On Succinctness of Geometric Greedy Routing in Euclidean Plane

A Survey on Small-Area Planar Graph Drawing

“…Partially positive results on the mentioned open problem were achieved by He and Zhang, who proved in [83] that succinct convex weekly greedy drawings exist for all triconnected planar graphs, where weekly greedy means that the distance between two vertices u and v in the drawing is not the usual Euclidean distance D(u, v) but a function H(u, v) such that D(u, v) ≤ H(u, v) ≤ 2 √ 2D(u, v). On the other hand, Cao et al proved in [22] that there exist triconnected planar graphs requiring exponential area in any convex greedy drawing in the Euclidean plane.…”

On Planar Greedy Drawings of 3-Connected Planar Graphs

“…In this paper, we refuted a claim by Cao et al [6] and re-opened the question of whether 3-connected planar graphs admit planar, and possibly convex, greedy drawings on a polynomial-size grid. Further, we provided some evidence for a positive answer by showing that every n-vertex Halin graph admits a convex greedy drawing on an O(n) × O(n) grid; in fact, our drawings are angle-monotone, which is a stronger property than greediness.…”

“…Our contributions. We show that every n-vertex graph in the family H defined by Cao et al [6] actually admits a convex angle-monotone drawing that respects the prescribed plane embedding and that lies on an O(n) × O(n) grid. This refutes their claim that every planar greedy drawing of an nvertex graph in H requires Ω(n) bits for representing the coordinates of some vertices and reopens the question about the existence of succinct planar greedy drawings of 3-connected planar graphs.…”

“…3 Angle-Monotone Drawings of Cao-Strelzoff-Sun Graphs Cao et al [6] defined the following family H of plane graphs. For every integer i ≥ 1, the plane graph H i ∈ H on 3i + 4 vertices is inductively defined as follows:…”

“…That is, does every 3-connected planar graph admit a planar, and possibly convex, greedy drawing on a polynomial-size grid? Cao, Strelzoff, and Sun [6] claimed a negative answer by exhibiting a family H of subdivisions of 3-connected plane graphs and by showing that, for any n-vertex graph in H, any planar greedy drawing that respects the prescribed plane embedding requires 2 Ω(n) area and hence Ω(n) bits for representing the coordinates of some vertices.…”