Graphs &amp; Digraphs

Chartrand, Gary; Lesniak, Linda; Zhang, Ping

doi:10.1201/b14892

Cited by 299 publications

(296 citation statements)

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“…Let K n , K h,n−h , K 1,n−1 , P n , W n and C n denote complete graph, complete bipartite graph, star, path, wheel and cycle, respectively, which the order of each is n. For undefined terminology and notation, we refer the reader to [7].…”

Section: Basic Terminologymentioning

confidence: 99%

Some Families of Graphs with Small Power Domination Number

Shahbaznejad,

Kazemi,

Pelayo

2021

Preprint

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Let G be a graph with the vertex set V (G) and S be a subset of V (G). Let cl(S) be the set of vertices built from S, by iteratively applying the following propagation rule: if a vertex and all of its neighbors except one of them are in cl(S), then the exceptional neighbor is also in cl(S). A set S is called a zero forcing set of G if cl(S) = V (G). The zero forcing number Z(G) of G is the minimum cardinality of a zero forcing set. Let cl(N [S]) be the set of vertices built from the closed neighborhood N [S] of S, by iteratively applying the previous propagation rule. A set S is called a power dominating set of G if cl(N [S]) = V (G). The power domination number γ p (G) of G is the minimum cardinality of a power dominating set. In this paper, we present some families of graphs that their power domination number is 1 or 2.

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Section: Basic Terminologymentioning

confidence: 99%

Some Families of Graphs with Small Power Domination Number

Shahbaznejad,

Kazemi,

Pelayo

2021

Preprint

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“…We provide some basic background for the paper in this section. We shall now list below some basic definitions and results of crisp graph, fuzzy subset and fuzzy graph from [5,25,20], respectively. We concern with a fuzzy graph which is defined on a crisp graph.…”

Section: Preliminariesmentioning

confidence: 99%

“…Since 1977, when Cockayne and Hedetniemi ( [7], Section 3, p. 249-251) presented a survey of domination results, domination theory has received considerable attention. A set S of vertices of G ( [5], Chap. 10, p. 302) is a dominating set if every vertex in V (G) − S is adjacent to at least one vertex in S. The minimum cardinality among the dominating sets of G is called the domination number of G and is denoted by γ(G).…”

Section: Introductionmentioning

confidence: 99%

Nikfar Domination in Fuzzy Graphs

Garrett

2019

Preprint

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We introduce a new variation on the domination theme which we call nikfar domination as reducing waste of time in transportation planning. We determine the nikfar domination number  v for several classes of fuzzy graphs. The bounds is obtained for it. We prove both of the Vizing's conjecture and the Grarier-Khelladi's conjecture are monotone decreasing fuzzy graph property for nikfar domination. We obtain Nordhaus-Gaddum (NG) type results for these parameters. Finally, we discuss about nikfar dominating set of a fuzzy tree by using the bridges and -strong edges equivalence.

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“…Our terminology and notation are standard except as indicated. A good reference for undefined terms is [3].…”

Section: Introductionmentioning

confidence: 99%

A generalization of a theorem of Nash-Williams

Bauer¹,

Lesniak²,

Schmeichel³

2021

Preprint

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In [4], Chvátal gave a well-known sufficient condition for a graphical sequence to be forcibly hamiltonian, and showed that in some sense his condition is best possible. Nash-Williams [7] gave examples of forcibly hamiltonian n−sequences that do not satisfy Chvátal's condition, for every n ≥ 5. In this note we generalize the Nash-Williams examples, and use this generalization to generate Ω( 2 n √ n ) forcibly hamiltonian n−sequences that do not satisfy Chvátal's condition.

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Graphs & Digraphs

Cited by 299 publications

References 0 publications

Some Families of Graphs with Small Power Domination Number

Some Families of Graphs with Small Power Domination Number

Nikfar Domination in Fuzzy Graphs

A generalization of a theorem of Nash-Williams

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