区间图上多项式时间的捕获
Capturing Polynomial Time on Interval Graphs
摘要：
我们证明了一个特征的所有多项式时间可计算查询的区间图类的不动点逻辑计数的句子。更确切地说，它表明，在无序区间图类，任何查询是多项式时间可计算的，当且仅当它是可定义的定点逻辑计数。这一结果是第一个建立捕获的多项式时间的一类图，这是由禁止诱导子图定义。为此，我们定义了一个标准形式的区间图使用的一种类型的模块化分解，这是不同的树分解的方法，用于大多数已知的捕获结果的其他图形类，特别是那些定义的禁止未成年人。该方法也可能因其概念简单而引起独立兴趣。此外，它表明，不动点逻辑计数是不够的表现力，以捕捉多项式时间的类弦图或不可比图。
英文摘要：
We prove a characterization of all polynomial-time computable queries on the class of interval graphs by sentences of fixed-point logic with counting. More precisely, it is shown that on the class of unordered interval graphs, any query is polynomial-time computable if and only if it is definable in fixed-point logic with counting. This result is one of the first establishing the capturing of polynomial time on a graph class which is defined by forbidden induced subgraphs. For this, we define a canonical form of interval graphs using a type of modular decomposition, which is different from the method of tree decomposition that is used in most known capturing results for other graph classes, specifically those defined by forbidden minors. The method might also be of independent interest for its conceptual simplicity. Furthermore, it is shown that fixed-point logic with counting is not expressive enough to capture polynomial time on the classes of chordal graphs or incomparability graphs.
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