We study Vertex Contact representations of Paths on a Grid (VCPG). In such a representation, the vertices of G are represented by a family of interiorly disjoint grid-paths on a square grid. Adjacencies are represented by contacts between an endpoint of one grid-path and an interior point of another grid-path. Defining u → v if the path of u ends on the path of v, we obtain an orientation on G from a VCPG. To control the bends of the grid paths the orientation is not enough. We therefore consider pairs (α,ψ): a 2-orientation α and a flow ψ in the angle graph. The 2-orientation describes the contacts of the ends of a grid-path and the flow describes the behavior of a grid-path between its two ends. We give a necessary and sufficient condition for such a pair (α,ψ) to be realizable as a VCPG. Using realizable pairs, we show that every planar (2,2)-tight graph admits a VCPG with at most 2 bends per path and that this bound is tight. In a similar way, we show that simple planar (2,1)-sparse graphs have a 4-bend representation and simple planar (2,0)-sparse graphs have 6-bend representation.
Abstract.A straight line triangle representation (SLTR) of a planar graph is a straight line drawing such that all the faces including the outer face have triangular shape. Such a drawing can be viewed as a tiling of a triangle using triangles with the input graph as skeletal structure. In this paper we present a characterization of graphs that have an SLTR that is based on flat angle assignments, i.e., selections of angles of the graph that have size π in the representation. We also provide a second characterization in terms of contact systems of pseudosegments. With the aid of discrete harmonic functions we show that contact systems of pseudosegments that respect certain conditions are stretchable. The stretching procedure is then used to get straight line triangle representations. Since the discrete harmonic function approach is quite flexible it allows further applications, we mention some of them.The drawback of the characterization of SLTRs is that we are not able to effectively check whether a given graph admits a flat angle assignment that fulfills the conditions. Hence it is still open to decide whether the recognition of graphs that admit straight line triangle representation is polynomially tractable.
Contact and intersection representations of graphs and particularly of planar graphs have been studied for decades. The by now best known result in the area may be the Koebe-Andreev-Thurston circle packing theorem. A more recent highlight in the area is a result of Chalopin and Gonçalves: every planar graph is an intersection graph of segments in the plane. This boosted the study of intersection and contact graphs of restricted classes of curves. In this paper we study planar graphs that are VCPG, i.e. graphs admitting a representation as Vertex Contact graph of Paths on a Grid. In such a representation the vertices of G are represented by a family of interiorly disjoint grid-paths. Adjacencies are represented by contacts between an endpoint of one grid-path and an interior point of another grid-path. Defining u → v if the path of u ends on path of v we obtain an orientation on G from a VCPG representation. To get hand on the bends of the grid path the 2-orientation is not enough. We therefore consider pairs (α, ψ): a 2-orientation α and a flow ψ in the angle graph. The 2-orientation describes the contacts of the ends of a grid-path and the flow describes the behavior of a grid-path between its two ends. We give a necessary and sufficient condition for such a pair (α, ψ) to be realizable as a VCPG.Using realizable pairs we show that every planar (2,2)-tight graph can be represented with at most 2 bends per path and that this is tight (i.e. there exist (2,2)-tight graphs that cannot be represented with at most one bend per path). Using the same methodology it is easy to show that loopless planar (2,1)-sparse graphs have a 4-bend representation and loopless planar (2,0)-sparse graphs have 6-bend representation. We do not believe that the latter two are tight, we conjecture that loopless planar (2,0)-sparse graphs have a 3-bend representation.
A straight line triangle representation (SLTR) of a planar graph is a straight line drawing such that all the faces including the outer face have triangular shape. Such a drawing can be viewed as a tiling of a triangle using triangles with the input graph as skeletal structure. In this paper we present a characterization of graphs that have an SLTR. The characterization is based on flat angle assignments, i.e., selections of angles of the graph that have size π in the representation. We also provide a second characterization in terms of contact systems of pseudosegments. With the aid of discrete harmonic functions we show that contact systems of pseudosegments that respect certain conditions are stretchable. The stretching procedure is then used to get straight line triangle representations. Since the discrete harmonic function approach is quite flexible it allows further applications, we mention some of them.The drawback of the characterization of SLTRs is that we are not able to effectively check whether a given graph admits a flat angle assignment that fulfills the conditions. Hence it is still open to decide whether the recognition of graphs that admit straight line triangle representation is polynomially tractable.
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