경로 커버를 가지며, k+1-연결되어 있는 경우에만 일대다 k-서로소인 경로 커버를 가짐을 증명하였다. 그리고 $k{\geq}3$일 때 진구간 그래프는 2k-1-연결되어 있는 경우에만 (쌍형) 다대다 k-서로소인 경로 커버를 가진다.

• KSII Transactions on Internet and Information Systems (TIIS)
• /
• 제6권4호
• /
• pp.1188-1202
• /
• 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.

• 대한수학회보
• /
• 제52권4호
• /
• pp.1169-1183
• /
• 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.

• 한국콘텐츠학회논문지
• /
• 제22권6호
• /
• pp.56-68
• /
• 2022

최근 실시간 처리의 요구가 증가하면서 시간에 따라서 변화하는 동적 그래프에 관한 연구가 활발하게 진행되고 있다. 동적 그래프를 분석하기 위한 알고리즘의 하나로 연결 요소가 있다. GPU는 높은 메모리 대역폭, 연산 성능으로 대규모의 그래프 계산에 적합하다. 그러나 동적 그래프의 연결 요소를 GPU를 이용하여 처리할 때, GPU의 제한된 메모리로 인해 실제 그래프 처리 시 CPU와 GPU 간에 잦은 데이터 교환이 발생한다. 본 논문에서는 동적 그래프에서 GPU 기반의 효율적인 점진적 연결 요소 처리 기법을 제안한다. 제안하는 기법은 Weighted-Quick-Union 알고리즘을 기반으로 연결 요소 레이블에 구성 요소의 개수를 이용하여 연결 요소를 빠르게 계산한다. 또한, 재계산할 부분을 판별하여 GPU로 전송할 데이터를 최소화하여 대규모 그래프에 대하여 CPU와 GPU 간의 데이터 교환 횟수를 감소시킨다. 뿐만 아니라 GPU와 CPU 간에 데이터 전송 시간 낭비를 줄이기 위해 GPU와 CPU가 비동기로 실행하는 처리 구조를 제안한다. 실제 데이터 집합을 사용한 성능 평가를 통해 제안하는 기법의 우수성을 입증한다.