A note on dominating sets and average distance

DeLaViña, Ermelinda; Pepper, Ryan; Waller, Bill

doi:10.1016/j.disc.2008.03.018

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2010

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Cited by 9 publications

(2 citation statements)

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“…A simple proof of the above lemma was given in [9]. The lemma below appears to be folklore (its proof is very similar to the proof of Lemma 2 in [8]).…”

Section: Lemmasmentioning

confidence: 81%

Lower bounds for the domination number

DeLaViña

Pepper

Waller

2010

Discuss. Math. Graph Theory

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Abstract. In this note, we prove several lower bounds on the domination number of simple connected graphs. Among these are the following: the domination number is at least two-thirds of the radius of the graph, three times the domination number is at least two more than the number of cut-vertices in the graph, and the domination number of a tree is at least as large as the minimum order of a maximal matching.

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“…A simple proof of the above lemma was given in [9]. The lemma below appears to be folklore (its proof is very similar to the proof of Lemma 2 in [8]).…”

Section: Lemmasmentioning

confidence: 81%

Lower bounds for the domination number

DeLaViña

Pepper

Waller

2010

Discuss. Math. Graph Theory

Self Cite

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“…Our result for k = 1, i.e., for the ordinary domination number, is one of the two main results in [5]. We note that a related result, a bound on the average distance in terms of the total domination number, was given in [8].…”

Section: Introductionmentioning

confidence: 84%

Average distance and generalised packing in graphs

Dankelmann

2010

Discrete Mathematics

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a b s t r a c tLet G be a connected finite graph. The average distance µ(G) of G is the average of the distances between all pairs of vertices of G. For a positive integer k, a k-packing of G is a subset S of the vertex set of G such that the distance between any two vertices in S is greater than k. The k-packing number β k (G) of G is the maximum cardinality of a k-packing of G. We prove upper bounds on the average distance in terms of β k (G) and show that for fixed k the bounds are, up to an additive constant, best possible. As a corollary, we obtain an upper bound on the average distance in terms of the k-domination number, the smallest cardinality of a set S of vertices of G such that every vertex of G is within distance k of some vertex of S.

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