Rook's Graph Research Articles

The general position avoidance game and hardness of general position games

Given a graph G, a set S of vertices in G is a general position set if no triple of vertices from S lie on a common shortest path in G. The general position achievement/avoidance game is played on a graph G by players A and B who alternately select vertices of G. A selection of a vertex by a player is a legal move if it has not been selected before and the set of selected vertices so far forms a general position set of G. The player who picks the last vertex is the winner in the general position achievement game and is the loser in the avoidance game. In this paper, we prove that the general position achievement/avoidance games are PSPACE-complete even on graphs with diameter at most 4. For this, we prove that the misère play of the classical Node Kayles game is also PSPACE-complete. As positive results, we obtain linear time algorithms to decide the winning player of the general position avoidance game in rook's graphs, grids, cylinders, and lexicographic products with complete second factors.

Geodesic packing in graphs

A geodesic packing of a graph G is a set of vertex-disjoint maximal geodesics. The maximum cardinality of a geodesic packing is the geodesic packing number gpack(G). It is proved that the decision version of the geodesic packing number is NP-complete. We also consider the geodesic transversal number, gt(G), which is the minimum cardinality of a set of vertices that hit all maximal geodesics in G. While gt(G)≥gpack(G) in every graph G, the quotient gt(G)/gpack(G) is investigated. By using the rook's graph, it is proved that there does not exist a constant C<3 such that gt(G)gpack(G)≤C would hold for all graphs G. If T is a tree, then it is proved that gpack(T)=gt(T), and a linear algorithm for determining gpack(T) is derived. The geodesic packing number is also determined for the strong product of paths.

On the Spectra of Simplicial Rook Graphs

On the Spectra of Simplicial Rook Graphs

On the Maximum Number of Non-attacking Rooks on a High-Dimensional Simplicial Chessboard

The simplicial rook graph \({\mathcal {SR}}(m,n)\) is the graph whose vertices are vectors in \( {\mathbb {N}}^m\) such that for each vector the summation of its coordinates is n and two vertices are adjacent if their corresponding vectors differ in exactly two coordinates. Martin and Wagner (Graphs Combin 31:1589–1611, 2015) asked about the independence number of \({\mathcal {SR}}(m,n)\) that is the maximum number of non-attacking rooks which can be placed on a \((m-1)\)-dimensional simplicial chessboard of side length \(n+1\). In this work, we solve this problem and show that \(\alpha ({\mathcal {SR}}(m,n))=\big (1-o(1)\big )\frac{\left( {\begin{array}{c}n+m-1\\ n\end{array}}\right) }{m}\). We also prove that for the domination number of rook graphs we have \(\gamma ({\mathcal {SR}}(m, n))= \varTheta (n^{m-2})\). Moreover we show that these graphs are Hamiltonian. The cyclic simplicial rook graph \({\mathcal {CSR}}(m,n)\) is the graph whose vertices are vectors in \({\mathbb {Z}}^{m}_{n}\) such that for each vector the summation of its coordinates modulo n is 0 and two vertices are adjacent if their corresponding vectors differ in exactly two coordinates. In this work we determine several properties of these graphs such as independence number, chromatic number and automorphism group. Among other results, we also prove that computing the distance between two vertices of a given \({\mathcal {CSR}}(m,n)\) is \( \mathbf {NP}\)-hard in terms of n and m.

A framework for optimal quantum spatial search using alternating phase-walks

We present a novel methodological framework for quantum spatial search, generalising the Childs & Goldstone () algorithm via alternating applications of marked-vertex phase shifts and continuous-time quantum walks. We determine closed form expressions for the optimal walk time and phase shift parameters for periodic graphs. These parameters are chosen to rotate the system about subsets of the graph Laplacian eigenstates, amplifying the probability of measuring the marked vertex. The state evolution is asymptotically optimal for any class of periodic graphs having a fixed number of unique eigenvalues. We demonstrate the effectiveness of the algorithm by applying it to obtain search on a variety of graphs. One important class is the n × n 3 rook graph, which has N = n 4 vertices. On this class of graphs the algorithm performs suboptimally, achieving only overlap after time . Using the new alternating phase-walk framework, we show that overlap is obtained in phase-walk iterations.

On the Gonality of Cartesian Products of Graphs

In this paper we provide the first systematic treatment of Cartesian products of graphs and their divisorial gonality, which is a tropical version of the gonality of an algebraic curve defined in terms of chip-firing. We prove an upper bound on the gonality of the Cartesian product of any two graphs, and determine instances where this bound holds with equality, including for the $m\times n$ rook's graph with $\min\{m,n\}\leq 5$. We use our upper bound to prove that Baker's gonality conjecture holds for the Cartesian product of any two graphs with two or more vertices each, and we determine precisely which nontrivial product graphs have gonality equal to Baker's conjectural upper bound. We also extend some of our results to metric graphs.

On the quantum symmetry of distance-transitive graphs

In this article, we study quantum automorphism groups of distance-transitive graphs. We show that the odd graphs, the Hamming graphs H(n,3), the Johnson graphs J(n,2) and the Kneser graphs K(n,2) do not have quantum symmetry. We also give a table with the quantum automorphism groups of all cubic distance-transitive graphs. Furthermore, with one graph missing, we can now decide whether or not a distance-regular graph of order ≤20 has quantum symmetry. Moreover, we prove that the Hoffman-Singleton graph has no quantum symmetry. On a final note, we present an example of a pair of graphs with the same intersection array (the Shrikhande graph and the 4×4 rook's graph), where one of them has quantum symmetry and the other one does not.

On the metric dimensions for sets of vertices

Resolving sets were originally designed to locate vertices of a graph one at a time. For the purpose of locating multiple vertices of the graph simultaneously, $\{\ell\}$-resolving sets were recently introduced. In this paper, we present new results regarding the $\{\ell\}$-resolving sets of a graph. In addition to proving general results, we consider $\{2\}$-resolving sets in rook's graphs and connect them to block designs. We also introduce the concept of $\ell$-solid-resolving sets, which is a natural generalisation of solid-resolving sets. We prove some general bounds and characterisations for $\ell$-solid-resolving sets and show how $\ell$-solid- and $\{\ell\}$-resolving sets are connected to each other. In the last part of the paper, we focus on the infinite graph family of flower snarks. We consider the $\ell$-solid- and $\{\ell\}$-metric dimensions of flower snarks. In two proofs regarding flower snarks, we use a new computer-aided reduction-like approach.

Tolerant location detection in sensor networks

Location detection in sensor networks can be handled with so called identifying codes. For an identifying code to work properly, it is required that no sensors are malfunctioning. Previously, malfunctioning sensors have been typically coped with robust identifying codes. However, they are rather large and, hence, imply high signal interference and energy consumption. To overcome these issues, collections of disjoint identifying codes have been proposed for coping with malfunctioning sensors. However, these collections have some problems regarding detection of malfunctioning sensors and, moreover, it seems unnecessary to restrict oneself to disjoint codes. In this paper, we discuss a certain type of identifying codes, for which the detection of malfunctioning sensors is easy, and based on these codes we design a collection of codes tolerant against malfunctions. We present some results on general graphs as well as optimal constructions in rook's graphs and binary Hamming spaces.

3-tuple total domination number of rook's graphs

A $k$-tuple total dominating set ($k$TDS) of a graph $G$ is a set $S$ of vertices in which every vertex in $G$ is adjacent to at least $k$ vertices in $S$. The minimum size of a $k$TDS is called the $k$-tuple total dominating number and it is denoted by $\gamma_{\times k,t}(G)$. We give a constructive proof of a general formula for $\gamma_{\times 3, t}(K_n \Box K_m)$.

Autoparatopism stabilized colouring games on rook's graphs

We introduce the autoparatopism variant of the autotopism stabilized colouring game on the n×n rook's graph as a natural generalization of the latter so that each board configuration is uniquely related to a partial Latin square of order n that respects a given autoparatopism (θ; π). To this end, we distinguish between π∈{Id,(12)} and π∈{(13),(23),(123),(132)}. The complexity of this variant is examined by means of the autoparatopism stabilized game chromatic number. Some illustrative examples and results are shown.

The Smith and Critical Groups of the Square Rook's Graph and its Complement

Let $R_{n}$ denote the graph with vertex set consisting of the squares of an $n \times n$ grid, with two squares of the grid adjacent when they lie in the same row or column. This is the square rook's graph, and can also be thought of as the Cartesian product of two complete graphs of order $n$, or the line graph of the complete bipartite graph $K_{n,n}$. In this paper we compute the Smith group and critical group of the graph $R_{n}$ and its complement. This is equivalent to determining the Smith normal form of both the adjacency and Laplacian matrix of each of these graphs. In doing so we verify a 1986 conjecture of Rushanan.

Notes on simplicial rook graphs

The simplicial rook graph $$\mathrm{SR}(m,n)$$SR(m,n) is the graph of which the vertices are the sequences of nonnegative integers of length m summing to n, where two such sequences are adjacent when they differ in precisely two places. We show that $$\mathrm{SR}(m,n)$$SR(m,n) has integral eigenvalues, and smallest eigenvalue $$s = \max \left( -n, -{m \atopwithdelims ()2}\right) $$s=max-n,-m2, and that this graph has a large part of its spectrum in common with the Johnson graph $$J(m+n-1,n)$$J(m+n-1,n). We determine the automorphism group and several other properties.

Resistance distances in composite graphs

The resistance distance between any two vertices of a connected graph is defined as the effective resistance between them in the electrical network constructed from the graph by replacing each edge with a (unit) resistor. Let , G × H, , and be the join, product, composition, direct product, strong product, corona and rooted product of two graphs G and H, respectively. In this paper, formulae for resistance distances of these composite graphs are given in terms of parameters of the parent graphs, and some properties are established. Explicit formulae are obtained for resistance distances of some classes of graphs, including rook graphs, diagonal mesh graphs and generalized double graphs.

On the Spectra of Simplicial Rook Graphs

The simplicial rook graph \(SR(d,n)\) is the graph whose vertices are the lattice points in the \(n\)th dilate of the standard simplex in \(\mathbb {R}^d\), with two vertices adjacent if they differ in exactly two coordinates. We prove that the adjacency and Laplacian matrices of \(SR(3,n)\) have integral spectrum for every \(n\). The proof proceeds by calculating an explicit eigenbasis. We conjecture that \(SR(d,n)\) is integral for all \(d\) and \(n\), and present evidence in support of this conjecture. For \(n<\left( {\begin{array}{c}d\\ 2\end{array}}\right) \), the evidence indicates that the smallest eigenvalue of the adjacency matrix is \(-n\), and that the corresponding eigenspace has dimension given by the Mahonian numbers, which enumerate permutations by number of inversions.

On the Spectra of Simplicial Rook Graphs

The $\textit{simplicial rook graph}$ $SR(d,n)$ is the graph whose vertices are the lattice points in the $n$th dilate of the standard simplex in $\mathbb{R}^d$, with two vertices adjacent if they differ in exactly two coordinates. We prove that the adjacency and Laplacian matrices of $SR(3,n)$ have integral spectra for every $n$. We conjecture that $SR(d,n)$ is integral for all $d$ and $n$, and give a geometric construction of almost all eigenvectors in terms of characteristic vectors of lattice permutohedra. For $n \leq \binom{d}{2}$, we give an explicit construction of smallest-weight eigenvectors in terms of rook placements on Ferrers diagrams. The number of these eigenvectors appears to satisfy a Mahonian distribution. Le $\textit{graphe des tours simplicials}$ $SR(d,n)$ est le graphe dont les sommets sont les points du réseau dans le $n$ième dilation du simplexe standard dans $\mathbb{R}^d$ ; deux sommets sont adjacents s’ils diffèrent dans exactement deux coordonnées. Nous montrons que tous les valeurs propres des matrices d’adjacence et laplacienne de $SR(3,n)$ sont entiers, pour tous les $n$. Nous conjecturons que les valeurs propres sont entiers pour tous $d$ et $n$, et donnons une construction géometrique de presque tous les vecteurs propres en termes des vecteurs caractéristiques de permutoèdres treillis. Pour $n \leq \binom{d}{2}$, nous donnons une construction explicite des vecteurs propres de plus petits poids en termes desplacements des tours sur diagrammes de Ferrers. Le nombre de ces vecteurs propres semble satisfaire une distribution Mahonian.

Spanning tree congestion of rook's graphs

Spanning tree congestion of rook's graphs

The Many Formulae for the Number of Latin Rectangles

A $k \times n$ Latin rectangle $L$ is a $k \times n$ array, with symbols from a set of cardinality $n$, such that each row and each column contains only distinct symbols. If $k=n$ then $L$ is a Latin square. Let $L_{k,n}$ be the number of $k \times n$ Latin rectangles. We survey (a) the many combinatorial objects equivalent to Latin squares, (b) the known bounds on $L_{k,n}$ and approximations for $L_n$, (c) congruences satisfied by $L_{k,n}$ and (d) the many published formulae for $L_{k,n}$ and related numbers. We also describe in detail the method of Sade in finding $L_{7,7}$, an important milestone in the enumeration of Latin squares, but which was privately published in French. Doyle's formula for $L_{k,n}$ is given in a closed form and is used to compute previously unpublished values of $L_{4,n}$, $L_{5,n}$ and $L_{6,n}$. We reproduce the three formulae for $L_{k,n}$ by Fu that were published in Chinese. We give a formula for $L_{k,n}$ that contains, as special cases, formulae of (a) Fu, (b) Shao and Wei and (c) McKay and Wanless. We also introduce a new equation for $L_{k,n}$ whose complexity lies in computing subgraphs of the rook's graph.

The rook problem on saw-toothed chessboards

A saw-toothed chessboard, or STC for short, is a kind of chessboard whose boundary forms two staircases from left down to right without any hole inside it. A rook at square ( i , j ) can dominate the squares in row i and in column j . The rook problem of an STC is to determine the minimum number of rooks that can dominate all squares of the STC. In this paper, we model an STC by two particular graphs: a rook graph and a board graph. We show that for an STC, the rook graph is the line graph of the board graph, and the board graph is a bipartite permutation graph. Thus, the rook problem on STCs can be solved by any algorithm for solving the edge domination problem on bipartite permutation graphs.