Abstract:
For a given graph G, a maximum internal spanning tree of G is a spanning tree of
G with maximum number of internal vertices. The Maximum Internal Spanning
Tree (MIST) problem is to find a maximum internal spanning tree of the given
graph. The MIST problem is a generalization of the Hamiltonian path problem. Since
the Hamiltonian path problem is NP-hard, even for bipartite and chordal graphs, two
important subclasses of graphs, the MIST problem also remains NP-hard for these
graph classes. In this paper, we propose linear-time algorithms to compute a maximum
internal spanning tree of cographs, block graphs, cactus graphs, chain graphs and
bipartite permutation graphs. The optimal path cover problem, which asks to find a
path cover of the given graph with maximum number of edges, is also a well studied
problem. In this paper, we also study the relationship between the number of internal
vertices in maximum internal spanning tree and number of edges in optimal path cover
for the special graph classes mentioned above.
