Claw-free graphs with strongly perfect complements. Fractional and integral version. Part I. Basic graphs

Strongly perfect graphs have been studied by several authors (e.g. Berge and Duchet [1], Ravindra [12], Wang [14]). In a series of two papers, the current paper being the first one, we investigate a fractional relaxation of strong perfection. Motivated by a wireless networking problem, we consider claw-free graphs that are fractionally strongly perfect in the complement. We obtain a forbidden induced subgraph characterization and display graph-theoretic properties of such graphs. It turns out that the forbidden induced subgraphs that characterize claw-free that are fractionally strongly perfect in the complement are precisely the cycle of length 6, all cycles of length at least 8, four particular graphs, and a collection of graphs that are constructed by taking two graphs, each a copy of one of three particular graphs, and joining them in a certain way by a path of arbitrary length. Wang [14] gave a characterization of strongly perfect clawfree graphs. As a corollary of the results in this paper, we obtain a characterization of claw-free graphs whose complements are strongly perfect. ∗Partially supported by NSF grant DMS-0758364. †This research was performed while the author was at Columbia University and at the University of Warwick. ‡This research was performed while the author was at Columbia University.