Abstract:
In this paper we present further studies of convex covers and
convex partitions of graphs. Let G
be a finite simple graph. A
set of vertices
S
of
G
is convex if all vertices lying on a shortest
path between any pair of vertices of
S
are in
S
. If 3
≤ |
S
| ≤
|
X
| −
1, then
S
is a nontrivial set. We prove that determining
the minimum number of convex sets and the minimum number
of nontrivial convex sets, which cover or partition a graph, is in
general NP-hard. We also prove that it is NP-hard to determine
the maximum number of nontrivial convex sets, which cover or
partition a graph.