Clique-Width for Graph Classes Closed under Complementation

Clique-width is an important graph parameter due to its algorithmic and structural properties. A graph class is hereditary if it can be characterized by a (not necessarily finite) set H of forbidden induced subgraphs. We initiate a systematic study into the boundedness of clique-width of hereditary graph classes closed under complementation. First, we extend the known classification for the |H| = 1 case by classifying the boundedness of clique-width for every set H of self-complementary graphs. We then completely settle the |H| = 2 case. In particular, we determine one new class of (H,H)-free graphs of bounded clique-width (as a side effect, this leaves only six classes of (H1,H2)free graphs, for which it is not known whether their clique-width is bounded). Once we have obtained the classification of the |H| = 2 case, we research the effect of forbidding self-complementary graphs on the boundedness of cliquewidth. Surprisingly, we show that for a set F of self-complementary graphs on at least five vertices, the classification of the boundedness of clique-width for ({H,H}∪F)-free graphs coincides with the one for the |H| = 2 case if and only if F does not include the bull (the only non-empty self-complementary graphs on fewer than five vertices are P1 and P4, and P4-free graphs have clique-width at most 2). Finally, we discuss the consequences of our results for Colouring and Graph Isomorphism.