SIAM Journal on Discrete Mathematics, Volume 34, Issue 2, Page 1472-1483, January 2020.
A hole in a graph is an induced cycle of length at least 4, and an antihole is the complement of an induced cycle of length at least 4. A hole or antihole is long if its length is at least 5. For an integer $k$, the $k$-prism is the graph consisting of two cliques of size $k$ joined by a matching. The complexity of Maximum (Weight) Independent Set (MWIS) in long-hole-free graphs remains an important open problem. In this paper we give a polynomial-time algorithm to solve MWIS in long-hole-free graphs with no $k$-prism (for any fixed integer $k$) and a subexponential algorithm for MWIS in long-hole-free graphs in general. As a special case this gives a polynomial-time algorithm to find a maximum weight clique in perfect graphs with no long antihole and no hole of length 6. The algorithms use the framework of minimal chordal completions and potential maximal cliques.

中文翻译：

---

无最小长度的不引起长度循环的图中的最大权重独立集问题

SIAM离散数学杂志，第34卷，第2期，第1472-1483页，2020年1月。
图中的空洞是长度为4的诱导周期，而反空洞是长度为4的诱导周期的补码。如果空洞或抗洞的长度至少为5，则它是长的。对于整数$ k $，$ k $ -prism是由两个大小为$ k $的团由一个匹配项组成的图。无长孔图中最大（权重）独立集（MWIS）的复杂性仍然是一个重要的开放问题。在本文中，我们给出了多项式时间算法来求解无$ k $ -prism（对于任何固定整数$ k $）的无长孔图中的MWIS，并给出了针对无孔长图中MWIS的次指数算法一般。作为特殊情况，这提供了多项式时间算法，可以在没有长反孔且长度为6的孔的完美图中找到最大权重集团。