真图和单位区间图的同时表示
Simultaneous Representation of Proper and Unit Interval Graphs
摘要：
在组合学和几何学的融合中，同时表示提供了一种实现共享共同结构的组合对象的方法。同时表征研究中的一个标准案例是向日葵案例，其中所有对象共享相同的公共结构。虽然一般的同时区间图的识别问题-可以说是研究最充分的图形类之一的同时版本-是NP完全的，但三个或更多同时区间图的向日葵案例的复杂性目前是开放的。在这项工作中，我们解决了这个问题，适当的区间图。我们给出了一个算法来识别同时正常区间图在线性时间的向日葵的情况下，我们允许任何数量的同时图。同时单位区间图是更加“刚性”的，因此在它们的表示中具有较少的自由度。我们发现，它们可以被识别的时间O（|V| * |E|）的任何数量的同时在向日葵的情况下，G =（V，E）是联合的同时图。我们进一步表明，这两个识别问题一般是NP-完全的，如果同时图的数量是不固定的。从这个意义上说，对向日葵案的限制是必要的。
英文摘要：
In a confluence of combinatorics and geometry, simultaneous representations provide a way to realize combinatorial objects that share common structure. A standard case in the study of simultaneous representations is the sunflower case where all objects share the same common structure. While the recognition problem for general simultaneous interval graphs -- the simultaneous version of arguably one of the most well-studied graph classes -- is NP-complete, the complexity of the sunflower case for three or more simultaneous interval graphs is currently open. In this work we settle this question for proper interval graphs. We give an algorithm to recognize simultaneous proper interval graphs in linear time in the sunflower case where we allow any number of simultaneous graphs. Simultaneous unit interval graphs are much more 'rigid' and therefore have less freedom in their representation. We show they can be recognized in time O(|V|*|E|) for any number of simultaneous graphs in the sunflower case where G = (V, E) is the union of the simultaneous graphs. We further show that both recognition problems are in general NP-complete if the number of simultaneous graphs is not fixed. The restriction to the sunflower case is in this sense necessary.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
