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## Filters

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Isometric Diamond Subgraphs
[chapter]

2009
*
Lecture Notes in Computer Science
*

Hexagons and Diamonds from Slices

doi:10.1007/978-3-642-00219-9_37
fatcat:5rascgkdnffwrlwmgxf2vaz3a4
*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. ...##
###
On the Natural Imprint Function of a Graph

2002
*
European journal of combinatorics (Print)
*

Finally, we propose

doi:10.1006/eujc.2001.0555
fatcat:kcock5plirgyleskgtrkd3mq4y
*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. ...##
###
Isometric subgraphs of Hamming graphs and d-convexity

1988
*
Cybernetics
*

Let us provide

doi:10.1007/bf01069520
fatcat:pxorcwevpzefbids7kjei25b2q
*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}. ...##
###
Isometric Diamond Subgraphs
[article]

2008
*
arXiv
*
pre-print

We describe polynomial time algorithms for determining whether an undirected

arXiv:0807.2218v1
fatcat:aujgk3kr6ncsbjnfmxn2p6lyly
*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. ...##
###
Tree-like isometric subgraphs of hypercubes

2003
*
Discussiones Mathematicae Graph Theory
*

Tree-like

doi:10.7151/dmgt.1199
fatcat:l6zd2krtrzccnef23iminpw2sy
*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. ...##
###
Two-Dimensional Partial Cubes

2020
*
Electronic Journal of Combinatorics
*

We investigate the structure

doi:10.37236/8934
fatcat:bu53cq6r3jbk7ihpkqkuzmdknm
*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. ...##
###
Two-dimensional partial cubes
[article]

2021
*
arXiv
*
pre-print

We investigate the structure

arXiv:1906.04492v4
fatcat:zgbcmjio5zhj5gzdyeuj5hqtu4
*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. ...##
###
On dually compact closed classes of graphs and BFS-constructible graphs

2003
*
Discussiones Mathematicae Graph Theory
*

Finally we show that the

doi:10.7151/dmgt.1207
fatcat:6uds3ef25fhuvkgjkj7ubxbhfi
*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. ...##
###
Characterizing almost-median graphs

2007
*
European journal of combinatorics (Print)
*

Almost-median and semi-median

doi:10.1016/j.ejc.2005.10.009
fatcat:r3qwgkaxdbd2fa7xsldsbxgdte
*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. ...##
###
On Distance Preserving and Sequentially Distance Preserving Graphs
[article]

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*. ...

##
###
On the rainbow connection of Cartesian products and their subgraphs

2012
*
Discussiones Mathematicae Graph Theory
*

It is shown that the rainbow connection number

doi:10.7151/dmgt.1644
fatcat:qtu4d7cuj5fzxp26nk2i4r7xxi
*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. ...##
###
Partial Hamming graphs and expansion procedures

2001
*
Discrete Mathematics
*

Consequently,

doi:10.1016/s0012-365x(00)00362-9
fatcat:gdhel54dhbetlhjkit4q3vvrua
*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. ...##
###
Hamming dimension of a graph—The case of Sierpiński graphs

2013
*
European journal of combinatorics (Print)
*

The Hamming dimension

doi:10.1016/j.ejc.2012.09.006
fatcat:eygetj4ijnbwlkecjdj25vsnj4
*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. ...##
###
Fast recognition algorithms for classes of partial cubes

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. ...

##
###
Tiled partial cubes

2002
*
Journal of Graph Theory
*

In the quest to better understand the connection between median

doi:10.1002/jgt.10031
fatcat:n7dh5tzx6zaflkeb5ay5opsbxu
*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*. ...
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