We prove that every graph $G$ contains either $k$ edge-disjoint $K_4$-subdivisions or a set $X$ of at most $\ensuremath{O}(k^8 \log k)$ edges such that $G-X$ does not contain any $K_4$-subdivision. This shows that $K_4$-subdivisions have the edge-Erdös--Pósa property.

In the 1960s, Erdös and Pósa proved that there is a packing-covering duality for cycles in graphs. As part of the graph minor project, Robertson and Seymour greatly extended this: there is such a duality for $H$-expansions in graphs if and only if $H$ is a planar graph (this includes the previous result for $H=K_3$). We consider vertex labeled graphs and minors and provide such a characterization for 2-connected labeled graphs $H$. In particular, this generalizes results of Kakimura, Kawarabayashi and Marx [ J. Combin. Theory Ser. B, 101 (2011), pp. 378--381] and Huynh, Joos, and Wollan [ Combinatorica, 39 (2019), pp. 91--133] up to weaker dependencies of the parameters.

A key feature of Simonovits' proof of the classic Erdös--Pósa theorem is a simple subgraph of the host graph, a frame, that determines the outcome of the theorem. We transfer this frame technique to $A$-paths. With it we deduce a simple proof of Gallai's theorem, although with a worse bound, and we verify the Erdös--Pósa property for long and for even $A$-paths. We also show that even $A$-paths do not have the edge-Erdös--Pósa property.

A graph is called $t$-perfect if its stable set polytope is fully described by nonnegativity, edge, and odd-cycle constraints. We characterize $P_5$-free $t$-perfect graphs in terms of forbidden $t$-minors. Moreover, we show that $P_5$-free $t$-perfect graphs can always be colored with three colors and that they can be recognized in polynomial time.

A graph is called $t$-perfect if its stable set polytope is defined by nonnegativity, edge, and odd-cycle inequalities. We show that it can be decided in polynomial time whether a given claw-free graph is $t$-perfect.

A connected graph G is called t-perfect if its stable set polytope is determined by the nonnegativity, edge, and odd-cycle inequalities. Moreover, G is called strongly t-perfect if this system is totally dual integral. It is an open problem whether t-perfection is equivalent to strong t-perfection. We prove the equivalence for the class of claw-free graphs.

A classical theorem by Tutte ensures the existence of a Hamilton cycle in every finite 4-connected planar graph. Extensions of this result to infinite graphs require a suitable concept of an infinite cycle. Such a concept was provided by Diestel and Kühn, who defined circles to be homeomorphic images of the unit circle in the Freudenthal compactification of the (locally finite) graph. With this definition we prove a partial extension of Tutte's result to locally finite graphs.