k-tuple total domination in cross products of graphs

Henning, Michael A.; Kazemi, Adel P.

doi:10.1007/s10878-011-9389-z

Cited by 31 publications

(33 citation statements)

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“…In minimum vertex cover problem (Min Vertex Cover), it is required to find a vertex cover of minimum cardinality and let Decide Min Vertex Cover be the decision version of Min Vertex Cover problem. Decide Min Vertex Cover is known to be NP-complete for general graphs [15]. We show that Decide Min k-Tuple Total Dom Set is NPcomplete for split graphs by providing a polynomial time transformation from the Decide Min Vertex Cover problem.…”

Section: Complexity Of Decide Min K-tuple Total Dom Set Problem In Grmentioning

confidence: 99%

“…Kothari Postdoctoral Fellowship (DSKPDF), India. k-tuple total domination has been introduced by Henning and Kazemi [13]. For a fixed positive integer k, a k-tuple total dominating set of a graph G is a subset T D k of V such that every vertex in V is adjacent to at least k vertices of T D k .…”

Section: Introductionmentioning

confidence: 99%

“…So k-tuple total domination is the generalization of the usual total domination. The case when k = 2 is called double total domination [13] where some upper bounds are obtained for the minimum cardinality of a double total dominating set. The k-tuple total domination number of a graph G, denoted by γ ×k,t (G), is the minimum cardinality of a k-tuple total dominating set of G. Upper bounds and lower bounds for γ ×k,t (G) are also proposed in [13,14] in terms of different graph parameters.…”

Section: Introductionmentioning

confidence: 99%

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Algorithmic aspects of k-tuple total domination in graphs

Pradhan

2012

Information Processing Letters

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Section: Complexity Of Decide Min K-tuple Total Dom Set Problem In Grmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Algorithmic aspects of k-tuple total domination in graphs

Pradhan

2012

Information Processing Letters

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“…For a graph

G = (V, E)

, the total k ‐ domination number of G ,

γ_{k}^{t} (G)

, is the cardinality of the smallest vertex set

D \subseteq V

such that

| N_{G} (v) \cap D | \geq k

for each vertex

v \in V

. The following theorem was proved by Henning and Kazemi : Theorem Suppose G is a graph with minimum degree

δ \geq k

, and

0 \leq p \leq 1

. Then

γ_{k}^{t} (G) \leq n (p + \sum_{i = 0}^{k - 1} (k - i) ((), \begin{matrix} δ \\ i \end{matrix}) p^{i} {(1 - p)}^{δ - i}) .

…”

Section: Preliminary Lemmasmentioning

confidence: 99%

Graphs of Large Linear Size Are Antimagic

Eccles¹

2015

Journal of Graph Theory

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Given a graph G = (V, E) and a colouring f : E → N, the induced colour of a vertex v is the sum of the colours at the edges incident with v. If all the induced colours of vertices of G are distinct, the colouring is called antimagic. If G has a bijective antimagic colouring f : E → {1, . . . , |E|}, the graph G is called antimagic. A conjecture of Hartsfield and Ringel states that all connected graphs other than K2 are antimagic. Alon, Kaplan, Lev, Roddity and Yuster proved this conjecture for graphs with minimum degree at least c log |V | for some constant c; we improve on this result, proving the conjecture for graphs with average degree at least some constant d0.

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“…Burchett, Lane, and Lachniet [6] and Burchett [5] found bounds and exact formulas for the k-tuple domination number and k-domination number of the rook's graph in square cases, i.e., K n K n (where kdomination is similar to k-tuple total domination, but only vertices outside of the domination set need to be dominated). The k-tuple total domination number is known for K n × K m [14] and bounds are given for supergeneralized Petersen graphs [21]. In [22], the authors showed that the graph K n K m is an extremal case in the study of kTDS of cartesian product of graphs, motivating the study of the class of rook's graphs.…”

Section: Introductionmentioning

confidence: 99%