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Digraphs with proper connection number two
Theoretical Computer Science ( IF 0.9 ) Pub Date : 2021-04-30 , DOI: 10.1016/j.tcs.2021.04.025
Luyi Li , Xueliang Li

A directed path in a digraph is proper if any two consecutive arcs on the path have distinct colors. An arc-colored digraph D is proper connected if for any two distinct vertices x and y of D, there are both proper (x,y)-directed paths and proper (y,x)-directed paths in D. The proper connection number pc(D) of a digraph D is the minimum number of colors that can be used to make D proper connected. Obviously, if a digraph has a proper connection number, it must be strongly connected, and pc(D)=1 if and only if D is complete. Magnant et al. showed that pc(D)3 for all strong digraphs D, and Ducoffe et al. proved that deciding whether a given digraph has proper connection number at most two is NP-complete. In this paper, we give a few classes of strong digraphs with proper connection number two, and from our proofs one can construct an optimal arc-coloring for a digraph of order n in time O(n3).



中文翻译:

有正确的第二连接的有向图

如果路径上的任何两个连续弧具有不同的颜色,则有向图中的有向路径是正确的。的圆弧各色有向图d被适当地连接,如果对任意两个不同的顶点Xÿd,有两个合适的Xÿ导向的路径和适当的 ÿXD中的定向路径。正确的连接号pCdD的“ D”是可用于使D正确连接的最小颜色数。显然,如果有向图的连接号正确,则必须将其牢固连接,并且pCd=1个当且仅当D完成时。Magnant等。表明pCd3对于所有强有向图D和Ducoffe等。证明确定给定的有向图最多具有两个正确的连接数是NP完全的。在本文中,我们给予适当的连接数两强有向图的几类,从我们的证明可以构造一个最佳的圆弧着色订单的有向图ñ的时间Øñ3

更新日期:2021-04-30
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