Lecture Notes in Computer Science
We introduce the family of k-gap-planar graphs for k ≥ 0, i.e., graphs that have a drawing in which each crossing is assigned to one of the two involved edges and each edge is assigned at most k of its crossings. This definition is motivated by applications in edge casing, as a k-gap-planar graph can be drawn crossing-free after introducing at most k local gaps per edge. We present results on the maximum density of k-gap-planar graphs, their relationship to other classes of beyond-planar
... characterization of k-gap-planar complete graphs, and the computational complexity of recognizing kgap-planar graphs. A drawing Γ of a graph G = (V, E) is a mapping of the vertices of V to distinct points, and of the edges of E to a continuous arcs connecting their corresponding endpoints such that no edge (arc) passes through any vertex, if two edges have a common interior point in Γ, then they cross transversely at that point, and no three edges cross at the same point. For a subset E ⊆ E, the restriction of Γ to the curves representing the edges of E is denoted by Γ [E ]. A drawing Γ is planar if no two edges cross. A graph is planar if it admits a planar drawing. A planar embedding of a planar graph G is an equivalence class of topologically equivalent (i.e., isotopic) planar drawings of G. A plane graph is a planar graph with a planar embedding. A planar drawing subdivides the plane into topologically connected regions, called faces. The unbounded region is the outer face. The crossing number cr(G) of a graph G is the smallest number of edge crossings over all drawings of G. The crossing graph C(Γ) of a drawing Γ is the graph having a vertex v e for each edge e of G, and an edge (v e , v f ) if and only if edges e and f cross in Γ. The planarization Γ * of Γ is the plane graph formed from Γ by inserting a dummy vertex at each crossing, and subdividing both edges with the dummy vertex. To avoid ambiguities, we call real vertices the vertices of Γ * that are in V (i.e., that are not dummy). A class of graphs is informally called "beyond-planar" if the graphs in this family admit drawings in which the intersection patterns of the edges are characterized by some forbidden configuration (see, e.g., [33, 35, 41] ). Research on such graph classes is attracting increasing attention in graph theory, graph algorithms, graph drawing, and computational geometry, as these graphs represent a natural generalization of planar graphs, and their study can provide significant insights for the design of effective methods to visualize realworld networks. Indeed, the motivation for this line of research stems from both the interest raised by the combinatorial and geometric properties of these graphs, and experiments showing how the absence of particular edge crossing patterns has a positive impact on the readability of a graph drawing  . Among the investigated families of beyond-planar graphs are: k-planar graphs (see, e.g., [12, 39, 43] ), which can be drawn in the plane with at most k crossings per edge; k-quasiplanar graphs (see, e.g., [2, 3, 25]), which can drawn without k pairwise crossing edges; fan-planar graphs (see, e.g., [9, 13, 37] ), which can be drawn such that no edge crosses two independent edges; fan-crossing-free graphs  , which can be drawn such that no edge crosses any two edges that are adjacent to each other; planarly-connected graphs , which can be drawn such that each pair of crossing edges is independent and there is a crossing-free edge that connects their endpoints; RAC graphs (refer, e.g., to  ), which admit a straight-line (or polyline with few bends) drawing where any two crossing edges are perpendicular to each other. Eppstein et al.  studied several optimization problems related to edge casing, assuming the input is a graph together with a fixed drawing. In particular, the problem of minimizing the maximum number of gaps per edge in a drawing can be solved in polynomial time (see also Section 2). We also note that a similar drawing paradigm is used by partial edge drawings (PEDs), in which the central part of each edge is erased, while the two remaining stubs are required to be crossing-free (see, e.g., [16, 17] ). Let Γ be a drawing of a graph G. Recall that exactly two edges of G cross in one point p of Γ, and we say that these two edges are responsible for p. A k-gap assignment of Γ maps each crossing point of Γ to one of its two responsible edges so that each edge is assigned with at most k of its crossings; see, e.g., Fig. 1(right) . A gap of an edge is a crossing assigned to it. An edge with at least one gap is gapped, else it is gap-free. A drawing is k-gap-planar if it admits a k-gap assignment. A graph is k-gap-planar if it has a k-gap-planar drawing. Note that a graph is planar if and only if it is 0-gap-planar, and that k-gap-planarity is a monotone property: every subgraph of a k-gap-planar graph is k-gap-planar. The summation of the number of gaps over all edges in a set E ⊂ E yields the following. Property 1. Let Γ be a k-gap-planar drawing of a graph G = (V, E). For every E ⊆ E, the subdrawing Γ[E ] contains at most k · |E | crossings. In fact, the converse of Property 1 also holds, and we obtain the following stronger result. Theorem 2. Let Γ be a drawing of a graph G = (V, E). The drawing Γ is k-gap-planar if and only if for each edge set E ⊆ E the subdrawing Γ[E ] contains at most k · |E | crossings.