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.
It is shown that the graph obtained by merging two vertices of two 4-cycles is not a $Θ$-graceful partial cube, thus answering in the negative a question by Brešar and Klavžar from [1], who asked whether every partial cube is $Θ$-graceful.
Peano partial cubes are the bipartite graphs whose geodesic interval spaces are (closed) join spaces. They are the partial cubes all of whose finite convex subgraphs have a pre-hull number which is at most 1. Special Peano partial cubes are median graphs, cellular bipartite graphs and netlike partial cubes. Analogous properties of these graphs are satisfied by Peano partial cubes. In particular the convex hull of any isometric cycle of such a graph is a gated quasi-hypertori (i.e., the Cartesian product of copies of K_2 and even cycles). Moreover, for any Peano partial cubes G that contains no…
We prove that a connected bipartite graph G is a partial cube if and only if the set of attaching points of any copoint of G is convex. A consequence of this result is that any connected bipartite graph with pre-hull number at most 1 is a partial cube. We show that the class of partial cubes with pre-hull number at most 1 is closed under gated subgraphs, gated amalgams and cartesian products.
In this paper we study the Szeged index of partial cubes and hence generalize the result proved by V. Chepoi and S. Klavžar, who calculated this index for benzenoid systems. It is proved that the problem of calculating the Szeged index of a partial cube can be reduced to the problem of calculating the Szeged indices of weighted quotient graphs with respect to a partition coarser than $Θ$-partition. Similar result for the Wiener index was recently proved by S. Klavžar and M. J. Nadjafi-Arani. Furthermore, we show that such quotient graphs of partial cubes are again partial cubes. Since the resu…
The dominating graph of a graph G is a graph whose vertices correspond to the dominating sets of G and two vertices are adjacent whenever their corresponding dominating sets differ in exactly one vertex. Studying properties of dominating graph has become an increasingly interesting subject in domination theory. On the other hand, median graphs and partial cubes are two fundamental graph classes in graph theory. In this paper, we make some new connections between domination theory and the theory of median graphs and partial cubes. As the main result, we show that the following conditions are eq…
Partial cubes are graphs isometrically embeddable into hypercubes. We analyze how isometric cycles in partial cubes behave and derive that every partial cube of girth more than 6 must have vertices of degree less than 3. As a direct corollary we get that every regular partial cube of girth more than 6 is an even cycle. Along the way we prove that every partial cube $G$ with girth more than 6 is a tree-zone graph and therefore $2n(G)-m(G)-i(G)+ce(G)=2$ holds, where $i(G)$ is the isometric dimension of $G$ and $ce(G)$ its convex excess.
Partial cubes are graphs isometrically embeddable into hypercubes. In this paper it is proved that every cubic, vertex-transitive partial cube is isomorphic to one of the following graphs: $K_2 \, \square \, C_{2n}$, for some $n\geq 2$, the generalized Petersen graph $G(10,3)$, the cubic permutahedron, the truncated cuboctahedron, or the truncated icosidodecahedron. This classification is a generalization of results of Brešar et al.~from 2004 on cubic mirror graphs, it includes all cubic, distance-regular partial cubes (Weichsel, 1992), and presents a contribution to the classification of all …
We prove that the combinatorial optimization problem of determining the hull number of a partial cube is NP-complete. This makes partial cubes the minimal graph class for which NP-completeness of this problem is known and improves some earlier results in the literature. On the other hand we provide a polynomial-time algorithm to determine the hull number of planar partial cube quadrangulations. Instances of the hull number problem for partial cubes described include poset dimension and hitting sets for interiors of curves in the plane. To obtain the above results, we investigate convexity in p…
The edge-hyper-Wiener index of a connected graph $G$ is defined as $WW_e(G) = \frac{1}{2}\sum_{\lbrace e,f\rbrace \subseteq E(G)}d(e,f) + \frac{1}{2}\sum_{\lbrace e,f\rbrace \subseteq E(G)}d(e,f)^2$. We develop a method for computing the edge-hyper-Wiener index of partial cubes, which constitute a large class of graphs with a lot of applications. It is also shown how the method can be applied to trees. Furthermore, an algorithm for computing the edge-hyper-Wiener index of benzenoid systems is obtained. Finally, the algorithm is used to correct already known closed formulas for the edge-Wiener …
In this note we determine the set of expansions such that a partial cube is planar if and only if it arises by a sequence of such expansions from a single vertex. This corrects a result of Peterin.
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 variet…
We investigate the structure of isometric subgraphs of hypercubes (i.e., partial cubes) which do not contain finite convex subgraphs contractible to the 3-cube minus one vertex $Q^-_3$ (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). Extending similar results for median and cellular graphs, we show that the convex hull of an isometric cycle of such a graph is gated and isomorphic to the Cartesian product of edges and even cycles. Furthermore, we show that our graphs are exactly the class of partial cubes in which any finite convex subgraph c…
Building on a recent characterization of tope graphs of Complexes of Oriented Matroids (COMs), we tackle and generalize several classical problems in Oriented Matroids (OMs), Lopsided Sets (aka ample set systems), and partial cubes via Metric Graph Theory. Our first main result is that every element of an OM from a class introduced by Mandel is incident to a simplicial tope, i.e, such OMs contain no mutation-free elements. This allows us to refute a conjecture of Mandel from 1983, that would have implied the famous Las Vergnas' simplex conjecture. Further, we show that the mutation graph of un…
We show how to construct a cubic partial cube from any simplicial arrangement of lines or pseudolines in the projective plane. As a consequence, we find nine new infinite families of cubic partial cubes as well as many sporadic examples.
A partial cube is a graph having an isometric embedding in a hypercube. Partial cubes are characterized by a natural equivalence relation on the edges, whose classes are called zones. The number of zones determines the minimal dimension of a hypercube in which the graph can be embedded. We consider the problem of covering the vertices of a partial cube with the minimum number of zones. The problem admits several special cases, among which are the problem of covering the cells of a line arrangement with a minimum number of lines, and the problem of finding a minimum-size fibre in a bipartite po…
Let X $\subseteq$ {0, 1} n. Then the daisy cube Q n (X) is introduced as the sub-graph of Q n induced by the intersection of the intervals I(x, 0 n) over all x $\in$ X. Daisy cubes are partial cubes that include Fibonacci cubes, Lucas cubes, and bipartite wheels. If u is a vertex of a graph G, then the distance cube polynomial D G,u (x, y) is introduced as the bivariate polynomial that counts the number of induced subgraphs isomorphic to Q k at a given distance from the vertex u. It is proved that if G is a daisy cube, then D G,0 n (x, y) = C G (x + y -- 1), where C G (x) is the previously inv…
In this paper, first it is shown that the "FSibonacci $(p,r)$-cube"(denoted as $IΓ_{n}^{(p,r)}$) studied in many papers, such as \cite{OZY}, \cite{K1}, \cite{OZ}, \cite{KR} and \cite{JZ}, is a new topological structure different from the original one (denoted as $OΓ_{n}^{(p,r)}$) presented by Egiazarian and Astola $\cite{EA}$. Then some topological properties of $IΓ_{n}^{(p,r)}$ and $OΓ_{n}^{(p,r)}$ are studied, including the recursive structure of them, the cubes $OΓ_{n}^{(p,r)}$ which are partial cubes and median graphs, some distance invariants of $IΓ_{n}^{(p,r)}$ and $OΓ_{n}^{(p,r)}$, and …
The `full' edge isoperimetric inequality for the discrete cube (due to Harper, Bernstein, Lindsay and Hart) specifies the minimum size of the edge boundary $\partial A$ of a set $A \subset \{0,1\}^n$, as a function of $|A|$. A weaker (but more widely-used) lower bound is $|\partial A| \geq |A| \log_2(2^n/|A|)$, where equality holds iff $A$ is a subcube. In 2011, the first author obtained a sharp `stability' version of the latter result, proving that if $|\partial A| \leq |A| (\log(2^n/|A|)+ε)$, then there exists a subcube $C$ such that $|A ΔC|/|A| = O(ε/\log(1/ε))$. The `weak' version of the e…
For a CAT(0) cube complex $\mathbf X$, we define a simplicial flag complex $\partial_Δ\mathbf X$, called the \emph{simplicial boundary}, which is a natural setting for studying non-hyperbolic behavior of $\mathbf X$. We compare $\partial_Δ\mathbf X$ to the Roller, visual, and Tits boundaries of $\mathbf X$ and give conditions under which the natural CAT(1) metric on $\partial_Δ\mathbf X$ makes it (quasi)isometric to the Tits boundary. $\partial_Δ\mathbf X$ allows us to interpolate between studying geodesic rays in $\mathbf X$ and the geometry of its \emph{contact graph} $Γ\mathbf X$, which is …
A lucasene is a hexagon chain that is similar to a fibonaccene, an $L$-fence is a poset the Hasse diagram of which is isomorphic to the directed inner dual graph of the corresponding lucasene. A new class of cubes, which named after matchable Lucas cubes according to the number of its vertices (or elements), are a series of directed or undirected Hasse diagrams of filter lattices of $L$-fences. The basic properties and several classes of polynomials, e.g.\ rank generating functions, cube polynomials and degree sequence polynomials, of matchable Lucas cubes are obtained. Some special conclusion…
Let $S$ be a set of arbitrary objects, and let $S^d=\{v_1...v_d\colon v_i\in S\}$. A polybox code is a set $V\subset S^d$ with the property that for every two words $v,w\in V$ there is $i\in [d]$ with $v_i'=w_i$, where a permutation $s\mapsto s'$ of $S$ is such that $s''=(s')'=s$ and $s'\neq s$. If $|V|=2^d$, then $V$ is called a cube tiling code. Cube tiling codes determine $2$-periodic cube tilings of $\mathbb{R}^d$ or, equivalently, tilings of the flat torus $\mathbb{T}^d=\{(x_1,\ldots ,x_d)({\rm mod} 2):(x_1,\ldots ,x_d)\in \mathbb{R}^d\}$ by translates of the unit cube as well as $r$-perf…
Explicit algorithms are developed for constructing odd order n pandiagonal latin cubes in 3 and 4 dimensions, and these are used to construct pandiagonal magic cubes and 4 dimensional hypercubes, respectively. It is established that these structures exist in 3 dimensions only for odd orders n not less than 11, and in 4 dimensions only for odd orders n not less than 17. Estimates for the number of such structures as functions of n in 3 and 4 dimensions are also presented.
The edge isoperimetric inequality in the discrete cube specifies, for each pair of integers $m$ and $n$, the minimum size $g_n(m)$ of the edge boundary of an $m$-element subset of $\{0,1\}^{n}$; the extremal families (up to automorphisms of the discrete cube) are initial segments of the lexicographic ordering on $\{0,1\}^n$. We show that for any $m$-element subset $\mathcal{F} \subset \{0,1\}^n$ and any integer $l$, if the edge boundary of $\mathcal{F}$ has size at most $g_n(m)+l$, then there exists an extremal family $\mathcal{G} \subset \{0,1\}^n$ such that $|\mathcal{F} Δ\mathcal{G}| \leq C…
Klavžar and Mollard introduced daisy cubes which are interesting isometric subgraphs of $ n$-cubes $Q_n$, induced with intervals between the maximal elements of a poset $ (V (Q_n),\leq)$ and the vertex $ 0^n \in V (Q_n)$. In this paper we show that the resonance graph, which reflects the interaction between Kekulé structures of aromatic hydrocarbon molecules, is a daisy cube, if the molecules considered can be modeled with the so called kinky benzenoid systems, i.e. catacondensed benzenoid systems without linear hexagons.
Well-graded families, extremal systems and maximum systems (the last two in the sense of VC-theory and Sauer-Shelah lemma on VC-dimension) are three important classes of set systems. This paper aims to study the notion of duality in the context of these classes of set systems and then use the obtained results for studying graphs. More specifically, we are concerned with the characterization of the finite set systems which themselves and their dual systems are both well-graded, extremal or maximum. On the way to this goal, and maybe also of independent interest, we study the structure of the we…
The connectivity is an important parameter to evaluate the robustness of a network. As a generalization, structure connectivity and substructure connectivity of graphs were proposed. For connected graphs $G$ and $H$, the $H$-structure connectivity $κ(G; H)$ (resp. $H$-substructure connectivity $κ^{s}(G; H)$) of $G$ is the minimum cardinality of a set of subgraphs $F$ of $G$ that each is isomorphic to $H$ (resp. to a connected subgraph of $H$) so that $G-F$ is disconnected or the singleton. As popular variants of hypercubes, the $n$-dimensional folded hypercubes $FQ_{n}$ and augmented cubes $AQ…
This paper considers completions of COMs (complexes oriented matroids) to ample partial cubes of the same VC-dimension. We show that these exist for OMs (oriented matroids) and CUOMs (complexes of uniform oriented matroids). This implies that OMs and CUOMs satisfy the sample compression conjecture -- one of the central open questions of learning theory. We conjecture that every COM can be completed to an ample partial cube without increasing the VC-dimension.
The Fibonacci cube of dimension n, denoted as $Γ\_n$, is the subgraph of the hypercube induced by vertices with no consecutive 1's. The irregularity of a graph G is the sum of |d(x)-d(y)| over all edges {x,y} of G. In two recent paper based on the recursive structure of $Γ\_n$ it is proved that the irregularity of $Γ\_n$ and $Λ\_n$ are two times the number of edges of $Γ\_{n-1}$ and $2n$ times the number of vertices of $Γ\_{n-4}$, respectively. Using an interpretation of the irregularity in terms of couples of incident edges of a special kind (Figure 2) we give a bijective proof of both result…
In this paper we study the resonance graphs of benzenoid systems, tubulenes, and fullerenes. The resonance graph reflects the interactions between the Kekul e structures of a molecule. The equivalence of the Zhang-Zhang polynomial (which counts Clar covers) of the molecular graph and the cube polynomial (which counts hypercubes) of its resonance graph is known for all three families of molecular graphs. Instead of considering only interactions between 6-cycles (Clar covers), we also consider 10-cycles, which contribute to the resonance energy of a molecule as well. Therefore, we generalize the…
The article studies power complexes and generalized power complexes, and investigates the algebraic structure of their automorphism groups. The combinatorial incidence structures involved are cube-like, in the sense that they have many structural properties in common with higher-dimensional cubes and cubical tessellations on manifolds. Power complexes have repeatedly appeared in applications.
The augmented cube $AQ_n$, proposed by Choudum and Sunitha [S. A. Choudum, V. Sunitha, Augmented cubes, Networks 40 (2) (2002) 71-84], is a $(2n-1)$-regular $(2n-1)$-connected graph $(n\ge 4)$. This paper determines that the 2-extra connectivity of $AQ_n$ is $6n-17$ for $n\geq 9$ and the 2-extra edge-connectivity is $6n-9$ for $n\geq 4$. That is, for $n\geq 9$ (respectively, $n\geq 4$), at least $6n-17$ vertices (respectively, $6n-9$ edges) of $AQ_n$ have to be removed to get a disconnected graph that contains no isolated vertices and isolated edges. When the augmented cube is used to model th…
Let k_1,...,k_d be positive integers, and D be a subset of [k_1]x...x[k_d], whose complement can be decomposed into disjoint sets of the form {x_1}x...x{x_{s-1}}x[k_s]x{x_{s+1}}x...x{x_d}. We conjecture that the number of elements of D can be represented as a linear combination of the numbers k_1,..., k_d with non-negative integer coefficients. A connexion of this conjecture with the structure of periodical cube tilings is revealed.
Inspired by Coxeter's notion of Petrie polygon for $d$-polytopes (see \cite{Cox73}), we consider a generalization of the notion of zigzag circuits on complexes and compute the zigzag structure for several interesting families of $d$-polytopes, including semiregular, regular-faced, Wythoff Archimedean ones, Conway's 4-polytopes, half-cubes, folded cubes. Also considered are regular maps and Lins triality relations on maps.
We define a notion of facets-pairing structure and its seal space on a nice manifold with corners. We will study facets-pairing structures on any cube in detail and investigate when the seal space of a facets-pairing structure on a cube is a closed manifold. In particular, for any binary square matrix $A$ with zero diagonal in dimension n, there is a canonical facets-pairing structure $F_A$ on the n-dimensional cube. We will show that all the closed manifolds that we can obtain from the seal spaces of such $F_A$'s are neither more nor less than all the generalized real Bott manifolds --- a spe…
\noindent The augmented cube graph $AQ_n$ is the Cayley graph of $\mathbb{Z}_2^n$ with respect to the set of $2n-1$ generators $\{e_1,e_2, \ldots,e_n, 00\ldots0011, 00\ldots0111, 11\ldots1111 \}$. It is known that the order of the automorphism group of the graph $AQ_n$ is $2^{n+3}$, for all $n \ge 4$. In the present paper, we obtain the structure of the automorphism group of $AQ_n$ to be \[ \Aut(AQ_n) \cong \mathbb{Z}_2^n \rtimes D_8~~(n \ge 4),\] where $D_8$ is the dihedral group of order 8. It is shown that the Cayley graph $AQ_3$ is non-normal and that $AQ_n$ is normal for all $n \ge 4$. We…
Nielsen, Plotkin, and Winskel (1981) proved that every 1-safe Petri net $N$ unfolds into an event structure $\mathcal{E}_N$. By a result of Thiagarajan (1996 and 2002), these unfoldings are exactly the trace regular event structures. Thiagarajan (1996 and 2002) conjectured that regular event structures correspond exactly to trace regular event structures. In a recent paper (Chalopin and Chepoi, 2017, 2018), we disproved this conjecture, based on the striking bijection between domains of event structures, median graphs, and CAT(0) cube complexes. On the other hand, in Chalopin and Chepoi (2018)…
The Rubik's Cube is perhaps the world's most famous and iconic puzzle, well-known to have a rich underlying mathematical structure (group theory). In this paper, we show that the Rubik's Cube also has a rich underlying algorithmic structure. Specifically, we show that the n x n x n Rubik's Cube, as well as the n x n x 1 variant, has a "God's Number" (diameter of the configuration space) of Theta(n^2/log n). The upper bound comes from effectively parallelizing standard Theta(n^2) solution algorithms, while the lower bound follows from a counting argument. The upper bound gives an asymptotically…
Unique Sink Orientations (USOs) of cubes can be used to capture the combinatorial structure of many essential algebraic and geometric problems. For various structural and algorithmic questions, including enumeration of USOs and algorithm analysis, it is crucial to have systematic constructions of USOs. While some construction methods for USOs already exist, each one of them has some significant downside. Most of the construction methods have limited expressivity -- USOs with some desired properties cannot be constructed. In contrast, the phase flips of Schurr can construct all USOs, but the op…
Graphical designs are an extension of spherical designs to functions on graphs. We connect linear codes to graphical designs on cube graphs, and show that the Hamming code in particular is a highly effective graphical design. We show that even in highly structured graphs, graphical designs are distinct from the related concepts of extremal designs, maximum stable sets in distance graphs, and $t$-designs on association schemes.
Median graphs are connected graphs in which for all three vertices there is a unique vertex that belongs to shortest paths between each pair of these three vertices. In this paper we provide several novel characterizations of planar median graphs. More specifically, we characterize when a planar graph $G$ is a median graph in terms of forbidden subgraphs and the structure of isometric cycles in $G$, and also in terms of subgraphs of $G$ that are contained inside and outside of 4-cycles with respect to an arbitrary planar embedding of $G$. These results lead us to a new characterization of plan…
The median of a set of vertices $P$ of a graph $G$ is the set of all vertices $x$ of $G$ minimizing the sum of distances from $x$ to all vertices of $P$. In this paper, we present a linear time algorithm to compute medians in median graphs, improving over the existing quadratic time algorithm. We also present a linear time algorithm to compute medians in the $\ell_1$-cube complexes associated with median graphs. Median graphs constitute the principal class of graphs investigated in metric graph theory and have a rich geometric and combinatorial structure, due to their bijections with CAT(0) cu…
