In 1981, Lubiw proved that the fixed point free automorphism problem (FPFAut) is NP-complete: given a graph G, determine whether there exists an automorphism that maps no vertex of G to itself. We revisit this problem and prove that FPFAut remains NP-complete when restricted to split, bipartite, k-subdivided, and H-free graphs, if H is not an induced subgraph of P_4. The class of P_4-free graphs receives the special name of cographs. We provide a polynomial time algorithm for three extensions of cographs: bounded modular-width graphs, tree-cographs and P_4-sparse graphs. Our approach uses the well known modular decomposition of graphs. As a consequence, we generalize a result of Abiad et. al. on the problem of computing 2-homogeneous equitable partitions.

翻译：1981年，Lubiw证明了无不动点自同构问题（FPFAut）是NP完全的：给定图G，判断是否存在一个自同构将G的每个顶点都映射到其他顶点。我们重新审视该问题，并证明当限制在分裂图、二分图、k-细分图以及H-自由图（其中H不是P_4的诱导子图）上时，FPFAut仍然是NP完全的。P_4-自由图具有特殊的名称——余图（cographs）。我们针对余图的三种扩展类：有界模宽度图、树余图和P_4-稀疏图，提出了多项式时间算法。我们的方法利用了著名的模分解理论。作为推论，我们推广了Abiad等人关于计算2-齐次公平划分的结果。