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Isometric Diamond Subgraphs
[chapter]
2009
Lecture Notes in Computer Science
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Hexagons and Diamonds from Slices of Lattices The three-dimensional points {(x, y, z) | x+y+z ∈ {0, 1}}, with edges connecting points at unit distance, form a 3-regular infinite graph (Fig. 1, left) in ...
We test in polynomial time whether a graph embeds in a distancepreserving way into the hexagonal tiling, the three-dimensional diamond structure, or analogous higher-dimensional structures. ...
All figures are by the author, remain the copyright of the author, and are used by permission. ...
doi:10.1007/978-3-642-00219-9_37
fatcat:5rascgkdnffwrlwmgxf2vaz3a4
On the Natural Imprint Function of a Graph
2002
European journal of combinatorics (Print)
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Finally, we propose a conjecture on amalgamation procedure for absolute C-median graphs, and prove the fixed box theorem for this class modulo the conjecture. ...
In this paper, some characterizations of median and quasi-median graphs are extended to general isometric subgraphs of Cartesian products using the concept of an imprint function as introduced by Tardif ...
I also thank an anonymous referee for useful comments and, in particular, for the short proof of Lemma 8. ...
doi:10.1006/eujc.2001.0555
fatcat:kcock5plirgyleskgtrkd3mq4y
Isometric subgraphs of Hamming graphs and d-convexity
1988
Cybernetics
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Let us provide a more constructive description of isometric subgraphs of Hamming graphs. ...
THEOREM 3 . 3 The graph G = (X, U) is isometrically embeddable in a Hamming graph if and only if it is obtained from a one-vertex graph by a sequence of isometric expansions.
C = {xi,...,Xan}. ...
doi:10.1007/bf01069520
fatcat:pxorcwevpzefbids7kjei25b2q
Isometric Diamond Subgraphs
[article]
2008
arXiv
pre-print
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We describe polynomial time algorithms for determining whether an undirected graph may be embedded in a distance-preserving way into the hexagonal tiling of the plane, the diamond structure in three dimensions ...
The graphs that may be embedded in this way form an interesting subclass of the partial cubes. ...
Theorem 1 . 1 A partial cube is an isometric subgraph of a generalized diamond graph if and only if all cuts formed by Djokovic-Winkler equivalence classes are coherent. ...
arXiv:0807.2218v1
fatcat:aujgk3kr6ncsbjnfmxn2p6lyly
Tree-like isometric subgraphs of hypercubes
2003
Discussiones Mathematicae Graph Theory
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Tree-like isometric subgraphs of hypercubes, or tree-like partial cubes as we shall call them, are a generalization of median graphs. ...
Just as median graphs they capture numerous properties of trees, but may contain larger classes of graphs that may be easier to recognize than the class of median graphs. ...
Acknowledgment We wish to thank one of the referees for several intriguing questions and suggestions. ...
doi:10.7151/dmgt.1199
fatcat:l6zd2krtrzccnef23iminpw2sy
Two-Dimensional Partial Cubes
2020
Electronic Journal of Combinatorics
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We investigate the structure of two-dimensional partial cubes, i.e., of isometric subgraphs of hypercubes whose vertex set defines a set family of VC-dimension at most 2. ...
Moreover, we point out relations to tope graphs of COMs of low rank and region graphs of pseudoline arrangements. ...
Acknowledgements We are grateful to the anonymous referees for a careful reading of the paper and numerous useful comments and improvements. ...
doi:10.37236/8934
fatcat:bu53cq6r3jbk7ihpkqkuzmdknm
Two-dimensional partial cubes
[article]
2021
arXiv
pre-print
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We investigate the structure of two-dimensional partial cubes, i.e., of isometric subgraphs of hypercubes whose vertex set defines a set family of VC-dimension at most 2. ...
The cell structure of two-dimensional partial cubes enables us to establish a variety of results. ...
Acknowledgements
Acknowledgements We are grateful to the anonymous referees for a careful reading of the paper and numerous useful comments and improvements. ...
arXiv:1906.04492v4
fatcat:zgbcmjio5zhj5gzdyeuj5hqtu4
On dually compact closed classes of graphs and BFS-constructible graphs
2003
Discussiones Mathematicae Graph Theory
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Finally we show that the class of intervalfinite pseudo-median graphs (and thus the one of median graphs) and the class of Helly graphs are dually compact closed, and that moreover every finite subgraph ...
Abstract A class C of graphs is said to be dually compact closed if, for every infinite G ∈ C, each finite subgraph of G is contained in a finite induced subgraph of G which belongs to C. ...
The class of Helly graphs is dually compact closed. More precisely, every finite subgraph H of a Helly graph G is contained in a finite isometric Helly subgraph of G. P roof. ...
doi:10.7151/dmgt.1207
fatcat:6uds3ef25fhuvkgjkj7ubxbhfi
Characterizing almost-median graphs
2007
European journal of combinatorics (Print)
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Almost-median and semi-median graphs are two natural generalizations of the well-known class of median graphs. ...
In this note we prove that a semi-median graph is almost-median if and only if it does not contain any convex cycle of length greater than four. ...
The author was supported by the Ministry of Science and Technology of Slovenia under the grant P1-0297. ...
doi:10.1016/j.ejc.2005.10.009
fatcat:r3qwgkaxdbd2fa7xsldsbxgdte
On Distance Preserving and Sequentially Distance Preserving Graphs
[article]
2017
arXiv
pre-print
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A graph H is an isometric subgraph of G if d_H(u,v)= d_G(u,v), for every pair u,v∈ V(H). A graph is distance preserving if it has an isometric subgraph of every possible order. ...
Next we consider the distance preserving property on graphs with a cut vertex. Finally, we define a family of non-distance preserving graphs constructed from cycles. ...
One solution to this problem would be to find subgraphs where the distances between all vertices is equal to their distance in the original graph. Such a subgraph is called isometric. ...
arXiv:1701.06404v1
fatcat:rni5ugw3lze5fau4w257375iwe
On the rainbow connection of Cartesian products and their subgraphs
2012
Discussiones Mathematicae Graph Theory
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It is shown that the rainbow connection number of an isometric subgraph of a hypercube is bounded above with the rainbow connection number of the hypercube. ...
The concept of c-strong rainbow coloring is introduced. In particular it is proved that the socalled Θ-coloring of an isometric subgraph of a hypercube is its unique optimal c-strong rainbow coloring. ...
Finally, a graph is a median graph if for any triple of its vertices u, v, w there exists a unique vertex that lies on a shortest u, v-path, on a shortest u, w-path and on a shortest v, w-path. ...
doi:10.7151/dmgt.1644
fatcat:qtu4d7cuj5fzxp26nk2i4r7xxi
Partial Hamming graphs and expansion procedures
2001
Discrete Mathematics
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Consequently, a relation on the edge set of a graph which is closely related to Winkler-Djokovià c's relation is introduced and used for a characterization of isometric subgraphs of Hamming graphs. ...
Structural properties of isometric subgraphs of Hamming graphs are presented, generalizing certain results on quasi-median graphs. ...
I also thank one of the referees who carefully read the article and suggested several improvements. In particular, he discovered a gap in the proof of the previous version of Theorem 8. ...
doi:10.1016/s0012-365x(00)00362-9
fatcat:gdhel54dhbetlhjkit4q3vvrua
Hamming dimension of a graph—The case of Sierpiński graphs
2013
European journal of combinatorics (Print)
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The Hamming dimension of a graph G is introduced as the largest dimension of a Hamming graph into which G embeds as an irredundant induced subgraph. ...
The canonical isometric representation of Sierpiński graphs is also explicitly described. ...
The first two authors are also with the Institute of Mathematics, Physics and Mechanics, Ljubljana. ...
doi:10.1016/j.ejc.2012.09.006
fatcat:eygetj4ijnbwlkecjdj25vsnj4
Fast recognition algorithms for classes of partial cubes
2003
Discrete Applied Mathematics
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Isometric subgraphs of hypercubes, or partial cubes as they are also called, are a rich class of graphs that include median graphs, subdivision graphs of complete graphs, and classes of graphs arising ...
In general, one can recognize whether a graph on n vertices and m edges is a partial cube in O(mn) steps, faster recognition algorithms are only known for median graphs. ...
Acknowledgements We wish to thank the referees for helpful remarks and for the question about the linearity of Algorithm (A) in n. ...
doi:10.1016/s0166-218x(02)00416-x
fatcat:z5tshewwrzhm3gxxnrs6u6yduy
Tiled partial cubes
2002
Journal of Graph Theory
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In the quest to better understand the connection between median graphs, trianglefree graphs and partial cubes, a hierarchy of subclasses of partial cubes has been introduced. ...
We also describe median graphs as tiled partial cubes without convex Q − 3 and extend an inequality for median graphs to a larger subclass of partial cubes. ...
Now, let G be a proper semi-median graph with at least one Θ-class. ...
doi:10.1002/jgt.10031
fatcat:n7dh5tzx6zaflkeb5ay5opsbxu
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