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$H$-Convex Graphs

For two vertices $u$ and $v$ in 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 convex if $I(S) = S$.
The convexity number $\con(G)$ is the maximum cardinality of
a proper convex set in $G$.
A convex set $S$ is maximum if $|S| = \con (G)$.
The cardinality of a maximum convex set in a graph $G$ is the
convexity number of $G$.
For a nontrivial
connected graph $H$, a connected graph $G$ is an $H$-convex
graph if $G$ contains a maximum convex
set $S$ whose induced subgraph
is $\langle{S}\rangle = H$. It is shown that for every
positive integer $k$, there exist $k$ pairwise
nonisomorphic graphs $H_1, H_2, \cdots, H_k$ of the same order
and a graph $G$ that is
$H_i$-convex for all $i$ ($1 \leq i \leq k$). Also, for every connected
graph $H$
of order $k \geq 3$ with
convexity number 2, it is shown that there exists an $H$-convex
graph of order $n$ for all $n \geq k+1$. More generally, it is shown that
for every nontrivial connected graph $H$, there exists a positive
integer $N$ and an $H$-convex graph of order $n$ for every integer $n \geq
N$.