The MAXIMUM PLANAR SUBGRAPH problem---given a graph G, find a largest planar subgraph of G---has applications in circuit layout, facility layout, and graph drawing. No previous polynomialtime approximation algorithm for this NP-Complete problem was known to achieve a performance ratio larger than 1=3, which is achieved simply by producing a spanning tree of G. We present the first approximation algorithm for MAXIMUM PLANAR SUBGRAPH with higher performance ratio (4=9 instead of 1=3). We also apply our algorithm to find large outerplanar subgraphs. Last, we show that both MAXIMUM PLANAR SUBGRAPH and its complement, the problem of removing as few edges as possible to leave a planar subgraph, are Max SNP-Hard.

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Abstract. We construct an optimal linear-time algorithm for the maximal planar subgraph problem: given a graph G, find a planar subgraph G ′ of G such that adding to G ′ an extra edge of G results in a non-planar graph. Our solution is based on a fast data structure for incremental planarity testing of triconnected graphs and a dynamic graph search procedure. Our algorithm can be transformed into a new optimal planarity testing algorithm. Key words. Planar graphs, planarity testing, incremental algorithms, graph planarization, data structures, triconnectivity. AMS subject classifications. 05C10, 05C85, 68R10, 68Q25, 68W40 1. Introduction. Agraphisplanar

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