Simple Realizability of Abstract Topological Graphs

Abstract An abstract topological graph ( AT-graph ) is a pair $$A=(G,\mathcal {X})$$ A = ( G , X ) , where $$G=(V,E)$$ G = ( V , E ) is a graph and $$\mathcal {X} \subseteq \left( {\begin{array}{c}E\\ 2\end{array}}\right) $$ X ⊆ E 2 is a set of pairs of edges of G . A realization of A is a drawing $$\Gamma _A$$ Γ A of G in the plane such that any two edges $$e_1,e_2$$ e 1 , e 2 of G cross in $$\Gamma _A$$ Γ A if and only if $$(e_1,e_2) \in \mathcal {X} $$ ( e 1 , e 2 ) ∈ X ; $$\Gamma _A$$ Γ A is simple if any two edges intersect at most once (either at a common endpoint or at a proper crossing). The AT-graph Realizability ( ATR ) problem asks whether an input AT-graph admits a realization. The version of this problem that requires a simple realization is called Simple AT-graph Realizability ( SATR ). It is a classic result that both ATR and SATR are -complete [18, 21]. In this paper, we study the SATR problem from a new structural perspective. More precisely, we consider the size $$\mathrm {\lambda }(A)$$ λ ( A ) of the largest connected component of the crossing graph of any realization of A , i.e., the graph $$\mathcal{C}(A) = (E, \mathcal {X})$$ C ( A ) = ( E , X )