Steven Chaplick

Planar Graphs as VPG-Graphs

A graph is Bk-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) 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. Submitted: December 2012 Reviewed: March 2013 Revised: April 2013 Accepted: May 2013 Final: July 2013 Published: July 2013 Article type: Regular paper Communicated by: W. Didimo and M. Patrignani An extended abstract of this paper was presented at the 20th International Symposium on Graph Drawing, in Redmond, USA, in September 2012 [7]. The research of the first author was supported by NSERC and partially by GraDR EUROGIGA project No. GIG/11/E023. The research of the second author was supported by GraDR EUROGIGA project No. GIG/11/E023. E-mail addresses: chaplick@kam.mff.cuni.cz (Steven Chaplick) torsten.ueckerdt@kit.edu (Torsten Ueckerdt) 476 Chaplick and Ueckerdt Planar Graphs as VPG-Graphs

Recognizing some subclasses of vertex intersection graphs of 0-bend paths in a grid

We investigate graphs that can be represented as vertex intersections of horizontal and vertical paths in a grid, known as B0 -VPG graphs. Recognizing these graphs is an NP-hard problem. In light of this, we focus on their subclasses. In the paper, we describe polynomial time algorithms for recognizing chordal B0 -VPG graphs, and for recognizing B0 -VPG graphs that have a representation on a grid with 2 rows.

Contact Representations of Planar Graphs: Extending a Partial Representation is Hard

Planar graphs are known to have geometric representations of various types, e.g. as contacts of disks, triangles or - in the bipartite case - vertical and horizontal segments. It is known that such representations can be drawn in linear time, we here wonder whether it is as easy to decide whether a partial representation can be completed to a representation of the whole graph. We show that in each of the cases above, this problem becomes NP-hard. These are the first classes of geometric graphs where extending partial representations is provably harder than recognition, as opposed to e.g. interval graphs, circle graphs, permutation graphs or even standard representations of plane graphs.

Equilateral L-Contact Graphs

We consider L-graphs, that is contact graphs of axis-aligned L-shapes in the plane, all with the same rotation. We provide several characterizations of L-graphs, drawing connections to Schnyder realizers and canonical orders of maximally planar graphs. We show that every contact system of L’s can always be converted to an equivalent one with equilateral L’s. This can be used to show a stronger version of a result of Thomassen, namely, that every planar graph can be represented as a contact system of square-based cuboids.

Bend-bounded path intersection graphs: sausages, noodles, and waffles on a grill

In this paper we study properties of intersection graphs of k-bend paths in the rectangular grid. A k-bend path is a path with at most k 90 degree turns. The class of graphs representable by intersections of k-bend paths is denoted by Bk-VPG. We show here that for every fixed k, Bk-VPG

On the structure of (pan, even hole)‐free graphs

\subsetneq

Max point-tolerance graphs

Bk+1-VPG and that recognition of graphs from Bk-VPG is NP-complete even when the input graph is given by a Bk+1-VPG representation. We also show that the class Bk-VPG (for k≥1) is in no inclusion relation with the class of intersection graphs of straight line segments in the plane.

Intersection Dimension of Bipartite Graphs

A hole is a chordless cycle with at least four vertices. A pan is a graph which consists of a hole and a single vertex with precisely one neighbor on the hole. An even hole is a hole with an even number of vertices. We prove that a (pan, even hole)-free graph can be decomposed by clique cutsets into essentially unit circular-arc graphs. This structure theorem is the basis of our

Edge Intersection Graphs of L-Shaped Paths in Grids

O(nm)

Grid Intersection Graphs and Order Dimension

-time certifying algorithm for recognizing (pan, even hole)-free graphs and for our