On the hull number of some graph classes
Theoretical Computer Science
On the Carathéodory number of interval and graph convexities
Theoretical Computer Science
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A set of vertices $C$ in a graph is convex if it contains all vertices which lie on shortest paths between vertices in $C$. The convex hull of a set of vertices $S$ is the smallest convex set containing $S$. The hull number $h(G)$ of a graph $G$ is the smallest cardinality of a set of vertices whose convex hull is the vertex set of $G$. For a connected triangle-free graph $G$ of order $n$ and diameter $d$ at least 4, we prove that $h(G)\leq(n-d+3)/3$ if $G$ has minimum degree at least 3 and that $h(G)\leq2(n-d+5)/7$, if $G$ is cubic. Furthermore for a connected graph $G$ of order $n$, girth $g$ at least 5, minimum degree at least 2, and diameter $d$, we prove $h(G)\leq2+(n-d-1)/\left\lceil\frac{g-1}{2}\right\rceil$. All bounds are best possible.