Fast recognition algorithms for classes of partial cubes

European Journal of Combinatorics, 1995

Graphs that can be isometrically embedded into the metric space 1~ are called /rgraphs. Halved cubes play an important role in the characterization of /i-graphs. We present an algorithm that recognizes halved cubes in O(n log 2 n) time. {~

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.

European Journal of Combinatorics

We fill in the details of the algorithm sketched in [6] and determine its complexity . As a part of this main algorithm , we also describe an algorithm which recognizes graphs which are isometric subgraphs of halved cubes . We discuss possible further applications of the same ideas and give a nice example of non-l 1 -graph allowing a highly isometric embedding into a halved cube .

European Journal of Combinatorics, 2007

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 $Q_3$ (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...

European Journal of Combinatorics, 1996

We fill in the details of the algorithm sketched in [6] and determine its complexity . As a part of this main algorithm , we also describe an algorithm which recognizes graphs which are isometric subgraphs of halved cubes . We discuss possible further applications of the same ideas and give a nice example of non-l 1 -graph allowing a highly isometric embedding into a halved cube .

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...

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.

Discrete Mathematics, 2008

Partial cubes are isometric subgraphs of hypercubes. Structures on a graph defined by means of semicubes, and Djoković's and Winkler's relations play an important role in the theory of partial cubes. These structures are employed in the paper to characterize bipartite graphs and partial cubes of arbitrary dimension. New characterizations are established and new proofs of some known results are given. The operations of Cartesian product and pasting, and expansion and contraction processes are utilized in the paper to construct new partial cubes from old ones. In particular, the isometric and lattice dimensions of finite partial cubes obtained by means of these operations are calculated.

AUSTRALASIAN JOURNAL OF …, 2003

All cubic partial cubes (i.e., cubic isometric subgraphs of hypercubes) up to 30 vertices and all edge-critical partial cubes up to 14 vertices are presented. The lists of graphs were confirmed by computer search to be complete. Non-trivial cubic partial cubes on 36, 42, and 48 vertices are also constructed.