Finding Hamiltonian Circuits in Interval Graphs

A simple circuit in a graph G = ( V , E ) is a sequence (v 1, v2, . . . , v k) of distinct vertices from V such that { v , , v i + l } ~ E for l ~ < i < k 1 and {v k, v I } ~ E. A Hamiltonian circuit in G is a simple circuit that includes all the vertices of G. The problem of deciding whether a graph has a Hamiltonian circuit has long been known to be NP-complete [5,11]. The Hamiltonian circuit problem remains NP-complete for planar 3-connected graphs [6], bipartite graphs [12], split graphs [8], edge graphs [1], planar bipartite graphs [10], and grid graphs [10]. Thus far, polynomial time algorithms for the problem have only been developed for 4-connected planar graphs [9] and proper interval graphs [2]. This paper presents a linear time algorithm for the Hamiltonian circuit problem in interval graphs. A graph is an intersection graph if there exists a one-to-one correspondence between its vertices and a family F of sets such that two vertices are adjacent in the graph if and only if their two corresponding sets intersect. If F is a family of intervals of the real line, then G is called an interval graph [8,13] and the family F is called the interval model for G.