Dual trees must share their ends

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• 作者：Reinhard Diestel ; Julian Pott
• 关键词：Graph ; End ; Spanning tree ; Freudenthal compactification ; Duality ; Infinite matroid
• 刊名：Journal of Combinatorial Theory, Series B
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：123
• 期：Complete
• 页码：32-53
• 全文大小：511 K
• 卷排序：123

文摘

We extend to infinite graphs the matroidal characterization of finite graph duality, that two graphs are dual iff they have complementary spanning trees in some common edge set. The naive infinite analogue of this fails.The key in an infinite setting is that dual trees must share between them not only the edges of their host graphs but also their ends: the statement that a set of edges is acyclic and connects all the vertices in one of the graphs iff the remaining edges do the same in its dual will hold only once each of the two graphs' common ends has been assigned to one graph but not the other, and ‘cycle’ and ‘connected’ are interpreted topologically in the space containing the respective edges and precisely the ends thus assigned.This property characterizes graph duality: if, conversely, the spanning trees of two infinite graphs are complementary in this end-sharing way, the graphs form a dual pair.