Finding the largest planar subgraph[edit]
Using incremental planarization for graph drawing is most effective when the first step of the process finds as large a planar graph as possible. Unfortunately, finding the planar subgraph with the maximum possible number of edges is NP-hard, and MaxSNP-hard, implying that there probably does not exist a polynomial time algorithm that solves the problem exactly or that approximates it arbitrarily well.[3]
In an n-vertex connected graph, the largest planar subgraph has at most 3n − 6 edges, and any spanning tree forms a planar subgraph with n − 1 edges. Thus, it is easy to approximate the maximum planar subgraph within an approximation ratio of one-third, simply by finding a spanning tree. A better approximation ratio, 9/4, is known, based on a method for finding a large partial 2-tree as a subgraph of the given graph.[1][3] Alternatively, if it is expected that the planar subgraph will include almost all of the edges of the given graph, leaving only a small number k of non-planar edges for the incremental planarization process, then one can solve the problem exactly by using a fixed-parameter tractable algorithm whose running time is linear in the graph size but non-polynomial in the parameter k.[4] The problem may also be solved exactly by a branch and cut algorithm, with no guarantees on running time, but with good performance in practice.[1][5]
There has also been some study of a related problem, finding the largest planar induced subgraph of a given graph. Again, this is NP-hard, but fixed-parameter tractable when all but a few vertices belong to the induced subgraph.[6] Edwards & Farr (2002) proved a tight bound of 3n/(Δ + 1) on the size of the largest planar induced subgraph, as a function of n, the number of vertices in the given graph, and Δ, its maximum degree; their proof leads to a polynomial time algorithm for finding an induced subgraph of this size.[7]
Finding the largest planar subgraph[edit]Using incremental planarization for graph drawing is most effective when the first step of the process finds as large a planar graph as possible. Unfortunately, finding the planar subgraph with the maximum possible number of edges is NP-hard, and MaxSNP-hard, implying that there probably does not exist a polynomial time algorithm that solves the problem exactly or that approximates it arbitrarily well.[3]In an n-vertex connected graph, the largest planar subgraph has at most 3n − 6 edges, and any spanning tree forms a planar subgraph with n − 1 edges. Thus, it is easy to approximate the maximum planar subgraph within an approximation ratio of one-third, simply by finding a spanning tree. A better approximation ratio, 9/4, is known, based on a method for finding a large partial 2-tree as a subgraph of the given graph.[1][3] Alternatively, if it is expected that the planar subgraph will include almost all of the edges of the given graph, leaving only a small number k of non-planar edges for the incremental planarization process, then one can solve the problem exactly by using a fixed-parameter tractable algorithm whose running time is linear in the graph size but non-polynomial in the parameter k.[4] The problem may also be solved exactly by a branch and cut algorithm, with no guarantees on running time, but with good performance in practice.[1][5]There has also been some study of a related problem, finding the largest planar induced subgraph of a given graph. Again, this is NP-hard, but fixed-parameter tractable when all but a few vertices belong to the induced subgraph.[6] Edwards & Farr (2002) proved a tight bound of 3n/(Δ + 1) on the size of the largest planar induced subgraph, as a function of n, the number of vertices in the given graph, and Δ, its maximum degree; their proof leads to a polynomial time algorithm for finding an induced subgraph of this size.[7]
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