The (theta, wheel)-free graphs Part II: structure theorem

A hole in a graph is a chordless cycle of length at least 4. A theta is a graph formed by three paths between the same pair of distinct vertices so that the union of any two of the paths induces a hole. A wheel is a graph formed by a hole and a node that has at least 3 neighbors in the hole. In this paper we obtain a decomposition theorem for the class of graphs that do not contain an induced subgraph isomorphic to a theta or a wheel, i.e. the class of (theta, wheel)-free graphs. The decomposition theorem uses clique cutsets and 2-joins. Clique cutsets are vertex cutsets that work really well in decomposition based algorithms, but are unfortunately not general enough to decompose more complex hereditary graph classes. A 2-join is an edge cutset that appeared in decomposition theorems of several complex classes, such as perfect graphs, even-hole-free graphs and others. In these decomposition theorems 2-joins are used together with vertex cutsets that are more general than clique cutsets, such as star cutsets and their generalizations (which are much harder to use in algorithms). This is a first example of a decomposition theorem that uses just the combination of ∗University of Belgrade, Faculty of Mathematics, Belgrade, Serbia. Partially supported by Serbian Ministry of Education, Science and Technological Development project 174033. E-mail: [email protected] †CNRS, LIP, ENS de Lyon. Partially supported by ANR project Stint under reference ANR-13-BS02-0007 and by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program Investissements d’Avenir (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). Also Université Lyon 1, université de Lyon. E-mail: [email protected] ‡School of Computing, University of Leeds, and Faculty of Computer Science (RAF), Union University, Belgrade, Serbia. Partially supported by EPSRC grants EP/K016423/1 and EP/N0196660/1, and Serbian Ministry of Education and Science projects 174033 and III44006. E-mail: [email protected]