弦图、可比图和置换图的同时隶属问题
The Simultaneous Membership Problem for Chordal, Comparability and Permutation graphs
摘要：
在本文中，我们介绍了“同时成员问题”，定义为任何图形类C的表征方面，例如任何类的交叉图。两个图G_1和G_2共享一些顶点X（和相应的诱导边），如果存在G_1和G_2在X上“相容”的表示R_1和R_2，则称这两个图是图类C的“同时成员”。等价地（对于我们考虑的类C）在G_1-X和G_2-X之间存在边E'使得G_1 \cup G_2 \cup E'属于类C。同时隶属度问题在任何情况下，它是可取的，始终表示两个相关的图，例如：区间图捕捉重叠的DNA片段的两个类似的生物体，或图形连接的时间，其中一个是另一个的更新版本的应用程序。同时隶属度问题涉及到同时平面嵌入、图三明治问题和探针图识别问题。本文给出了弦图、可比图和置换图上的同时隶属度问题的有效算法。这些结果意味着，图三明治问题的上述类是听话的一个有趣的特殊情况下：当一组可选的边缘形成一个完全的二部图。我们的研究结果补充了最近的多项式时间识别算法的探针弦，可比性，置换图，其中的一组可选的边缘形成一个集团。
英文摘要：
In this paper we introduce the 'simultaneous membership problem', defined for any graph class C characterized in terms of representations, e.g. any class of intersection graphs. Two graphs G_1 and G_2, sharing some vertices X (and the corresponding induced edges), are said to be 'simultaneous members' of graph class C, if there exist representations R_1 and R_2 of G_1 and G_2 that are "consistent" on X. Equivalently (for the classes C that we consider) there exist edges E' between G_1-X and G_2-X such that G_1 \cup G_2 \cup E' belongs to class C. Simultaneous membership problems have application in any situation where it is desirable to consistently represent two related graphs, for example: interval graphs capturing overlaps of DNA fragments of two similar organisms; or graphs connected in time, where one is an updated version of the other. Simultaneous membership problems are related to simultaneous planar embeddings, graph sandwich problems and probe graph recognition problems. In this paper we give efficient algorithms for the simultaneous membership problem on chordal, comparability and permutation graphs. These results imply that graph sandwich problems for the above classes are tractable for an interesting special case: when the set of optional edges form a complete bipartite graph. Our results complement the recent polynomial time recognition algorithms for probe chordal, comparability, and permutation graphs, where the set of optional edges form a clique.
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