加强了图的韧性的一些复杂性结果
Strengthening some complexity results on toughness of graphs
摘要：
设$t$为正实数。一个图被称为$t$ -tough，如果删除任何断开图的顶点集$S$，最多只留下$|S|/t$组件。图的韧性是图是$t$ -tough的最大$t$。本文的主要结果如下。对于任何正有理数$t \le 1$和任何$k \ge 2$和$r \ge 6$整数，识别$t$ -坚韧二部图是coNP-完全的（$t=1$的情况是已知的），并且这个问题对于$k$ -连通二部图仍然是coNP-完全的，识别1-坚韧r-正则二部图的问题也是如此。为了证明这些陈述，我们也处理其他相关的韧性复杂性问题。%在本文中，我们证明了以下内容。对于任意正有理数$t$，判定$τ(G)=t$是否DP-完全，如果是$t < 1$，则该问题对于二部图仍是DP-完全的.对任意整数$k \ge 2$和正有理数$t \le 1$，识别$t$ -坚韧$k$ -连通二部图是coNP-完全的.对于任意整数$r \ge 5$，识别$1/2$ -坚韧$r$ -正则图是coNP-完全的.对于任意整数$r \ge 6$，识别1-坚韧$r$ -正则二部图是coNP-完全的.对于任意正有理数$t < 2/3$，给出了识别具有韧性$t$的3-正则图的多项式时间算法.最后，我们证明了每个连通4-正则图都是1/2-坚韧的。
英文摘要：
Let $t$ be a positive real number. A graph is called $t$-tough if the removal of any vertex set $S$ that disconnects the graph leaves at most $|S|/t$ components. The toughness of a graph is the largest $t$ for which the graph is $t$-tough. The main results of this paper are the following. For any positive rational number $t \le 1$ and for any $k \ge 2$ and $r \ge 6$ integers recognizing $t$-tough bipartite graphs is coNP-complete (the case $t=1$ was already known), and this problem remains coNP-complete for $k$-connected bipartite graphs, and so does the problem of recognizing 1-tough r-regular bipartite graphs. To prove these statements we also deal with other related complexity problems on toughness. % In this paper we prove the following. For any positive rational number $t$, deciding whether $τ(G)=t$ is DP-complete and if $t < 1$, this problem remains DP-complete for bipartite graphs. For any integer $k \ge 2$ and positive rational number $t \le 1$, recognizing $t$-tough $k$-connected bipartite graphs is coNP-complete. For any integer $r \ge 5$, recognizing $1/2$-tough $r$-regular graphs is coNP-complete. For any integer $r \ge 6$, recognizing 1-tough $r$-regular bipartite graphs is coNP-complete. For any positive rational number $t < 2/3$ we give a polynomial time algorithm for recognizing 3-regular graphs with toughness $t$. Finally, we prove that every connected 4-regular graph is 1/2-tough.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
