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On diameter critical graphs

✍ Scribed by Louis Caccetta; Roland Häggkvist

Publisher
Elsevier Science
Year
1979
Tongue
English
Weight
520 KB
Volume
28
Category
Article
ISSN
0012-365X

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📜 SIMILAR VOLUMES

On diameter 2-critical graphs
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✍ Genghua Fan 📂 Article 📅 1987 🏛 Elsevier Science 🌐 English ⚖ 296 KB

A graph is diameter 2-critical if the graph has diameter 2 and the deletion of any edge increases its diameter. We prove that if G is diameter 2-critical graph on n vertices and e edges, then (i) e ~< [14n2 ] for n <~ 24, and (ii) e < !4n2 + (n 2 -16.2 n + 56)/320 (<0.2532 n2), for n/> 25.

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## Abstract Let __G__ be connected simple graph with diameter __d__(__G__). __G__ is said __v__^+^‐critical if __d__(__G__−__v__) is greater than __d__(__G__) for every vertex __v__ of __G__. Let D′ = max {__d__(__G__−__v__) : __v__ ∈ __V__(__G__)}. Boals et al. [Congressus Numerantium 72 (1990), 1

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## Abstract We prove that the minimum number of edges in a vertex‐diameter‐2‐critical graph on __n__ ≥ 23 vertices is (5__n__ − 17)/2 if __n__ is odd, and is (5__n__/2) − 7 if __n__ is even. © 2005 Wiley Periodicals, Inc. J Graph Theory

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## Abstract A graph is __k__‐domination‐critical if γ(__G) = k__, and for any edge __e__ not in __G__, γ(__G + e) = k__ − 1. In this paper we show that the diameter of a domination __k__‐critical graph with __k__ ≧ 2 is at most 2__k__ − 2. We also show that for every __k__ ≧ 2, there is a __k__‐dom

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A graph is vertex-critical if deleting any vertex increases its diameter. We construct, for each & 5 except &=6, a vertex-critical graph of diameter two on & vertices with at least , where c 2 is some constant. We also construct, for each & 5 except &=6, a vertex-critical graph of diameter two on &