Abstract. Ramsey proved that for every positive integer [Formula: see text], every sufficiently large graph contains an induced [Formula: see text] or [Formula: see text]. Among the many extensions of Ramsey’s theorem, there is an analogue for connected graphs: For every positive integer [Formula: see text], every sufficiently large connected graph contains an induced [Formula: see text], [Formula: see text], or [Formula: see text]. In this paper, we establish an analogue for 2-connected graphs. In particular, we prove that for every integer exceeding two, every sufficiently large 2-connected graph contains one of the following as an induced subgraph: [Formula: see text], a subdivision of [Formula: see text], a subdivision of [Formula: see text] with an edge between the two vertices of degree [Formula: see text], and a well-defined structure similar to a ladder.

Wavelengths and converters are shared by communication requests in optical networks. The usage of converters increases the utilization of wavelengths and allows more requests to succeed. The converters usage problem (CUP) is to determine the minimum number of converter so that each node can send messages to all the others (broadcasting). In this paper, we study the CUP in sparse conversion networks with bounded treewidth. A converter wavelength-dominates a node if there is a uniform wavelength path between them. The minimal wavelength dominating set problem (MWDSP) is to locate the minimum number of converters so that all the other nodes in the network are wavelength-dominated. We use a linear complexity dynamic programming algorithm to solve the MWDSP for networks with bounded treewidth. One such solution provides a low bound for the optimal solution to the CUP.