• Journal of the Korean Data and Information Science Society
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• v.22 no.6
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• pp.1275-1281
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• 2011

Many authors studied cooperative games that arise from variants of dominating set games on graphs. In wireless networks, the connected dominating set is used to reduce routing table size and communication cost. In this paper, we introduce a connected dominating set game to model the cost allocation problem arising from a connected dominating set on a given graph and study its core. In addition, we give a polynomial time algorithm for determining the balancedness of the game on a tree, for finding a element of the core.

• Communications of the Korean Mathematical Society
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• v.14 no.2
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• pp.295-299
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• 1999

Every quasi-random graph G(n) on n vertices consists of a giant component plus o(n) vertices, and every quasi-random graph G(n) with minimum degree (1+o(1))\ulcorner is connected.

• Kyungpook Mathematical Journal
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• v.55 no.2
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• pp.279-287
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• 2015

A subset S of V (G), where G is a graph without isolated vertices, is a double dominating set of G if for each $x{{\in}}V(G)$, ${\mid}N_G[x]{\cap}S{\mid}{\geq}2$. This paper, shows that any positive integers a, b and n with $2{\leq}a<b$, $b{\geq}2a$ and $n{\geq}b+2a-2$, can be realized as domination number, double domination number and order, respectively. It also characterize the double dominating sets in the Cartesian and tensor products of two graphs and determine sharp bounds for the double domination numbers of these graphs. In particular, it show that if G and H are any connected non-trivial graphs of orders n and m respectively, then ${\gamma}_{{\times}2}(G{\square}H){\leq}min\{m{\gamma}_2(G),n{\gamma}_2(H)\}$, where ${\gamma}_2$, is the 2-domination parameter.

• Bulletin of the Korean Mathematical Society
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• v.55 no.5
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• pp.1397-1404
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• 2018

For a connected graph G, an r-maximum edge-coloring is an edge-coloring f defined on E(G) such that at every vertex v with $d_G(v){\geq}r$ exactly r incident edges to v receive the maximum color. The r-maximum index $x^{\prime}_r(G)$ is the least number of required colors to have an r-maximum edge coloring of G. In this paper, we show how the r-maximum index is affected by adding an edge or a vertex. As a main result, we show that for each $r{\geq}3$ the r-maximum index function over the graphs admitting an r-maximum edge-coloring is unbounded and the range is the set of natural numbers. In other words, for each $r{\geq}3$ and $k{\geq}1$ there is a family of graphs G(r, k) with $x^{\prime}_r(G(r,k))=k$. Also, we construct a family of graphs not admitting an r-maximum edge-coloring with arbitrary maximum degrees: for any fixed $r{\geq}3$, there is an infinite family of graphs ${\mathcal{F}}_r=\{G_k:k{\geq}r+1\}$, where for each $k{\geq}r+1$ there is no r-maximum edge-coloring of $G_k$ and ${\Delta}(G_k)=k$.

• Journal of applied mathematics & informatics
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• v.24 no.1_2
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• pp.141-154
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• 2007

The connectivity problem is a fundamental problem in graph theory. The best known algorithm to solve the connectivity problem on general graphs with n vertices and m edges takes $O(K(G)mn^{1.5})$ time, where K(G) is the vertex connectivity of G. In this paper, an efficient algorithm is designed to solve vertex connectivity problem, which takes $O(n^2)$ time and O(n) space for a trapezoid graph.

• 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.

• Journal of the Korean Mathematical Society
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• v.42 no.6
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• pp.1251-1264
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• 2005

Let D be an acyclic digraph. The competition graph of D has the same set of vertices as D and an edge between vertices u and v if and only if there is a vertex x in D such that (u, x) and (v, x) are arcs of D. The competition number of a graph G, denoted by k(G), is the smallest number k such that G together with k isolated vertices is the competition graph of an acyclic digraph. It is known to be difficult to compute the competition number of a graph in general. Even characterizing the graphs with competition number one looks hard. In this paper, we continue the work done by Cho and Kim[3] to characterize the graphs with one hole and competition number one. We give a sufficient condition for a graph with one hole to have competition number one. This generates a huge class of graphs with one hole and competition number one. Then we completely characterize the graphs with one hole and competition number one that do not have a vertex adjacent to all the vertices of the hole. Also we show that deleting pendant vertices from a connected graph does not change the competition number of the original graph as long as the resulting graph is not trivial, and this allows us to construct infinitely many graph having the same competition number. Finally we pose an interesting open problem.

• Journal of applied mathematics & informatics
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• v.30 no.5_6
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• pp.793-807
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• 2012

If G is any connected graph of order p; then the thorn graph $G_p^*$ with code ($n_1$, $n_2$, ${\cdots}$, $n_p$) is obtained by adding $n_i$ pendent vertices to each $i^{th}$ vertex of G. By treating the pendent edge of a thorn graph as $P_2$, $K_2$, $K_{1,1}$, $K_1{\circ}K_1$ or $P_1{\circ}K_1$, we generalize a thorn graph by replacing $P_2$ by $P_m$, $K_2$ by $K_m$, $K_{1,1}$ by $K_{m,n}$, $K_1{\circ}K_1$ by $K_m{\circ}K_1$ and $P_1{\circ}K_1$ by $P_m{\circ}K_1$ and their respective generalized thorn graphs are denoted by $G_P$, $G_K$, $G_B$, $G_{KK}$ and $G_{PK}$ respectively. Many chemical compounds can be treated as $G_P$, $G_K$, $G_B$, $G_{KK}$ and $G_{PK}$ of some graphs in graph theory. In this paper, we obtain the bounds of the wiener index for these generalization of thorn graphs.

• Honam Mathematical Journal
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• v.41 no.1
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• pp.67-87
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• 2019

The main aim of this study is to characterize new classes of multicone graphs which are determined by their adjacency spectra, their Laplacian spectra, their complement with respect to signless Laplacian spectra and their complement with respect to their adjacency spectra. A multicone graph is defined to be the join of a clique and a regular graph. If n is a positive integer, a friendship graph $F_n$ consists of n edge-disjoint triangles that all of them meet in one vertex. It is proved that any connected graph cospectral to a multicone graph $F_n{\nabla}F_n=K_2{\nabla}nK_2{\nabla}nK_2$ is determined by its adjacency spectra as well as its Laplacian spectra. In addition, we show that if $n{\neq}2$, the complement of these graphs are determined by their adjacency spectra. At the end of the paper, it is proved that multicone graphs $F_n{\nabla}F_n=K_2{\nabla}nK_2{\nabla}nK_2$ are determined by their signless Laplacian spectra and also we prove that any graph cospectral to one of multicone graphs $F_n{\nabla}F_n$ is perfect.

• Journal of applied mathematics & informatics
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• v.41 no.3
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• pp.511-524
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• 2023

An L(p₁, p₂, p₃, . . . , p_m)-labeling of a graph G is an assignment of non-negative integers, called as labels, to the vertices such that the vertices at distance i should have at least pi as their label difference. If p₁ = 4, p₂ = 3, p₃ = 2, p₄ = 1, then it is called a L(4, 3, 2, 1)-labeling which is widely studied in the literature. A L(4, 3, 2, 1)-path coloring of graphs, is a labeling g : V (G) → Z⁺ such that there exists at least one path P between every pair of vertices in which the labeling restricted to this path is a L(4, 3, 2, 1)-labeling. This concept was defined and results for some simple graphs were obtained by the same authors in an earlier article. In this article, we study the concept of L(4, 3, 2, 1)-path coloring for complete bipartite graphs, 2-edge connected split graph, Cartesian product and join of two graphs and prove an existence theorem for the same.