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Moo Young Sohn - One of the best experts on this subject based on the ideXlab platform.
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Bounds on locating total domination number of the Cartesian Product of cycles and paths
Information Processing Letters, 2015Co-Authors: Huaming Xing, Moo Young SohnAbstract:The problem of placing monitoring devices in a system in such a way that every site in the safeguard system (including the monitors themselves) is adjacent to a monitor site can be modeled by total domination in graphs. Locating-total dominating sets are of interest when the intruder/fault at a vertex precludes its detection in that location. A total dominating set S of a graph G with no isolated vertex is a locating-total dominating set of G if for every pair of distinct vertices u and v in V - S are totally dominated by distinct subsets of the total dominating set. The locating-total domination number of a graph G is the minimum cardinality of a locating-total dominating set of G. In this paper, we study the bounds on locating-total domination numbers of the Cartesian Product C m ? P n of cycles C m and paths P n . Exact values for the locating-total domination number of the Cartesian Product C 3 ? P n are found, and it is shown that for the locating-total domination number of the Cartesian Product C 4 ? P n this number is between ? 3 n 2 ? and ? 3 n 2 ? + 1 with two sharp bounds. We show that the locating-total domination number of the Cartesian Product C 3 ? P n is equal to n + 1 .We show that for the locating-total domination number of the Cartesian Product C 4 ? P n this number is between ? 3 n 2 ? and ? 3 n 2 ? + 1 with two sharp bounds.
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Bounds on locating-total domination number of the Cartesian Product of cycles
Ars Combinatoria, 2014Co-Authors: Huaming Xing, Moo Young SohnAbstract:A total dominating set S of a graph G with no isolated vertex is a locating-total dominating set of G if for every pair of distinct vertices u and v in V - S are totally dominated by distinct subsets of the total dominating set. The minimum cardinality of a locating-total dominating set is the locating-total domination number. In this paper, we obtain new upper bounds for locating-total domination numbers of the Cartesian Product of cycles C-m and C-n and prove that for any positive integer n >= 3, the locating-total domination numbers of the Cartesian Product of cycles C-3 and C-n is equal to n for n equivalent to 0 (mod 6) or n + 1 otherwise.
Huaming Xing - One of the best experts on this subject based on the ideXlab platform.
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Bounds on locating total domination number of the Cartesian Product of cycles and paths
Information Processing Letters, 2015Co-Authors: Huaming Xing, Moo Young SohnAbstract:The problem of placing monitoring devices in a system in such a way that every site in the safeguard system (including the monitors themselves) is adjacent to a monitor site can be modeled by total domination in graphs. Locating-total dominating sets are of interest when the intruder/fault at a vertex precludes its detection in that location. A total dominating set S of a graph G with no isolated vertex is a locating-total dominating set of G if for every pair of distinct vertices u and v in V - S are totally dominated by distinct subsets of the total dominating set. The locating-total domination number of a graph G is the minimum cardinality of a locating-total dominating set of G. In this paper, we study the bounds on locating-total domination numbers of the Cartesian Product C m ? P n of cycles C m and paths P n . Exact values for the locating-total domination number of the Cartesian Product C 3 ? P n are found, and it is shown that for the locating-total domination number of the Cartesian Product C 4 ? P n this number is between ? 3 n 2 ? and ? 3 n 2 ? + 1 with two sharp bounds. We show that the locating-total domination number of the Cartesian Product C 3 ? P n is equal to n + 1 .We show that for the locating-total domination number of the Cartesian Product C 4 ? P n this number is between ? 3 n 2 ? and ? 3 n 2 ? + 1 with two sharp bounds.
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Bounds on locating-total domination number of the Cartesian Product of cycles
Ars Combinatoria, 2014Co-Authors: Huaming Xing, Moo Young SohnAbstract:A total dominating set S of a graph G with no isolated vertex is a locating-total dominating set of G if for every pair of distinct vertices u and v in V - S are totally dominated by distinct subsets of the total dominating set. The minimum cardinality of a locating-total dominating set is the locating-total domination number. In this paper, we obtain new upper bounds for locating-total domination numbers of the Cartesian Product of cycles C-m and C-n and prove that for any positive integer n >= 3, the locating-total domination numbers of the Cartesian Product of cycles C-3 and C-n is equal to n for n equivalent to 0 (mod 6) or n + 1 otherwise.
Mohammadreza Doostmohammadian - One of the best experts on this subject based on the ideXlab platform.
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Recovering the Structural Observability of Composite Networks via Cartesian Product
IEEE Transactions on Signal and Information Processing over Networks, 2020Co-Authors: Mohammadreza DoostmohammadianAbstract:Observability is a fundamental concept in system inference and estimation. This article is focused on structural observability analysis of Cartesian Product networks. Cartesian Product networks emerge in variety of applications including in parallel and distributed systems. We provide a structural approach to extend the structural observability of the constituent networks (referred as the factor networks) to that of the Cartesian Product network. The structural approach is based on graph theory and is generic. We introduce certain structures which are tightly related to structural observability of networks, namely parent Strongly-Connected-Component (parent SCC), parent node, and contractions. The results show that for particular type of networks (e.g. the networks containing contractions) the structural observability of the factor network can be recovered via Cartesian Product. In other words, if one of the factor networks is structurally rank-deficient, using the other factor network containing a spanning cycle family, then the Cartesian Product of the two networks is structurally full-rank. We define certain network structures for structural observability recovery. On the other hand, we derive the number of observer nodes–the node whose state is measured by an output– in the Cartesian Product network based on the number of observer nodes in the factor networks. An example illustrates the graph-theoretic analysis in the article.
Xuding Zhu - One of the best experts on this subject based on the ideXlab platform.
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Total weight choosability of Cartesian Product of graphs
European Journal of Combinatorics, 2012Co-Authors: Tsai-lien Wong, Xuding ZhuAbstract:A graph G=(V,E) is called (k,k^')-choosable if the following is true: for any total list assignment L which assigns to each vertex x a set L(x) of k real numbers, and assigns to each edge e a set L(e) of k^' real numbers, there is a mapping f:V@?E->R such that f(y)@?L(y) for any y@?V@?E and for any two adjacent vertices x,x^', @?"e"@?"E"("x")f(e)+f(x) @?"e"@?"E"("x"^"'")f(e)+f(x^'). In this paper, we prove that if G is the Cartesian Product of an even number of even cycles, or the Cartesian Product of an odd number of even cycles and at least one of the cycles has length 4n for some positive integer n, then G is (1,3)-choosable. In particular, hypercubes of even dimension are (1,3)-choosable. Moreover, we prove that if G is the Cartesian Product of two paths or the Cartesian Product of a path and an even cycle, then G is (1,3)-choosable. In particular, Q"3 is (1,3)-choosable.
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game coloring the Cartesian Product of graphs
Journal of Graph Theory, 2008Co-Authors: Xuding ZhuAbstract:This article proves the following result: Let G and G′ be graphs of orders n and n′, respectively. Let G* be obtained from G by adding to each vertex a set of n′ degree 1 neighbors. If G* has game coloring number m and G′ has acyclic chromatic number k, then the Cartesian Product GsG′ has game chromatic number at most k(k + m - 1). As a consequence, the Cartesian Product of two forests has game chromatic number at most 10, and the Cartesian Product of two planar graphs has game chromatic number at most 105. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 261–278, 2008
Xuluo Yin - One of the best experts on this subject based on the ideXlab platform.
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the thickness of the Cartesian Product of two graphs
Canadian Mathematical Bulletin, 2016Co-Authors: Yichao Chen, Xuluo YinAbstract:The thickness of a graph is the minimum number of planar subgraphs whose union is . A -minimal graph is a graph of thickness that contains no proper subgraph of thickness . In this paper, upper and lower bounds are obtained for the thickness, , of the Cartesian Product of two graphs and , in terms of the thickness and . Furthermore, the thickness of the Cartesian Product of two planar graphs and of a -minimal graph and a planar graph are determined. By using a new planar decomposition of the complete bipartite graph , the thickness of the Cartesian Product of two complete bipartite graphs and is also given for .
