Graph Planarization
نویسندگان
چکیده
A graph is said to be planar if it can be drawn on the plane in such a way that no two of its edges cross. Given a graph G = (V,E) with vertex set V and edge set E, the objective of graph planarization is to find a minimum cardinality subset of edges F ⊆ E such that the graph G = (V,E \ F ), resulting from the removal of the edges in F from G, is planar. This problem is also known as the maximum planar subgraph problem. A related and simpler problem is that of finding a maximal planar subgraph, which is a planar subgraph G = (V,E) of G such that the addition of any edge e ∈ E \E to G destroys its planarity. Graph planarization is known to be NP-hard [15]. The proof of NP-completeness of its decision version is based on a transformation from the Hamiltonian path problem restricted to bipartite graphs. Although exact methods for solving the maximum planar subgraph problem have been recently proposed, most algorithms to date attempt to find good approximate solutions. In this article, we survey graph planarization and related problems. In the next section, we describe variants and applications of the basic problem formulated above. Next, we describe the branch-and-cut algorithm of Jünger and Mutzel [11]. We then review work on heuristics based on planarity testing and those based on two-phase procedures. Finally, computational results are considered.
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