Domination number of Cartesian products of directed cycles

Zhang, Xindong; Liu, Juan; Chen, Xing; Meng, Jixiang

doi:10.1016/j.ipl.2010.10.001

Cited by 15 publications

(11 citation statements)

References 8 publications

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“…Cartesian products of paths) are widely studied because they represent interconnection models of multiprocessors in VLSI systems. The domination numbers of the Cartesian products, for several fixed values of k, were computed for P n 2P k in [1,8,11,19,20], for C n 2C k in [14,35,52] and for P n 2C k in [42]. A general O(log n) algorithm based on path algebra in [36], can be used to compute the domination number of P n 2P k , for any fixed k. This algorithm can also be used to compute distance based invariants [32] and domination numbers [53] in polygraphs in constant time, that is, the algorithm can find closed formulas for arbitrary values of n. The existence of an algorithm that can provide closed formulas for the domination numbers of all grid graphs (P n 2P k ) has been observed or claimed in [17,40].…”

Section: Introductionmentioning

confidence: 99%

Roman Domination Number of the Cartesian Products of Paths and Cycles

Pavlič

Žerovnik

2012

Electron. J. Combin.

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Roman domination is an historically inspired variety of domination in graphs, in which vertices are assigned a value from the set $\{0,1,2\}$ in such a way that every vertex assigned the value 0 is adjacent to a vertex assigned the value 2. The Roman domination number is the minimum possible sum of all values in such an assignment. Using an algebraic approach we present an $O(C)$-time algorithm for computing the Roman domination numbers of special classes of graphs called polygraphs, which include rotagraphs and fasciagraphs. Using this algorithm we determine formulas for the Roman domination numbers of the Cartesian products of the form $P_n\Box P_k$, $P_n\Box C_k$, for $k\leq8$ and $n \in {\mathbb N}$, and $C_n\Box P_k$ and $C_n\Box C_k$, for $k\leq 6$ and $n \in {\mathbb N}$, for paths $P_n$ and cycles $C_n$. We also find all special graphs called Roman graphs in these families of graphs.

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Section: Introductionmentioning

confidence: 99%

Roman Domination Number of the Cartesian Products of Paths and Cycles

Pavlič

Žerovnik

2012

Electron. J. Combin.

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“…e domination number [38][39][40][41], respectively, the total domination number [42] of the Cartesian product of two directed cycles are investigated.…”

Section: Introductionmentioning

confidence: 99%

Feedback Arc Number and Feedback Vertex Number of Cartesian Product of Directed Cycles

Chen

2019

Discrete Dynamics in Nature and Society

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For a digraph D, the feedback vertex number τD, (resp. the feedback arc number τ′D) is the minimum number of vertices, (resp. arcs) whose removal leaves the resultant digraph free of directed cycles. In this note, we determine τD and τ′D for the Cartesian product of directed cycles D=Cn1→□Cn2→□…Cnk→. Actually, it is shown that τ′D=n1n2…nk∑i=1k1/ni, and if nk≥…≥n1≥3 then τD=n2…nk.

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“…Klavžar and Seifter (1995) computed the exact value of γ (C n C m ) for 3 ≤ m ≤ 5. The domination number of Cartesian product of two directed cycles has been recently investigated Zhang et al 2010;Shaheen 2009;Mollard 2013). Even more recently, Liu et al (2011) began the study of the domination number of Cartesian product of two directed paths − → P n and − → P n .…”

Section: Introductionmentioning

confidence: 99%

Total and paired domination numbers of $$C_m$$ C m bundles over a cycle $$C_n$$ C n

Sohn

Chen

2015

J Comb Optim

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Let G = (V, E) be a simple graph without isolated vertices. A set S of vertices is a total dominating set of a graph G if every vertex of G is adjacent to some vertex in S. A paired dominating set of G is a dominating set whose induced subgraph has a perfect matching. The minimum cardinality of a total dominating set (respectively, a paired dominating set) is the total domination number (respectively, the paired domination number). Hu and Xu (J Combin Optim 27(2):369-378, 2014) computed the exact values of total and paired domination numbers of Cartesian product C n C m for m = 3, 4. Graph bundles generalize the notions of covering graphs and Cartesian products. In this paper, we generalize these results given in Hu and Xu (J Combin Optim 27(2):369-378, 2014) to graph bundle and compute the total domination number and the paired domination number of C m bundles over a cycle C n for m = 3, 4. Moreover, we give the exact value for the total domination number of Cartesian product C n C 5 and some upper bounds of C m bundles over a cycle C n where m ≥ 5.

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Domination number of Cartesian products of directed cycles

Cited by 15 publications

References 8 publications

Roman Domination Number of the Cartesian Products of Paths and Cycles

Roman Domination Number of the Cartesian Products of Paths and Cycles

Feedback Arc Number and Feedback Vertex Number of Cartesian Product of Directed Cycles

Total and paired domination numbers of $$C_m$$ C m bundles over a cycle $$C_n$$ C n

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