The complexity of the Bondage problem in planar graphs
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A set S⊆ V(G) of a graph G is a dominating set if each vertex has a neighbor in S or belongs to S. Let γ(G) be the cardinality of a minimum dominating set in G. The bondage number b(G) of a graph G is the smallest number of edges A⊆ E(G), such that γ(G-A)=γ(G)+1. The problem of finding b(G) for a graph G is known to be NP-hard even for bipartite graphs. In this paper, we show that deciding if b(G)=1 is NP-hard, while deciding if b(G)=2 is coNP-hard, even when G is restricted to one of the following classes: planar 3-regular graphs, planar claw-free graphs with maximum degree 3, planar bipartite graphs of maximum degree 3 with girth k, for any fixed k≥ 3.
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