Strong T-Perfection of Bad-K4 -Free Graphs

Abstract

We show that each graph not containing a bad subdivision of -K4 as a subgraph is strongly t-perfect. Here a graph G=(V,E) is strongly t-perfect if, for each weight function $w:V\to\mathbb{Z}_+$, the maximum weight of a stable set is equal to the minimum (total) cost of a family of vertices, edges, and circuits covering any vertex v at least w(v) times. By definition, the cost of a vertex or edge is 1, and the cost of a circuit C is $\lfloor\frac{1}{2}|VC|\rfloor$. A subdivision of K4 is called bad if each triangle has become an odd circuit and if it is not obtained by making the edges in a 4-circuit of K4 . evenly subdivided, while the other two edges are not subdivided. The theorem generalizes earlier results of Gerards [J. Combin. Theory Ser. B, 47 (1989), pp. 330--348] on the strong t-perfection of odd-K4 -free graphs and of Gerards and Shepherd [SIAM J. Discrete Math., 11 (1998), pp. 524--545] on the t-perfection of bad-K4 -free graphs.