The longest cycle problem is the problem of finding a cycle with maximal vertices in a graph. Although it is solvable in polynomial time on few trivial graph classes, the longest cycle problem is well known as NP-complete. A lot of efforts have been devoted to the longest cycle problem. To the best of our knowledge however, there are no polynomial algorithms that can solve any of the non-trivial graph classes. Interval graphs, the intersection of chordal graphs and asteroidal triple-free graphs, are known to be the non-trial graph classes that have polynomial algorithm of the longest cycle problem. In 2009, K. Ioannidou, G.B. Mertzios and S.D. Nikolopoulos presented a polynomial algorithm for the longest path problem on interval graphs in Ioannidou et al. (2009) [19]. Inspired by their work, we investigate the longest cycle problem of interval graphs. In this paper, we present the first polynomial algorithm for the longest cycle problem on interval graphs. A dynamic programming approach is proposed in the polynomial algorithm that runs in $O(n^8)$ time, where n is the number of vertices of the input graph. Using a similar approach, we design a polynomial algorithm to solve the longest k-thick subgraph problem on interval graphs which will be presented in another separate work. According to the interesting properties of k-thick interval graphs that we discovered (e.g., an interval graph G is traceable if and only if G is 1-thick, G is hamiltonian if and only if G is 2-thick, G is hamiltonian connected if and only if G is 3-thick and so on), the algorithm presented in this paper can be important in studying the spanning connectivity on interval graphs.

最长循环问题是找到图中最大顶点的循环的问题。尽管在几个平凡的图类上可以在多项式时间内解决，但最长的循环问题众所周知为NP-完全。已经针对最长周期问题做出了许多努力。据我们所知，没有多项式算法可以解决任何非平凡的图类。间隔图，即弦图与小行星三重自由图的相交，是具有最长循环问题的多项式算法的非试验图类。2009年，K。Ioannidou，GB Mertzios和SD Nikolopoulos在Ioannidou等人的论文中提出了一种用于区间图上最长路径问题的多项式算法。（2009）[19]。受到他们工作的启发，我们研究了区间图的最长周期问题。在本文中，我们提出了区间图上最长周期问题的第一个多项式算法。在多项式算法中提出了一种动态规划方法。$Ø（{ñ}^8）$时间，其中n是输入图的顶点数。使用类似的方法，我们设计了一种多项式算法来解决区间图上最长的k厚子图问题，该问题将在另一份单独的工作中提出。据的令人感兴趣的性质ķ厚的间隔的曲线图，我们发现（例如，区间图ģ是可追踪当且仅当ģ是1-厚，ģ是Hamilton当且仅当ģ是2-厚，ģ是Hamilton连接当且仅当G 是3层等），本文中提出的算法对于研究区间图上的跨度连通性很重要。