# The Strong Convexity Spectra of Grids

Ontology type: schema:ScholarlyArticle      Open Access: True

### Article Info

DATE

2017-06-01

AUTHORS ABSTRACT

Let D be a connected oriented graph. A set S⊆V(D)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S \subseteq V(D)$$\end{document} is convex in D if, for every pair of vertices x,y∈S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x, y \in S$$\end{document}, the vertex set of every xy-geodesic, (xy shortest directed path) and every yx-geodesic in D is contained in S. The convexity number, con(D)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {con}(D)$$\end{document}, of a non-trivial oriented graph, D, is the maximum cardinality of a proper convex set of D. The strong convexity spectrum of the graph G, SSC(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{SC} (G)$$\end{document}, is the set {con(D):Dis a strong orientation of G}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\hbox {con}(D) :\ D \hbox { is a strong orientation of G} \}$$\end{document}. In this paper we prove that the problem of determining the convexity number of an oriented graph is NP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {NP}$$\end{document}-complete, even for bipartite oriented graphs of arbitrary large girth, extending previous known results for graphs. We also determine SSC(Pn□Pm)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{SC} (P_n \Box P_m)$$\end{document}, for every pair of integers n,m≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n,m \ge 2$$\end{document}. More... »

PAGES

689-713

### References to SciGraph publications

• 2002-05. The Convexity Number of a Graph in GRAPHS AND COMBINATORICS
• 2003-09. Some Remarks on the Convexity Number of a Graph in GRAPHS AND COMBINATORICS
• 2011-04-22. On the Convexity Number of Graphs in GRAPHS AND COMBINATORICS
• ### Journal

TITLE

Graphs and Combinatorics

ISSUE

4

VOLUME

33

### Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s00373-017-1805-4

DOI

http://dx.doi.org/10.1007/s00373-017-1805-4

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1085738672

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