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Abstract: A graph is a segment graph if its vertices can be mapped to line segments in the plane such that two vertices are adjacent if and only if their corresponding line segments intersect. Kratochvíl and Kuběna asked the question of whether the complements of planar graphs, called co-planar graphs, are segment graphs. We show here that the complements of all partial 2-trees are segment graphs. [Copyright &y& Elsevier]

We study the computational complexity of graph planarization via edge contraction. The problem Contract asks whether there exists a set S of at most k edges that when contracted produces a planar graph. We work with a more general problem called P - RestrictedContract in which S , in addition, is required to satisfy a fixed MSOL formula P ( S , G ) . We give an FPT algorithm in time O ( n 2 f ( k ) ) which solves P - RestrictedContract , where n is number of vertices of the graph and P ( S , G ) is (i) inclusion-closed and (ii) inert contraction-closed (where inert edges are the edges non-incident to any inclusion-minimal solution S ). As a specific example, we can solve the ℓ -subgraph contractibility problem in which the edges of the set S are required to form disjoint connected subgraphs of size at most ℓ . This problem can be solved in time O ( n 2 f ′ ( k , ℓ ) ) using the general algorithm. We also show that for ℓ ≥ 2 the problem is NP -complete. [ABSTRACT FROM AUTHOR]