Well-Quasi-Order for Permutation Graphs Omitting a Path and a Clique

We consider well-quasi-order for classes of permutation graphs which omit both a path and a clique. Our principle result is that the class of permutation graphs omitting P5 and a clique of any size is well-quasi-ordered. This is proved by giving a structural decomposition of the corresponding permutations. We also exhibit three infinite antichains to show that the classes of permutation graphs omitting {P6,K6}, {P7,K5}, and {P8,K4} are not well-quasi-ordered. Atminas and Lozin gratefully acknowledge support from DIMAP – the Center for Discrete Mathematics and its Applications at the University of Warwick. Brignall, Korpelainen, and Vatter were partially supported by EPSRC Grant EP/J006130/1. Lozin was partially supported by EPSRC Grants EP/I01795X/1 and EP/L020408/1. Vatter was partially supported by the National Security Agency under Grant Number H98230-12-10207 and the National Science Foundation under Grant Number DMS-1301692. The United States Government is authorized to reproduce and distribute reprints not-withstanding any copyright notation herein. the electronic journal of combinatorics 22 (2015), #P00 1