Planar order on vertex poset

2019

We give a combinatorial characterization of upward planar graphs in terms of upward planar orders, which are special linear extensions of edge posets.

Lecture Notes in Computer Science, 2012

A universal point-set supports a crossing-free drawing of any planar graph. For a planar graph with n vertices, if bends on edges of the drawing are permitted, universal point-sets of size n are known, but only if the bend-points are in arbitrary positions. If the locations of the bend-points must also be specified as part of the point-set, we prove that any planar graph with n vertices can be drawn on a universal set S of O(n 2 / log n) points with at most one bend per edge and with the vertices and the bend points in S. If two bends per edge are allowed, we show that O(n log n) points are sufficient, and if three bends per edge are allowed, Θ(n) points are sufficient. When no bends on edges are permitted, no universal point-set of size o(n 2 ) is known for the class of planar graphs. We show that a set of n points in balanced biconvex position supports the class of maximum degree 3 series-parallel lattices.

1990

We give finite list of orders such that an order of width two is planar if and only if it does not contain a subdiagram a homeomorph of a member of this list. The proof of our result yields an O(n 3/2 ) algorithm to test planarity of orders of width two with n vertices.

2011

A geometric graph G(bar) is a simple graph drawn in the plane, on points in general position, with straight-line edges. We call G(bar) a geometric realization of the underlying abstract graph G. A geometric homomorphism is a vertex map that preserves adjacencies and crossings (but not necessarily non-adjacencies or non-crossings). This work uses geometric homomorphisms to introduce a partial order

1998

Discrete Mathematics, 2009

Journal of Graph Algorithms and Applications, 2013

A graph is B k -VPG when it has an intersection representation by paths in a rectangular grid with at most k bends (turns). It is known that all planar graphs are B3-VPG and this was conjectured to be tight. We disprove this conjecture by showing that all planar graphs are B2-VPG. We also show that the 4-connected planar graphs constitute a subclass of the intersection graphs of Z-shapes (i.e., a special case of B2-VPG). Additionally, we demonstrate that a B2-VPG representation of a planar graph can be constructed in O(n 3/2 ) time. We further show that the triangle-free planar graphs are contact graphs of: L-shapes, Γ-shapes, vertical segments, and horizontal segments (i.e., a special case of contact B1-VPG). From this proof we obtain a new proof that bipartite planar graphs are a subclass of 2-DIR.

Journal of Graph Theory, 2006

A graph G is k-ordered if for every ordered sequence of k vertices, there is a cycle in G that encounters the vertices of the sequence in the given order. We prove that if G is a connected graph distinct from a path, then there is a number t G such that for every t ≥ t G the t-iterated line graph of G, L t (G), is (δ(L t (G))+1)-ordered. Since there is no graph H which is (δ(H)+2)-ordered, the result is best possible.

Graphs and Combinatorics, 2014

Kelly showed that there exist planar posets of arbitrarily large dimension, and Streib and Trotter showed that the dimension of a poset with a planar cover graph is bounded in terms of its height. Here we continue the study of conditions that bound the dimension of posets with planar cover graphs. We show that if P is poset with a planar comparability graph, then the dimension of P is at most four. We also show that if P has an outerplanar cover graph, then the dimension of P is at most four. Finally, if P has an outerplanar cover graph and the height of P is two, then the dimension of P is at most three. These three inequalities are all best possible.

arXiv (Cornell University), 2023

We give a more coherent and apparent definition of upward planar order.