• 대한수학회지
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• 제56권5호
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• pp.1247-1263
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• 2019

In ordered trees, two randomly chosen vertices are said to be dependent if one lies under the other. If not, we say that they are independent. We consider several classes of ordered trees with uniform updegree requirements and find the generating functions for the trees with two marked dependent/independent vertices. As a result, we compute the probability for two vertices being dependent/independent. We also count such trees by the distance between two independent vertices.

• 한국전산응용수학회:학술대회논문집
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• 한국전산응용수학회 2003년도 KSCAM 학술발표회 프로그램 및 초록집
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• pp.16.1-16
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• 2003

A subset S of vertices of a graph G is independent if no two vertices of S are adjacent by an edge in G. Also we say that S is maximal independent if it is contained In no larger independent set in G. A planted plane tree is a tree that is embedded in the plane and rooted at an end-vertex. A (k＋1) -valent tree is a planted plane tree in which each vertex has degree one or (k＋1). We classify maximal independent sets of (k＋1) -valent trees into two groups, namely, type A and type B maximal independent sets and consider specific independent sets of these trees. We study relations among these three types of independent sets. Using the relations, we count the number of all maximal independent sets of (k＋1) -valent trees with n vertices of degree (k＋1).

• 대한수학회보
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• 제33권1호
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• pp.119-126
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• 1996

In this paper we study the asymptotic behavior of the edge independence number of a random (n,n)-tree. The tools we use include the matrix-tree theorem, the probabilistic method and Hall's theorem. We begin with some definitions. An (n,n)_tree T is a connected, acyclic, bipartite graph with n light and n dark vertices (see [Pa92]). A subset M of edges of a graph is called independent(or matching) if no two edges of M are adfacent. A subset S of vertices of a graph is called independent if no two vertices of S are adjacent. The edge independence number of a graph T is the number $\beta_1(T)$ of edges in any largest independent subset of edges of T. Let $\Gamma(n,n)$ denote the set of all (n,n)-tree with n light vertices labeled 1, $\ldots$, n and n dark vertices labeled 1, $\ldots$, n. We give $\Gamma(n,n)$ the uniform probability distribution. Our aim in this paper is to find bounds on $\beta_1$(T) for a random (n,n)-tree T is $\Gamma(n,n)$.

• 한국수학교육학회지시리즈A:수학교육
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• 제43권2호
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• pp.177-186
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• 2004

In this paper, we introduce new functional definitions for school geometry based on DGS (dynamic geometry system) teaching-learning environment. For the vertices forming a geometric figure, we first consider the relationship between the independent vertices and dependent vertices, and using this relationship and educational considerations in DGS, we introduce functional definitions for the geometric figures in terms of its independent vertices. For this purpose, we design a new DGS called JavaMAL MicroWorld. Based on the needs of new definitions in DGS environment for the student's construction activities in learning geometry, we also design a new DGS based geometry curriculum in which the definitions of the school geometry are newly defined and reconnected in a new way. Using these funct onal definitions, we have taught the new geometry contents emphasizing the sequential expressions for the student's geometric activities.

• 대한수학회논문집
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• 제37권4호
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• pp.969-975
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• 2022

Let R be a finite commutative ring with nonzero unity and let Z(R) be the zero divisors of R. The total graph of R is the graph whose vertices are the elements of R and two distinct vertices x, y ∈ R are adjacent if x + y ∈ Z(R). The total graph of a ring R is denoted by 𝜏(R). The independence number of the graph 𝜏(R) was found in [11]. In this paper, we again find the independence number of 𝜏(R) but in a different way. Also, we find the independent dominating number of 𝜏(R). Finally, we examine when the graph 𝜏(R) is well-covered.

• 한국인터넷방송통신학회논문지
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• 제19권3호
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• pp.193-199
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• 2019

본 논문은 지금까지 NP-완전인 난제로 알려진 최대 독립집합(MIS) 문제를 선형시간 복잡도로 해결한 알고리즘을 제안하였다. 제안된 알고리즘은 "MIS 집합의 모든 정점들은 상호간에 연결되지 않는다"는 기본 성질을 적용하여 n개의 정점으로 구성된 그래프에서 최소 차수 ${\delta}(G)$ 정점 ${\nu}$를 선택하고 부속 간선을 제거하였을 때 차수가 변하지 않는 정점들을 차수 오름차순으로 계속적으로 선택하는 단순한 방법을 적용하였다. 제안된 알고리즘을 22개 그래프에 적용한 결과, 시각적으로 그래프를 보면서도 MIS를 쉽게 찾을 수 있는 장점을 갖고 있으며, 알고리즘은 항상 MIS 집합의 원소 개수인 ${\alpha}(G)$회를 수행하여 알고리즘 복잡도는 O(n)으로 선형 알고리즘이다. 결국, 제안된 MIS 알고리즘은 MIS의 최적 해를 도출하는 일반적인 알고리즘으로 적용할 수 있을 것이다.

• 한국수학교육학회지시리즈A:수학교육
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• 제12권2호
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• pp.1-4
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• 1974

Matroid theory, which was first introduced in 1935 by Whitney (2), is a branch of combinational mathematics which has some very much to the fore in the last few years. H. Whitney had just spent several years working in the field of graph theory, and had noticed several similarities between the ideas of independence and rank in graph theory and those of linear independence and dimension in the study of vector spaces. A matroid is essentially a set with some kind of 'independence structure' defined on it. There are several known results concerning how matroids can be induced from given matroid by a digraph. The purpose of this note is to show that, given a matroid M$_{0}$ (N) and a digraph $\Gamma$(N), then a new matroid M(N) is induced, where A⊆N is independent in M(N) if and only if A is the set of initial vertices of a family of pairwise-vertex-disjoint paths with terminal vertices independent in M$_{0}$ (N).(N).

• 대한수학회지
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• 제39권2호
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• pp.205-219
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• 2002

We deride formulas for the expected values $\mu$(n) of the independent domination numbers of a random planted plane tree and a random trivalent tree with n vertices, respectively, and we determine the asymptotic behavior of $\mu$(n) as n goes to infinity.

• 대한수학회지
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• 제41권5호
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• pp.921-931
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• 2004

We derive a formula for the expected value $\mu$(n) of the independent domination number of a random directed rooted tree with n labeled vertices and determine the asymptotic behavior of $\mu$(n) as n goes to infinity.

• 한국컴퓨터정보학회논문지
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• 제16권7호
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• pp.85-93
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• 2011

본 논문은 지금까지 NP-완전인 난제로 알려진 정점 색칠 문제를 선형시간 복잡도로 해결한 알고리즘을 제안하였다. 제안된 알고리즘은 그래프 G=(V,E)의 최소 채색수 ${\chi}(G)$=k를 결정하기 위해 사전에 k값을 알지 못한다는 가정에 기반하고 있다. 단지 주어진 그래프를 독립집합 $\overline{C}$와 정점 피복 집합 C로 정확히 양분하여 $\overline{C}$에 색을 배정하는 방법을 적용하였다. 독립집합 $\overline{C}$의 원소는 ${\delta}(G)$인 정점 ${\upsilon}$가, C의 원소는 정점 ${\upsilon}$의 인접 정점들 u가배정된다. 축소된 그래프 C는 다시 $\overline{C}$와 C로 양분되며, 이 과정을 C의 간선이 없을 때까지 수행한다. 26개의 다양한 그래프를 대상으로 제안된 알고리즘을 적용한 결과 정점 ${\upsilon}$를 선택하는 횟수는 정점의 수 n보다 작은 값을 나타내었으며, ${\chi}(G)$=k를 찾는데 성공하였다.