Partitioning a Graph into Complementary Subgraphs

In the PARTITION INTO COMPLEMENTARY SUBGRAPHS (COMP-SUB) problem we are given a graph G = (V, E), and an edge set property Pi, and asked whether G can be decomposed into two graphs, H and its complement (H) over bar, for some graph H, in such a way that the edge cut-set (of the cut) [V(H), V((H) over bar)] satisfies property Pi. 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((empty set)). In this paper we are particularly interested in the case COMP-SUB((empty set)), where the decomposition also partitions the set of edges of G into E(H) and E((H) over bar). When the input is a chordal graph, we show that COMP-SUB((empty set)) is GI-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 Pi is a general edge set property and the case when the cut-set M induces a complete bipartite graph.