Abstract
For a property π on graphs, the corresponding edge-contraction problem P EC( π) is defined as follows: Given a graph G, find a set of edges of minimum cardinality whose contraction results in a graph satisfying property π. In this paper we show that the edge-contraction problem P EC( π) is NP-hard if π is hereditary on contractions and is determined by the biconnected components. Moreover, this problem is NP-hard even if we restrict ourselves to the class of planar graphs. Furthermore, if we add a condition that π is determined by the 3-connected components, then P EC( π) is NP-hard even if restricted to 3-connected graphs and to bipartite graphs.
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