Linear-time certifying algorithms for the path cover and Hamiltonian cycle problems on interval graphs

A certifying algorithm for a problem is an algorithm that provides a certificate with each answer that it produces. The certificate is an evidence that can be used to authenticate the correctness of the answer. A Hamiltonian cycle in a graph is a simple cycle in which each vertex of the graph appears exactly once. The Hamiltonian cycle problem is to test whether a graph has a Hamiltonian cycle. A path cover of a graph is a family of vertex-disjoint paths that covers all vertices of the graph. The path cover problem is to find a path cover of a graph with minimum cardinality. This paper presents O ( n ) -time certifying algorithms for the above two problems on interval graphs given a set of n intervals with endpoints sorted. The certificates provided by our algorithms can be authenticated in O ( n ) time.