密林
Thick Forests
AI总结：
提出厚图与厚森林概念，证明厚森林是最大完美厚图类且可多项式识别，并分析了独立集与着色计数问题的复杂性及参数化扩展。
摘要：
我们考虑一类图，称之为厚图，其对应薄图的顶点被替换为团，边被替换为补二部图。特别地，我们考察了厚森林的情况，并证明其为最大类的完美厚图。除非薄图类是无三角的，否则识别一个图是否属于某厚图类是NP完全的，因此我们重点关注这一情况。即便如此，该成员判定仍可能是NP完全的。然而，我们证明厚森林类可在多项式时间内被识别。我们考察了厚图上两个被广泛研究的组合问题：独立集和正常着色。由于确定完美图的独立数或色数已知是可处理的，我们检验了厚森林中计数所有独立集和着色的复杂性。最后，我们考虑向更大类厚图的两个参数化扩展：参数为薄图大小的情况，以及参数为其树宽的情况。
英文摘要：
We consider classes of graphs, which we call thick graphs, that have the vertices of a corresponding thin graph replaced by cliques and the edges replaced by cobipartite graphs In particular, we consider the case of thick forests, which we show to be the largest class of perfect thick graphs. Recognising membership of a class of thick graphs is NP-complete unless the class of thin graphs is triangle-free, so we focus on this case. Even then membership can be NP-complete. However, we show that the class of thick forests can be recognised in polynomial time. We consider two well-studied combinatorial problems on thick graphs, independent sets and proper colourings. Since determining the independence or chromatic number of a perfect graph is known to be tractable, we examine the complexity of counting all independent sets and colourings in thick forests. Finally, we consider two parametric extensions to larger classes of thick graphs: where the parameter is the size of the thin graph, and where the parameter is its treewidth.
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