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The Convexity Number of a Graph

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 $G$, while
for a set $S$
of vertices of $G$, the set $I(S)$ is the union of all
sets $I(u,v)$ for $u, v \in S$. A set $S$ is convex if $I(S) = S$.
The convexity number $\con(G)$ of $G$ is
the maximum cardinality of a proper convex set of $G$.
The clique number $\om(G)$ is the maximum
cardinality of a clique in $G$.
A convex set of cardinality $\con(G)$ is called a maximum
convex set, while a set of $\om(G)$ vertices
that induces a clique is called a maximum clique set.
If $G$ is a connected
graph of order $n$ that is not complete, then $n \geq 3$ and $2 \leq
\om(G) \leq \con(G) \leq n-1$.
It is shown that for every triple $\ell, k,n$ of integers with
$n \geq 3$ and $2 \leq \ell \leq k \leq n-1$,
there exists a noncomplete connected graph
$G$ of order $n$ with $\om (G) = \ell$ and $\con (G) = k$.
Other results are presented concerning graphs containing maximum clique sets
and maximum convex sets satisfying a variety of conditions.