Connected Proper Interval Graphs and the Guard Problem in Spiral Polygons

The main purpose of this paper is to study the hamiltonicity of proper interval graphs and applications of these graphs to the guard problem in spiral polygons. The Hamiltonian path (circuit) problem is, given an undirected graph G = (V ,E), to determine whether G contains a Hamiltonian path (circuit). These two problems are well-known N P-complete problems. In the rst part of this paper, we shall derive necessary and suucient conditions for a proper interval graph to have a Hamiltonian path, have a Hamiltonian circuit, and be Hamiltonian-connected, respectively. See 1,4,5]. Let c(G) denote the number of components of G. The scattering number s(G) of a graph G is max fc(G ? S)? jSj j S V and c(G ? S) 6 = 1g. The closed neighborhood N v] of a vertex v is the set of vertices adjacent to v plus v it self. Theorem 2. For any proper interval graph G = (V; E) of n k + 1 vertices, the following statements are equivalent for any positive integer k.