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Critical concepts in domination

โœ Scribed by David P. Sumner


Publisher
Elsevier Science
Year
1990
Tongue
English
Weight
771 KB
Volume
86
Category
Article
ISSN
0012-365X

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The smallest cardinality of any such dominating set is called the domination number of G and is denoted by y(G). The purpose of this paper is to initiate an investigation of those graphs which are critical in the following sense: For each v, u E V(G) with v not adjacent to u, y(G + vu) < y(G). Thus

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for any vertex x in G. This work considers properties of k-distance domination-critical graphs and establishes a best possible upper bound on the diameter of a 2-distance domination-critical graph G, that is, d(G) โ‰ค 3(ฮณ 2 -1) for ฮณ 2 โ‰ฅ 2.

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We show that for each k L 4 there exists a connected k-domination critical graph with independent domination number exceeding k, thus disproving a conjecture of Sumner and Blitch ( J Cornbinatorial Theory B 34 (19831, 65-76) in all cases except k = 3.