The Graphs with All Subgraphs T-Perfect

The richest class of t-perfect graphs known so far consists of the graphs with no so-called odd-K. Clearly, these graphs have the special property that they are hereditary t-perfect in the sense that every subgraph is also t-perfect, but they are not the only ones. In this paper we characterize hereditary t-perfect graphs by showing that any non--t-perfect graph contains a non--t-perfect subdivision of K4, called a bad-K4. To prove the result we show which "weakly 3-connected" graphs contain no bad-K4; as a side-product of this we get a polynomial time recognition algorithm. It should be noted that our result does not characterize t-perfection, as that is not maintained when taking subgraphs but only when taking induced subgraphs.