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The Convexity Spectra and the Strong Convexity Spectra of Graphs

Given a connected oriented graph D, we say that a subset S of V(D) is convex in D if, for every pair of vertices x, y in S, the vertex set of every x-y geodesic (x-y shortest dipath) and y-x geodesic in D is contained in S. The convexity number con (D) of a nontrivial connected oriented graph D is the maximum cardinality of a proper convex set of D.
Let S_{C}(K_{n})={con(D)|D is an orientation of K_{n}} and S_{SC}(K_{n})={con(D)|D is a strong orientation of K_{n}}. We show that S_{C}(K_{3})={1,2} and S_{C}(K_{n})={1,3,4,...,n-1} if n >= 4. We also have that S_{SC}(K_{3})={1} and S_{SC}(K_{n})={1,3,4,...,n-2} if n >= 4 .
We also show that every triple n, m, k of integers with n >= 5, 3 <= k <= n-2, and n+1 <= m <= n(n-1)/2, there exists a strong connected oriented graph D of order n with |E(D)|=m and con (D)=k.

Identiferoai:union.ndltd.org:NSYSU/oai:NSYSU:etd-0728105-104156
Date28 July 2005
CreatorsYen, Pei-lan
ContributorsSen-Peng Eu, Tsai-Lien Wong, Li-Da Tong, Xuding Zhu
PublisherNSYSU
Source SetsNSYSU Electronic Thesis and Dissertation Archive
LanguageEnglish
Detected LanguageEnglish
Typetext
Formatapplication/pdf
Sourcehttp://etd.lib.nsysu.edu.tw/ETD-db/ETD-search/view_etd?URN=etd-0728105-104156
Rightsunrestricted, Copyright information available at source archive

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