Reconstructing the shape of a tree from observed dissimilarity data
Advances in Applied Mathematics
From copair hypergraphs to median graphs with latent vertices
Discrete Mathematics
Arboricity and bipartite subgraph listing algorithms
Information Processing Letters
A triangle-free circle graph with chromatic number 5
Discrete Mathematics
Graphs of acyclic cubical complexes
European Journal of Combinatorics - Special issue on discrete metric spaces
European Journal of Combinatorics - Special issue on discrete metric spaces
Every circle graph of girth at least 5 is 3-colourable
Discrete Mathematics
Regular Article: Graphs of Some CAT(0) Complexes
Advances in Applied Mathematics
Triangle-free graphs with large chromatic numbers
Discrete Mathematics
Decomposition and l1-embedding of weakly median graphs
European Journal of Combinatorics
Center and diameter problems in plane triangulations and quadrangulations
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
A Fast Algorithm for Generating Nonisomorphic Chord Diagrams
SIAM Journal on Discrete Mathematics
Median problem in some plane triangulations and quadrangulations
Computational Geometry: Theory and Applications
The lattice dimension of a graph
European Journal of Combinatorics
Distance and routing labeling schemes for non-positively curved plane graphs
Journal of Algorithms
A self-stabilizing algorithm for the median problem in partial rectangular grids and their relatives
SIROCCO'07 Proceedings of the 14th international conference on Structural information and communication complexity
Shortest path problem in rectangular complexes of global nonpositive curvature
Computational Geometry: Theory and Applications
On embeddings of CAT(0) cube complexes into products of trees via colouring their hyperplanes
Journal of Combinatorial Theory Series B
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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 one 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 embedded into the Cartesian product of five trees. This embedding result can also be extended to the infinite case without reference to an embedding in the plane and without any cardinality restriction when formulated for median graphs free of cubes and further finite obstructions. Further, we exhibit a class of squaregraphs that can be embedded into the product of three trees, and we characterize those squaregraphs that are embeddable into the product of just two trees. Finally, finite squaregraphs enjoy a number of algorithmic features that do not extend to arbitrary median graphs. For instance, we show that minimum-size median-generating sets of finite squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the corresponding problem for median graphs turns out to be NP-hard. Finite squaregraphs can be recognized in linear time by a Breadth-First-Search.