Compact Grid Representation of Graphs

Abstract

Abstract.A graph G is said to be grid locatable if it admits a representation such that vertices are mapped to grid points and edges to line segments that avoid grid points but the extremes. Additionally G is said to be properly embeddable in the grid if it is grid locatable and the segments representing edges do not cross each other. We study the area needed to obtain those representations for some graph families.

Grid Drawings and the Chromatic Number

“…As a consequence we obtain that drawing/locating of graphs on a bounded number of columns is a difficult task answering the open question in [2]. This problem was already partially solved, as Chappell, Gimbel and Hartman [3] proved that determining whether G can be colored with at least two path colors is NPcomplete.…”

“…al. [2] showed that every outerplanar graph can be drawn (and located) on two columns. In the same paper there is an example of a planar graph which is not (2, 1)-colorable.…”

“…According to Proposition 1 and 2 it suffices to show by induction on d that the assumption in the theorem implies that G is 0, 2 dcolorable. For d = 1 we can use the proof of a theorem proved in [2] which says that every graph with maximum degree at most three can be located on two columns in Z 2 .…”

Grid representations and the chromatic number

“…, V 2 d such that every induced subgraph G [V i ] is isomorphic to a linear forest. As the basis of the induction we use the proof of a weaker theorem proven in [8].…”

“…al. [8] showed that every outerplanar graph can be drawn (and located) on two columns. In the same paper there is introduced an example of a planar graph which is not (2, 1)-colorable.…”