The k-fixed-endpoint path partition problem

The Hamiltonian path problem is to determine whether a graph has a Hamiltonian path. This problem is NP-complete in general. The path partition problem is to determine the minimum number of vertex-disjoint paths required to cover a graph. Since this problem is a generalization of the Hamiltonian path problem, it is also NP-complete in general. The k-fixed-endpoint path partition problem is to determine the minimum number of vertex-disjoint paths required to cover a graphG such that each vertex in a set T of k vertices is an endpoint of a path. Since this problem is a generalization of the Hamiltonian path problem and path partition problem, it is also NP-complete in general. For certain classes of graphs, there exist efficient algorithms for the k-fixed-endpoint path partition problem. We consider this problem restricted to trees, threshold graphs, block graphs, and unit interval graphs and show min-max theorems which characterize the k-fixed-endpoint pathpartition number

Forbidden subgraphs for Hamiltonian problems on 2-trees

The Hamiltonian path problem is a well-known NP-complete graph theory problem which is to determine whether or not it is possible to find a spanning path in a graph. Some variations on this problem include the 1HP and 2HP problems, which are to determine whether or not it is possible to find a Hamiltonian path in a graph if one or two endpoints of the path are fixed, respectively. Both problems are also NP-complete for graphs in general, though like the Hamiltonian path problem, they are polynomially solvable on certain types of graphs. 2-trees are a specific type of graph for which the 1HP, 2HP, and traditional Hamiltonian path problems are polynomially solvable. It is known that 2-trees have a Hamiltonian cycle if and only if they are 1-tough. However, the analogous statement for Hamiltonian paths does not hold. We will structurally characterize 2HP on 2-trees, and then use these results to structurally characterize 1HP and HP on 2-trees. We will define a family of 2-trees such that any 2-tree has a Hamiltonian path if and only if it does not contain any graph from that family as an induced graph