In the Partition Into Complementary Subgraphs (Comp-Sub) problem we are given a graph \(G=(V,E)\), and an edge set property \(\varPi \), and asked whether G can be decomposed into two graphs, H and its complement \(\overline{H}\), for some graph H, in such a way that the edge cut-set (of the cut) \([V(H),V(\overline{H})]\) satisfies property \(\varPi \). Such a problem is motivated by the fact that several parameterized problems are trivially fixed-parameter tractable when the input graph G is decomposable into two complementary subgraphs. In addition, it generalizes the recognition of complementary prism graphs, and it is related to graph isomorphism when the desired cut-set is empty, Comp-Sub(\(\emptyset \)). In this paper we are particularly interested in the case Comp-Sub(\(\emptyset \)), where the decomposition also partitions the set of edges of G into E(H) and \(E(\overline{H})\). When the input is a chordal graph, we show that Comp-Sub(\(\emptyset \)) is \({\textsf {GI}}{\text{-}}{\textit{complete}}\), that is, polynomially equivalent to Graph Isomorphism. But it becomes more tractable than Graph Isomorphism for several subclasses of chordal graphs. We present structural characterizations for split, starlike, block, and unit interval graphs. We also obtain complexity results for permutation graphs, cographs, comparability graphs, co-comparability graphs, interval graphs, co-interval graphs and strongly chordal graphs. Furthermore, we present some remarks when \(\varPi \) is a general edge set property and the case when the cut-set M induces a complete bipartite graph.

在分区成互补子图（Comp-Sub）问题中，我们得到了一个图\（G =（V，E）\）和一个边集属性\（\ varPi \），并询问G是否可以分解为两个图H和它的补数\（\ overline {H} \），对于某些图H，以（切割的）边切割集\（[V（H），V（\ overline {H }）] \）满足属性\（\ varPi \）。这个问题是由以下事实引起的：当输入图G时，几个参数化问题都是固定参数易处理的可分解为两个互补的子图。此外，它概括了对互补棱镜图的识别，并且与所需的割集为空时的图同构Comp-Sub（\（\ emptyset \））有关。在本文中，我们对Comp-Sub（\（\ emptyset \））情况特别感兴趣，其中分解还将G的边集划分为E（H）和\（E（\ overline {H}）\ ）。当输入是一个弦图时，我们表明Comp-Sub（\（\ emptyset \））是\（{\ textsf {GI}} {\ text {-}} {\ textit {complete}} \}，即，在多项式上等同于图同构。但是对于和弦图的几个子类，它比图同构更容易处理。我们介绍了分裂图，星形图，块图和单位间隔图的结构特征。我们还获得了置换图，cograph，可比性图，co-comparability图，区间图，co-interval图和强和弦图的复杂性结果。此外，当\（\ varPi \）是常规边集属性时，以及在割集M诱导出完整的二部图的情况下，我们将提供一些说明。