Hamiltonian Cycle Problem on Distance-Hereditary Graphs

All graphs considered in this paper are finite and undirected, without loops or multiple edges. Let G = (V, E) be a graph. Throughout this paper, let m and n denote the numbers of edges and vertices of graph G, respectively. A connected graph is distance-hereditary if the distance between every two vertices in a connected induced subgraph is the same as that in the original graph. Distance-hereditary graphs were introduced by Howorka [21]. Bandelt and Mulder showed that a distance-hereditary graph can be constructed from an isolated vertex by adding vertices one by one through operations called one-vertex extensions [2]. Furthermore, Hammer and Maffray proposed a linear time recognition algorithm that constructs a sequence of one-vertex extensions for a distance-hereditary graph [20]. Chang et al. gave a recursive definition for distance-hereditary graphs [9]. Further properties and optimization problems in these graphs have been studied in [1, 2, 4-6, 8, 15-17, 23, 24, 30-32]. Distance-hereditary graphs are a subclass of parity graphs [11] and a superclass of cographs [12, 13] and Ptolemaic graphs [22]. A Hamiltonian cycle of a graph G is a simple cycle that passes through each vertex exactly once. The Hamiltonian cycle problem involves testing whether a graph contains a Hamiltonian cycle. It is well known that this problem is NP-complete for general graphs [18] and NP-complete even for special classes of graphs, such as bipartite graphs [27], split graphs [19], circle graphs [14], and grid graphs [25]. Chang et al. [7] solved the Hamiltonian cycle problem on Ptolemaic graphs in O(n + m) time. Nicolai [29] presented the first polynomial time algorithm to solve the Hamiltonian cycle problem on distance-hereditary graphs. His algorithm runs in O(n) time. In this paper, we present an O(n) time algorithm to solve the Hamiltonian cycle problem on distance-hereditary graphs.