Matched Drawings of Planar Graphs

Abstract

Abstract.A natural way to draw two planar graphs whose vertex sets are matched is to assign each matched pair a unique y-coordinate. In this paper we introduce the concept of such matched drawings, which are a relaxation of simultaneous geometric embeddings with mapping. We study which classes of graphs allow matched drawings and show that (i) two 3-connected planar graphs or a 3-connected planar graph and a tree may not be matched drawable, while (ii) two trees or a planar graph and a planar graph of some special families-such as unlabeled level planar (ULP) graphs or the family of "carousel graphs"-are always matched drawable.

Matched drawability of graph pairs and of graph triples

“…In this paper, we extend the results of [9] in different directions. Namely, we present both new pairs of matched drawable planar graphs, and we initiate the study of matched drawings of triples of planar graphs.…”

“…In [9], it has been proven that every pair T 0 , T 1 where both T 0 and T 1 are trees is matched drawable. A matched drawing of two trees is constructed by a vertex addition strategy that alternately chooses the next vertex to be drawn from T 0 and from T 1 : T 0 determines which vertex is placed in odd steps at y-coordinates n − 1, n − 2, .…”

“…We extend this result by presenting new pairs of matched drawable graphs. This is done by using a drawing approach that looks similar but is also quite different from the one of [9].…”

“…The known results about matched drawability of graph pairs are described in [9]. The authors prove that not all planar pairs are matched drawable and also describe meaningful pairs of planar graphs that always admit a matched drawing.…”

“…The concept of matched drawability of a pair of planar graphs was first defined and studied in [9]. Matched drawings are a variant of simultaneous geometric embeddings first defined by Brass et al in [2] and then studied in several papers, including [1,3,4,10,11,13,15,17].…”