Partial cubes: structures, characterizations, and constructions (original) (raw)
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The Electronic Journal of Combinatorics, 2020
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. Equivalently, those are the partial cubes which are not contractible to the 3-cube Q_3Q_3Q_3 (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). We show that our graphs can be obtained from two types of combinatorial cells (gated cycles and gated full subdivisions of complete graphs) via amalgams. The cell structure of two-dimensional partial cubes enables us to establish a variety of results. In particular, we prove that all partial cubes of VC-dimension 2 can be extended to ample aka lopsided partial cubes of VC-dimension 2, yielding that the set families defined by such graphs satisfy the sample compression conjecture by Littlestone and Warmuth (1986) in a strong sense. The latter is a central conjecture of the area of computational machine learning, that is far from...
Partial cubes as subdivision graphs and as generalized Petersen graphs
Discrete Mathematics, 2003
Isometric subgraphs of hypercubes are known as partial cubes. The subdivision graph of a graph G is obtained from G by subdividing every edge of G. It is proved that for a connected graph G its subdivision graph is a partial cube if and only if every block of G is either a cycle or a complete graph. Regular partial cubes are also considered. In particular it is shown that among the generalized Petersen graphs P (10, 3) and P (2n, 1), n ≥ 2, are the only (regular) partial cubes.
On the k-subgraphs of the generalized n-cubes
Graphs are used in modeling interconnections networks and measuring their properties. Knowing and understanding the graph theoretical/combinatorial properties of the underlying networks are necessary in developing more efficient parallel algorithms as well as fault-tolerant communication/routing algorithms [1] The hypercube is one of the most versatile and efficient networks yet discovered for parallel computation. One generalization of the hypercube is the n-cube Q(n,m) which is a graph whose vertices are all the binary n-tuples, such that two vertices are adjacent whenever they differ in exactly m coordinates. The k-subgraph of the Generalized n-cube Q k (n,m) is the induced subgraph of the n-cube Q(n,m) where q=2, such that a vertex v ∈ V(Q k (n,m)) if and only if v ∈ V(Q(n,m)) and v is of parity k. This paper presents some degree properties of Q k (n,m) as well as some isomorphisms it has with other graphs, namely: 1)) 2 , (1 n Q n− is isomorphic to Kn 2)) 2 , (i n Q k is isomor...
Discuss. Math. Graph Theory, to appear
The Θ-graph Θ(G) of a partial cube G is the intersection graph of the equivalence classes of the Djoković-Winkler relation. Θ-graphs that are 2-connected, trees, or complete graphs are characterized. In particular, Θ(G) is complete if and only if G can be obtained from K 1 by a sequence of (newly introduced) dense expansions. Θ-graphs are also compared with familiar concepts of crossing graphs and τ -graphs.
Tree-like isometric subgraphs of hypercubes
Discussiones Mathematicae Graph Theory, 2003
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. We investigate the structure of treelike partial cubes, characterize them, and provide examples of similarities with trees and median graphs. For instance, we show that the cube graph of a tree-like partial cube is dismantlable. This in particular implies that every tree-like partial cube G contains a cube that is invariant under every automorphism of G. We also show that weak retractions preserve tree-like partial cubes, which in turn implies that * Supported by the Ministry of Education, Science and Sport of Slovenia under the grants Z1-3073, and 0101-P-504, respectively. 228 B. Brešar, W. Imrich and S. Klavžar every contraction of a tree-like partial cube fixes a cube. The paper ends with several Frucht-type results and a list of open problems.
On Theta-graphs of partial cubes
2007
The Θ-graph Θ(G) of a partial cube G is the intersection graph of the equivalence classes of the Djoković-Winkler relation. Θ-graphs that are 2-connected, trees, or complete graphs are characterized. In particular, Θ(G) is complete if and only if G can be obtained from K 1 by a sequence of (newly introduced) dense expansions. Θ-graphs are also compared with familiar concepts of crossing graphs and τ -graphs.
On regular subgraphs of augmented cubes
AKCE International Journal of Graphs and Combinatorics, 2020
The n-dimensional augmented cube AQ n is a variation of the hypercube Q n : It is a ð2n À 1Þ-regular and ð2n À 1Þ-connected graph on 2 n vertices. One of the fundamental properties of AQ n is that it is pancyclic, that is, it contains a cycle of every length from 3 to 2 n : In this paper, we generalize this property to k-regular subgraphs for k ¼ 3 and k ¼ 4: We prove that the augmented cube AQ n with n ! 4 contains a 4-regular, 4-connected and pancyclic subgraph on l vertices if and only if 8 l 2 n : Also, we establish that for every even integer l from 4 to 2 n , there exists a 3-regular, 3-connected and pancyclic subgraph of AQ n on l vertices.
Edge-Critical Isometric Subgraphs of Hypercubes
Ars Combinatoria Waterloo Then Winnipeg, 2001
Isometric subgraphs of hypercubes are known as partial cubes. Edge-critical partial cubes are introduced as the partial cubes G for which G?e is not a partial cube for any edge e of G. An expansion theorem is proved by means of which one can generate many edge-critical partial cubes. Edge-critical partial cubes are characterized among the Cartesian product graphs. We also show that the 3-cube and the subdivision graph of K 4 are the only edge-critical partial cubes on at most 10 vertices.
Journal of Graph …, 2002
Abstract: In the quest to better understand the connection between median graphs, triangle-free graphs and partial cubes, a hierarchy of sub-classes of partial cubes has been introduced. In this article, we study the role of tiled partial cubes in this scheme. For instance, we prove ...