# Article

 Title: The forcing convexity number of a graph  (English) Author: Chartrand, Gary Author: Zhang, Ping Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 Volume: 51 Issue: 4 Year: 2001 Pages: 847-858 Summary lang: English . Category: math . Summary: For two vertices $u$ and $v$ of a connected graph $G$, the set $I(u, v)$ consists of all those vertices lying on a $u$–$v$ geodesic in $G$. For a set $S$ of vertices of $G$, the union of all sets $I(u,v)$ for $u, v \in S$ is denoted by $I(S)$. A set $S$ is a convex set if $I(S) = S$. The convexity number $\mathop {\mathrm con}(G)$ of $G$ is the maximum cardinality of a proper convex set of $G$. A convex set $S$ in $G$ with $|S| = \mathop {\mathrm con}(G)$ is called a maximum convex set. A subset $T$ of a maximum convex set $S$ of a connected graph $G$ is called a forcing subset for $S$ if $S$ is the unique maximum convex set containing $T$. The forcing convexity number $f(S, \mathop {\mathrm con})$ of $S$ is the minimum cardinality among the forcing subsets for $S$, and the forcing convexity number $f(G, \mathop {\mathrm con})$ of $G$ is the minimum forcing convexity number among all maximum convex sets of $G$. The forcing convexity numbers of several classes of graphs are presented, including complete bipartite graphs, trees, and cycles. For every graph $G$, $f(G, \mathop {\mathrm con}) \le \mathop {\mathrm con}(G)$. It is shown that every pair $a$, $b$ of integers with $0 \le a \le b$ and $b \ge 3$ is realizable as the forcing convexity number and convexity number, respectively, of some connected graph. The forcing convexity number of the Cartesian product of $H \times K_2$ for a nontrivial connected graph $H$ is studied. Keyword: convex set Keyword: convexity number Keyword: forcing convexity number MSC: 05C12 idZBL: Zbl 0995.05044 idMR: MR1864046 . Date available: 2009-09-24T10:47:42Z Last updated: 2012-05-31 Stable URL: http://hdl.handle.net/10338.dmlcz/127690 . Reference: [bh:dg] F. Buckley and F. Harary: Distance in Graphs.Addison-Wesley, Redwood City, CA, 1990. MR 1045632 Reference: [cwz:cn] G. Chartrand, C. E. Wall, and P. Zhang: The convexity number of a graph.(to appear). MR 1913663 Reference: [cz:cs] G. Chartrand and P. Zhang: $H$-convex graphs.Math. Bohem. 126 (2001), 209–220. MR 1826483 Reference: [hn:cg] F. Harary and J. Nieminen: Convexity in graphs.J. Differential Geom. 16 (1981), 185–190. MR 0638785 Reference: [m] H. M. Mulder: The Interval Function of a Graph.Mathematisch Centrum, Amsterdam, 1980. Zbl 0446.05039, MR 0605838 Reference: [n1] L. Nebeský: A characterization of the interval function of a connected graph.Czechoslovak Math. J. 44 (119) (1994), 173–178. MR 1257943 Reference: [n2] L. Nebeský: Characterizing of the interval function of a connected graph.Math. Bohem. 123 (1998), 137–144. MR 1673965 .

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