Abstract
A natural way to draw two planar 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 is a relaxation of simultaneous geometric embeddings with mapping. We study which classes of allow matched drawings and show that (i) two 3-connected planar or a 3-connected planar graph and a tree may not be matched drawable, while (ii) two trees or a planar graph and a sufficiently restricted planar grap - such as an unlabeled level planar (ULP) graph or a graph of the family of carousel graphs - are always matched drawable.
Highlights
The visual comparison of two graphs whose vertex sets are associated in some way requires drawings of these graphs that highlight their association in a clear manner
A simultaneous geometric embedding with mapping of G1 and G2 is a pair of straight-line planar drawings Γ1 and Γ2 of G1 and G2, respectively, such that for any pair of matched vertices u ∈ V1 and v ∈ V2 the position of u in Γ1 is the same as the position of v in Γ2
We prove that two matched trees are always matched drawable. These results show that matched drawings do allow larger classes of graphs to be drawn than simultaneous geometric embeddings with mapping
Summary
The visual comparison of two graphs whose vertex sets are associated in some way requires drawings of these graphs that highlight their association in a clear manner. Most importantly, the positions of associated vertices in the two drawings should be “close” This makes it possible for the user to identify structurally identical and structurally different portions of the two graphs, or to maintain her “mental map” [17]. A simultaneous geometric embedding with mapping (introduced by Brass et al in [3]) of G1 and G2 is a pair of straight-line planar drawings Γ1 and Γ2 of G1 and G2, respectively, such that for any pair of matched vertices u ∈ V1 and v ∈ V2 the position of u in Γ1 is the same as the position of v in Γ2. Kaufmann, and Vrt’o [15] recently proved that even a pair of trees may not have a simultaneous geometric embedding with mapping.
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