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Isometric Diamond Subgraphs [chapter]

David Eppstein
2009 Lecture Notes in Computer Science  
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

Boštjan Brešar
2002 European journal of combinatorics (Print)  
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

V. D. Chepoi
1988 Cybernetics  
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]

David Eppstein
2008 arXiv   pre-print
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

Boštjan Brešar, Wilfried Imrich, Sandi Klavžar
2003 Discussiones Mathematicae Graph Theory  
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

Victor Chepoi, Kolja Knauer, Manon Philibert
2020 Electronic Journal of Combinatorics  
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]

Victor Chepoi, Kolja Knauer, Manon Philibert
2021 arXiv   pre-print
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

Norbert Polat
2003 Discussiones Mathematicae Graph Theory  
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

Boštjan Brešar
2007 European journal of combinatorics (Print)  
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]

Jason P Smith, Emad Zahedi
2017 arXiv   pre-print
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

Sandi Klavžar, Gašper Mekiš
2012 Discussiones Mathematicae Graph Theory  
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

Boštjan Brešar
2001 Discrete Mathematics  
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

Sandi Klavžar, Iztok Peterin, Sara Sabrina Zemljič
2013 European journal of combinatorics (Print)  
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

Boštjan Brešar, Wilfried Imrich, Sandi Klavžar
2003 Discrete Applied Mathematics  
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

Bo?tjan Bre?ar, Wilfried Imrich, Sandi Klav?ar, Henry Martyn Mulder, Riste ?krekovski
2002 Journal of Graph Theory  
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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