Graph Hamiltonicity Parameterized by Proper Interval Deletion Set

The Path Cover and Cycle Cover problems are well-known generalizations of the classical Hamiltonian Path and Hamiltonian Cycle problems. Here, we are given an undirected graph on n vertices and a positive integer r and the task is to check if there are r vertex-disjoint paths (cycles) that together visit all the vertices of the graph exactly once. Path Cover and Cycle Cover remain NP-hard even when restricted to chordal graphs (Information Processing Letters 1986) but are polynomial-time solvable on proper interval graphs (Discrete Mathematics 1993 and Proceedings of WADS 2019). In this paper, we study the complexity of Path Cover and Cycle Cover with respect to a structural parameter, namely, distance to proper interval graphs. In particular, we show that Path Cover and Cycle Cover are fixed-parameter tractable (FPT) when parameterized by k, the size of a proper interval deletion set (a set of vertices whose deletion results in a proper interval graph). For this purpose, we design an algorithm with O(2^{O ( k log k )}n^{O ( 1 )}) running time for each of these problems. Our algorithms use several interesting properties of proper interval graphs and a dynamic programming procedure over clique partitions to solve these problems in the mentioned time. As a consequence, we get the same fixed-parameter tractability results for Hamiltonian Cycle and Hamiltonian Path problems with the same parameterization. Recently, Chaplick et al. (Proceedings of WADS 2019) obtained polynomial kernels and compression algorithms for Path Cover and Cycle Cover parameterized by a different measure of similarity with proper interval graphs. Our FPT algorithms also adds to this study of structural parameterizations for these classical problems.