Partial Cubes and Crossing Graphs

Abstract

Abstract. Partial cubes are defined as isometric subgraphs of hypercubes. For a partial cube G, its crossing graph G # is introduced as the graph whose vertices are the equivalence classes of the Djoković-Winkler relation Θ, two vertices being adjacent if they cross on a common cycle. It is shown that every graph is the crossing graph of some median graph and that a partial cube G is 2-connected if and only if G # is connected. A partial cube G has a triangle-free crossing graph if and only if G is a cube-free median graph. This result is used to characterize the partial cubes having a tree or a forest as its crossing graph. An expansion theorem is given for the partial cubes with complete crossing graphs. Cartesian products are also considered. In particular, it is proved that G # is a complete bipartite graph if and only if G is the Cartesian product of two trees.

When Can Splits be Drawn in the Plane?

“…The first step of the proof is to show that if an internal face has edge e on its boundary then it has exactly one more boundary edge in the same class. Let C be the boundary of the internal face and let e ∈ C. From Lemma 2.3 of Klavzar and Mulder (2002), C contains at least one other edge in the same edge class as e, and since all the edges in this class are parallel and non-adjacent there can be at most two on the boundary of any convex region. Hence C contains zero or two edges from each edge class and any two edges from the same class will be on opposite sides of the cycle.…”

The Fibonacci Dimension of a Graph

“…The crossing graph G # of a partial cube G has the Θ-classes of G as its nodes, where two nodes of G # are joined by an edge whenever they cross as Θ-classes in G; see [35]. More precisely, if W (a,0) , W (a,1) and W (b,0) , W (b,1) are pairs of complementary semicubes corresponding to Θ-classes E and F , then E and F cross if each semicube has a nonempty intersection with the semicubes from the other pair; that is, it holds that A characterization of complete crossing graphs in terms of the expansion procedure is given in [35]: G # is complete if and only if G can be obtained from K 1 by a sequence of all-color expansions. We also note that among median graphs only hypercubes have complete crossing graphs [38].…”

Θ-graceful labelings of partial cubes

“…A subgraph H of a graph G is called isometric if for every two vertices u, v of H there exists a shortest u, v-path that lies in H. Isometric subgraphs of hypercubes are called partial cubes, [16] and have been extensively studied in recent years [2][3][4][5]7,8,19]. Probably the most important characterization of partial cubes is due to Winkler [23], and involves the following edge-parallelism property.…”