Combinatorics and Geometry of Finite and Infinite Squaregraphs

Abstract. Squaregraphs were originally defined as finite plane graphs in which all inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e., the vertices not incident with the outer face) have degrees larger than three. The planar dual of a finite squaregraph is determined by a triangle-free chord diagram of the unit disk, which could alternatively be viewed as a triangle-free line arrangement in the hyperbolic plane. This representation carries over to infinite plane graphs with finite vertex degrees in which the balls are finite squaregraphs. Algebraically, finite squaregraphs are median graphs for which the duals are finite circular split systems. Hence squaregraphs are at the crosspoint of two dualities, an algebraic and a geometric one, and thus lend themselves to several combinatorial interpretations and structural characterizations. With these and the 5-colorability theorem for circle graphs at hand, we prove that every squaregraph can be isometrically embedd...

• Type: Journal
• Year: 2010
• Where: SIAMDM
• Authors: Hans-Jürgen Bandelt, Victor Chepoi, David Eppstein
• Keywords: Finite Squaregraphs, Median Graphs, Plane Graphs, SIAMDM 2010