Abstract (EN)

Stronglyperfectgraphs have been studied by several authors (e.g. Berge and Duchet (1984) [1], Ravindra (1984) [12] and Wang (2006) [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-freegraphs that are fractionally stronglyperfect 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-freegraphs that are fractionally stronglyperfect 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 (2006) [14] gave a characterization of stronglyperfectclaw-freegraphs. As a corollary of the results in this paper, we obtain a characterization of claw-freegraphs whose complements are stronglyperfect.