A Dichotomy for 1-Planarity with Restricted Crossing Types Parameterized by Treewidth
📝 Abstract
A drawing of a graph is 1-planar if each edge participates in at most one crossing and adjacent edges do not cross. Up to symmetry, each crossing in a 1-planar drawing belongs to one out of six possible crossing types, where a type characterizes the subgraph induced by the four vertices of the crossing edges. Each of the 63 possible nonempty subsets $\mathcal{S}$ of crossing types gives a recognition problem: does a given graph admit an $\mathcal{S} $-restricted drawing, that is, a 1-planar drawing where the crossing type of each crossing is in $\mathcal{S} $? We show that there is a set $\mathcal{S}_{\rm bad}$ with three crossing types and the following properties: If $\mathcal{S}$ contains no crossing type from $\mathcal{S}_{\rm bad} $, then the recognition of graphs that admit an $\mathcal{S} $-restricted drawing is fixed-parameter tractable with respect to the treewidth of the input graph. If $\mathcal{S}$ contains any crossing type from $\mathcal{S}_{\rm bad} $, then it is NP-hard to decide whether a graph has an $\mathcal{S} $-restricted drawing, even when considering graphs of constant pathwidth. We also extend this characterization of crossing types to 1-planar straight-line drawings and show the same complexity behaviour parameterized by treewidth.
💡 Analysis
A drawing of a graph is 1-planar if each edge participates in at most one crossing and adjacent edges do not cross. Up to symmetry, each crossing in a 1-planar drawing belongs to one out of six possible crossing types, where a type characterizes the subgraph induced by the four vertices of the crossing edges. Each of the 63 possible nonempty subsets $\mathcal{S}$ of crossing types gives a recognition problem: does a given graph admit an $\mathcal{S} $-restricted drawing, that is, a 1-planar drawing where the crossing type of each crossing is in $\mathcal{S} $? We show that there is a set $\mathcal{S}_{\rm bad}$ with three crossing types and the following properties: If $\mathcal{S}$ contains no crossing type from $\mathcal{S}_{\rm bad} $, then the recognition of graphs that admit an $\mathcal{S} $-restricted drawing is fixed-parameter tractable with respect to the treewidth of the input graph. If $\mathcal{S}$ contains any crossing type from $\mathcal{S}_{\rm bad} $, then it is NP-hard to decide whether a graph has an $\mathcal{S} $-restricted drawing, even when considering graphs of constant pathwidth. We also extend this characterization of crossing types to 1-planar straight-line drawings and show the same complexity behaviour parameterized by treewidth.
📄 Content
Drawings of graphs are encountered early and used often: children are challenged to draw K 3,3 without crossings, drawings of graphs are regularly used when teaching graphs, they appear in any graph-related textbook, and we use them in our research discussions. Drawings
of graphs naturally led to the concept of planar graphs and, more generally, an interest to control the crossings in the drawings. Planar graphs enjoy a rich structural theory, have spurred research in graph theory and graph algorithms, and are relatively well understood. However, they are a quite restrictive class of graphs, and non-planar graphs still have to be drawn. For this reason, a range of beyond-planarity concepts have been suggested and investigated [14,23,32]. In this work, we will focus on 1-planarity, one of the extensions of planarity that has attracted much interest [24]. A drawing of a graph is 1-planar (or has local crossing number 1) if each edge participates in at most one crossing and adjacent edges do not cross1 ; see Figure 1, left, for an example. A graph is 1-planar iff it admits a 1-planar drawing.
Generally speaking, the class of 1-planar graphs is not well understood. We know that recognizing 1-planar graphs is computationally hard [18,25], even for graphs that are obtained from a planar graph by adding a single edge [11], or for graphs with constant treewidth, pathwidth or even bandwidth [1]. A fixed-parameter algorithm for deciding the existence of 1-planar drawings has been obtained recently by Hamm and Hlinený [21] (and actually is an easy consequence of [19]) using the total number of crossings in the drawing as a parameter.
Biedl and Murali [4] noticed that the crossings in a 1-planar drawing can be classified into different types, and the (non-)existence of some types may have important consequences. We will call these different types crossing types, and they describe adjacencies between the endpoints of the edges involved in that crossing. Consequently, there are six different crossing types (up to symmetry), for which we follow the terminology from [4]: (1) A full crossing is one in which each endpoint of an edge involved in that crossing is connected to each other such endpoint ( ), (2) an almost full crossing is one in which all but one pair of endpoints of the edges involved in that crossing are connected to each other ( ), (3) a bowtie crossing is one in which the graph induced by the endpoints of the edges involved in that crossing is a 4-cycle ( ), (4) an arrow crossing is one in which there is precisely one endpoint of an edge involved in that crossing that is connected to all other such endpoints ( ), (5) a chair crossing is one in which all but one pair of endpoints of the edges involved in that crossing are independent if one were to remove the edges involved in the crossing ( ) (6) a × crossing is one in which all endpoints of the edges involved in that crossing are independent if one were to remove the edges involved in the crossing ( ). See Figure 1, left, for an example.
For each non-empty S ⊆ { , , , , , }, we say that a drawing is S-restricted 1-planar if it is 1-planar and the type of each crossing belongs to S, and a graph is S-restricted 1-planar if it admits an S-restricted 1-planar drawing.
We already know that limiting the crossing types can have consequences. For instance, Biedl and Murali [4] described an algorithm to compute the vertex connectivity of a graph that is given with a 1-planar drawing without crossings. This was extended by Biedl, Bose and Murali [3] to allow some crossings in a controlled manner. Bose et al. [8] have shown that graphs that have a 1-planar -crossing-free drawing have bounded cop-number. However, it was not clear how difficult it is to decide if a given graph has such a 1-planar drawing without crossings, and this is posed explicitly as an open problem in [4].
On the other end of the spectrum when it comes to crossing types, Brandenburg [9,10] has shown that the class of 1-planar graphs that admit 1-planar drawings where all crossings are -crossings is equivalent to the class of 4-map graphs with holes, and thus { }-restricted 1-planar graphs are recognizable in polynomial time. Münch and Rutter [27] show that, for 1-Planarity with Restricted Crossing Types 3-connected graph by subdividing some edges once. This step is quite standard but tedious, especially in the case of geometric drawings, where we have to slightly generalize the problem we consider so that a prescribed vertex has to lie on the outer face of the target 1-planar drawing. The reductions do use that the crossing types are a subset of { , , }.
To solve the instances with enough connectivity, we use the fact that we target treewidth parameterizations to be able to invoke Courcelle’s theorem [12]. However, expressing 1planarity is something that is not possible efficiently in MSO (the logic required to formulate a problem in so that Courcelle’s theorem implies a fixed-parameter algor
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