On the Bend-Number of Planar and Outerplanar Graphs
نویسندگان
چکیده
The bend-number b(G) of a graph G is the minimum k such that G may be represented as the edge intersection graph of a set of grid paths with at most k bends. We confirm a conjecture of Biedl and Stern showing that the maximum bend-number of outerplanar graphs is 2. Moreover we improve the formerly known lower and upper bound for the maximum bend-number of planar graphs from 2 and 5 to 3 and 4, respectively.
منابع مشابه
Graph Drawings with One Bend and Few Slopes
We consider drawings of graphs in the plane in which edges are represented by polygonal paths with at most one bend and the number of different slopes used by all segments of these paths is small. We prove that d 2 e edge slopes suffice for outerplanar drawings of outerplanar graphs with maximum degree ∆ > 3. This matches the obvious lower bound. We also show that d 2 e + 1 edge slopes suffice ...
متن کاملEdge-intersection graphs of grid paths: the bend-number
We investigate edge-intersection graphs of paths in the plane grid regarding a parameter called the bend-number. The bend-number is related to the interval-number and the track-number of a graph. We provide new upper and lower bounds of the bend-number of any given simple graph in terms of the coloring number, edge clique covers and the maximum degree. We show that the bend-number of an outerpl...
متن کاملA note on simultaneous embedding of planar graphs
Let G1 and G2 be a pair of planar graphs such that V (G1) = V (G2) = V . A simultaneous embedding [6] Ψ = (Γ1,Γ2) of G1 and G2 is a pair of crossing-free drawings Γ1 and Γ2 of G1 and G2, respectively, such that for every vertex v ∈ V we have Γ1(v) = Γ2(v). If every edge e ∈ E(G1) ∩ E(G2) is represented with the same simple open Jordan curve both in Γ1 and in Γ2 we say that Ψ is a simultaneous e...
متن کاملEdge-Intersection Graphs of k-Bend Paths in Grids
Edge-intersection graphs of paths in grids are graphs that can be represented with vertices as paths in grids and edges between the vertices of the graph exist whenever two grid paths share a grid edge. This type of graphs is motivated by applications in conflict resolution of paths in grid networks. In this paper, we continue the study of edge-intersection graphs of paths in a grid, which was ...
متن کاملEmbedding Vertices at Points: Few Bends Suffice for Planar Graphs
The existing literature gives efficient algorithms for mapping trees or less restrictively outerplanar graphs on a given set of points in a plane, so that the edges are drawn planar and as straight lines. We relax the latter requirement and allow very few bends on each edge while considering general plane graphs. Our results show two algorithms for mapping four-connected plane graphs with at mo...
متن کامل