一种新的测量线性度的图形参数
A New Graph Parameter To Measure Linearity
摘要：
考虑LexBFS顶点排序σ1，σ2，. . .其中每个排序σi用于打破σi+1的平局。因为有限图的顶点序的总数是有限的，所以这个序列必须以顶点序的循环结束。这一周期的可能长度是这项工作的主要议题。直觉上，我们证明了对于具有已知线性概念的图（例如，区间图，其区间表示在实线上），这个圈不能太大，无论我们从哪个顶点排序开始。更确切地说，在[9]中证明了对于协可比图，这个圈的大小总是2，与起始顺序无关。此外[27]问是否任意图，这样的循环的大小总是由图的星状数限制。在这项工作中，虽然我们回答后一个问题否定，我们提供了支持的猜想上cocomparitygraphs证明它的子类的多米诺骨牌免费cocomparitygraphs。这个子类包含余图、真区间、区间和余偶图。我们还提供了更简单的独立证明，这些情况下，导致更强的结果，这个子类。
英文摘要：
Consider a sequence of LexBFS vertex orderings σ1, σ2, . . . where each ordering σi is used to break ties for σi+1. Since the total number of vertex orderings of a finite graph is finite, this sequence must end in a cycle of vertex orderings. The possible length of this cycle is the main subject of this work. Intuitively, we prove for graphs with a known notion of linearity (e.g., interval graphs with their interval representation on the real line), this cycle cannot be too big, no matter which vertex ordering we start with. More precisely, it was conjectured in [9] that for cocomparability graphs, the size of this cycle is always 2, independent of the starting order. Furthermore [27] asked whether for arbitrary graphs, the size of such a cycle is always bounded by the asteroidal number of the graph. In this work, while we answer this latter question negatively, we provide support for the conjecture on cocomparability graphs by proving it for the subclass of domino-free cocomparability graphs. This subclass contains cographs, proper interval, interval, and cobipartite graphs. We also provide simpler independent proofs for each of these cases which lead to stronger results on this subclasses.
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