Discrete Mathematics
Arboricity and bipartite subgraph listing algorithms
Information Processing Letters
A note on packing paths in planar graphs
Mathematical Programming: Series A and B
Edge-disjoint placement of three trees
European Journal of Combinatorics
An Algorithm for Subgraph Isomorphism
Journal of the ACM (JACM)
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Edge partition of planar sraphs into two outerplanar graphs
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
On simultaneous planar graph embeddings
Computational Geometry: Theory and Applications
Packing trees into planar graphs
Journal of Graph Theory
Planar packing of trees and spider trees
Information Processing Letters
Embedding graphs simultaneously with fixed edges
GD'06 Proceedings of the 14th international conference on Graph drawing
Two trees which are self–intersecting when drawn simultaneously
GD'05 Proceedings of the 13th international conference on Graph Drawing
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In the graph packing problem we are given several graphs and have to map them into a single host graph G such that each edge of G is used at most once. Much research has been devoted to the packing of trees, especially to the case where the host graph must be planar. More formally, the problem is: Given any two trees T1 and T2 on n vertices, we want a simple planar graph G on n vertices such that the edges of G can be colored with two colors and the subgraph induced by the edges colored i is isomorphic to Ti, for i∈{1,2}. A clear exception that must be made is the star tree which cannot be packed together with any other tree. But a popular hypothesis states that this is the only exception, and all other pairs of trees admit a planar packing. Previous proof attempts lead to very limited results only, which include a tree and a spider tree, a tree and a caterpillar, two trees of diameter four and two isomorphic trees. We make a step forward and prove the hypothesis for any two binary trees. The proof is algorithmic and yields a linear time algorithm to compute a plane packing, that is, a suitable two-edge-colored host graph along with a planar embedding for it. In addition we can also guarantee several nice geometric properties for the embedding: vertices are embedded equidistantly on the x-axis and edges are embedded as semi-circles.