A set of vertices in a graph is called independent if no two vertices of the set are connected by an edge. In this paper we use the state matrix recursion algorithm, developed by Oh, to enumerate independent vertex sets in a grid graph and even further to provide the generating function with respect to the number of vertices. We also enumerate bipartite independent vertex sets in a grid graph. The asymptotic behavior of their growth rates is presented.
This paper provides an alternate proof to parts of the Goulden-Slofstra formula for enumerating two vertex maps by genus, which is an extension of the famous Harer-Zagier formula that computes the Euler characteristic of the moduli space of curves. This paper also shows a further simplification to the Goulden-Slofstra formula. Portions of this alternate proof will be used in a subsequent paper, where it forms a basis for a more general result that applies for a certain class of maps with an arbitrary number of vertices.
In this paper we demonstrate a method for counting the number of solutions to various logic puzzles. Specifically, we remove all of the "clues" from the puzzle which help the solver to a unique solution, and instead start from an empty grid. We then count the number of ways to fill in this empty grid to a valid solution. We fix the number of rows $r$, vary the number of columns $k$, and then compute the sequence $A_r(k)$, which gives the number of solutions on an empty grid of size $r \times k$.
Over some types of trees with a given number of vertices, which trees minimize or maximize the total number of subtrees or leaf containing subtrees are studied. Here are some of the main results:\ (1)\, Sharp upper bound on the total number of subtrees (resp. leaf containing subtrees) among $n$-vertex trees with a given matching number is determined; as a consequence, the $n$-vertex tree with domination number $γ$ maximizing the total number of subtrees (resp. leaf containing subtrees) is characterized. (2)\, Sharp lower bound on the total number of leaf containing subtrees among $n$-vertex tr…
We enumerate a certain class of monomino-domino coverings of square grids, which conform to the \emph{tatami} restriction; no four tiles meet. Let $\mathbf T_{n}$ be the set of monomino-domino tatami coverings of the $n\times n$ grid with the maximum number, $n$, of monominoes, oriented so that they have a monomino in each of the top left and top right corners. We give an algorithm for exhaustively generating the coverings in $\mathbf T_{n}$ with exactly $v$ vertical dominoes in constant amortized time, and an explicit formula for counting them. The polynomial that generates these counts has t…
By extending the notion of grid classes to include infinite grids, we establish a structural characterisation of the simple permutations in Av(4231, 35142, 42513, 351624), a pattern class which has three different connections with algebraic geometry, including the specification of indices of Schubert varieties defined by inclusions. This characterisation leads to the enumeration of the class.
The method of graphical vertex-condensation for enumerating perfect matchings of plane bipartite graph was found by Propp (Theoret. Comput. Sci. 303(2003), 267-301), and was generalized by Kuo (Theoret. Comput. Sci. 319 (2004), 29-57) and Yan and Zhang (J. Combin. Theory Ser. A, 110(2005), 113-125). In this paper, by a purely combinatorial method some explicit identities on graphical vertex-condensation for enumerating perfect matchings of plane graphs (which do not need to be bipartite) are obtained. As applications of our results, some results on graphical edge-condensation for enumerating p…
Recently designed biomolecular approaches to build single chain polypeptide polyhedra as molecular origami nanostructures have risen high interest in various double traces of the underlying graphs of these polyhedra. Double traces are walks that traverse every edge of the graph twice, usually with some additional conditions on traversal direction and vertex neighborhood coverage. Given that double trace properties are intimately related to theefficiency of polypeptide polyhedron construction, enumerating all different possible double traces and analyzing their properties is an important step i…
In 1983 Kalai proved an incredible generalisation of Cayley's formula for the number of trees on a labelled vertex set to a formula for a class of $r$-dimensional simplicial complexes. These simplicial complexes generalise trees by means of being homologically $\Bbb Q$-acyclic. In this text we consider a different generalisation of trees to the class of pure dimensional simplicial complexes that \textit{minimally connect a vertex set} (in the sense that the removal of any top dimensional face disconnects the complex). Our main result provides an upper and lower bound for the number of these mi…
There are local operators on (labeled) graphs $G$ with labels $(g_{ij})$ coming from a finite field. If the filed is binary, in other words, if the graph is ordinary, the operation is just the local complementation. That is, to choose a vertex and complement the subgraph induced by its neighbors. But, in the general case, there are two different types of operators. The first type is the following. Let $v$ be a vertex of the graph and $a\in \mathbf{F}_q$, the finite field of $q$ elements. The operator is to obtain a graph with labels $g'_{ij}=g_{ij}+ag_{vi}g_{vj}$. For the second type of operat…
Let $G = (V, E)$ be a graph. A set $S \subseteq V$ is a restrained dominating set (RDS) if every vertex not in $S$ is adjacent to a vertex in $S$ and to a vertex in $V - S$. The restrained domination number of $G$, denoted by $γ_r(G)$, is the smallest cardinality of a restrained dominating set of $G$. Finding the restrained domination number is NP-hard for bipartite and chordal graphs. Let $G_n^i$ be the family of restrained dominating sets of a graph $G$ of order $n$ with cardinality $i$, and let $d_r(G_n, i)=|G_n^i|$. The restrained domination polynomial (RDP) of $G_n$, $D_r(G_n, x)$ is defi…
For $n\geq 3$ and $r=r(n) \geq 3$, let $\boldsymbol{k} =\boldsymbol{k}(n)=(k_1, \ldots, k_n)$ be a sequence of non-negative integers with sum $M(\boldsymbol{k})=\sum_{j=1}^{n} k_j$. We assume that $M(\boldsymbol{k})$ is divisible by $r$ for infinitely many values of $n$, and restrict our attention to these values. Let $X=X(n)$ be a simple $r$-uniform hypergraph on the vertex set $V=\{v_1,v_2, \ldots, v_n\}$ with $t$ edges and maximum degree $x_{\max}$. We denote by $\mathcal{H}_r(\boldsymbol{k})$ the set of all simple $r$-uniform hypergraphs on the vertex set $V$ with degree sequence $\boldsym…
Problems from metric graph theory like Metric Dimension, Geodetic Set, and Strong Metric Dimension have recently had a strong impact in parameterized complexity by being the first known problems in NP to admit double-exponential lower bounds in the treewidth, and even in the vertex cover number for the latter, assuming the Exponential Time Hypothesis. We initiate the study of enumerating minimal solution sets for these problems and show that they are also of great interest in enumeration. Specifically, we show that enumerating minimal resolving sets in graphs and minimal geodetic sets in split…
Motivated to find the answers to some of the questions that have occurred in recent papers dealing with Hamiltonian cycles (abbreviated HCs) in some special classes of grid graphs we started the investigation of spanning unions of cycles, the so-called 2-factors, in these graphs (as a generalizations of HCs). For all the three types of graphs from the title and for any integer $m \geq 2$ we propose an algorithm for obtaining a specially designed (transfer) digraph ${\cal D}^*_m$. The problem of enumeration of 2-factors is reduced to the problem of enumerating oriented walks in this digraph. Co…
We investigate graphs that can be represented as vertex intersections of horizontal and vertical paths in a grid, the so called $B_0$-VPG graphs. Recognizing this class is an NP-complete problem. Although, there exists a polynomial time algorithm for recognizing chordal $B_0$-VPG graphs. In this paper, we present a minimal forbidden induced subgraph characterization of $B_0$-VPG graphs restricted to block graphs. As a byproduct, the proof of the main theorem provides an alternative certifying recognition and representation algorithm for $B_0$-VPG graphs in the class of block graphs.
For a graph, $G$, and a vertex $v \in V(G)$, let $N[v]$ be the set of vertices adjacent to and including $v$. A set $D \subseteq V(G)$ is a vertex identifying code if for any two distinct vertices $v_1, v_2 \in V(G)$, the vertex sets $N[v_1] \cap D$ and $N[v_2] \cap D$ are distinct and non-empty. We consider the minimum density of a vertex identifying code for the infinite hexagonal grid. In 2000, Cohen et al. constructed two codes with a density of $3/7 \approx 0.428571$, and this remains the best known upper bound. Until now, the best known lower bound was $12/29 \approx 0.413793$ and was pr…
We propose a randomized algorithm for enumerating the vertices of a zonotope, which is a low-dimensional linear projection of a hypercube. The algorithm produces a pair of the zonotope's vertices by sampling a random linear combination of the zonotope generators, where the combination's weights are the signs of the product between the zonotope's generator matrix and random vectors with normally distributed entries. We study the probability of recovering particular vertices and relate it to the vertices' normal cones. This study shows that if we terminate the randomized algorithm before all ver…
A graph $G$ is equitably $k$-list arborable if for any $k$-uniform list assignment $L$, there is an equitable $L$-colouring of $G$ whose each colour class induces an acyclic graph. The smallest number $k$ admitting such a coloring is named equitable list vertex arboricity and is denoted by $ρ_l^=(G)$. Zhang in 2016 posed the conjecture that if $k \geq \lceil (Δ(G)+1)/2 \rceil$ then $G$ is equitably $k$-list arborable. We give some new tools that are helpful in determining values of $k$ for which a general graph is equitably $k$-list arborable. We use them to prove the Zhang's conjecture for $d…
PhD Thesis--A compilation of the papers: "Lower Bounds for Identifying Codes in Some Infinite Grids", "Improved Bounds for r-identifying Codes of the Hex Grid", and "Vertex Identifying Codes for the n-dimensional Lattics" along with some other results
A pebbling move on a graph removes two pebbles from a vertex and adds one pebble to an adjacent vertex. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using pebbling moves. The optimal pebbling number $π_{opt}$ is the smallest number m needed to guarantee a pebble distribution of m pebbles from which any vertex is reachable. The optimal pebbling number of the square grid graph $P_n\square P_m$ was investigated in several papers. In this paper, we present a new method using some recent ideas to give a lower bound on $π_{opt}$. We apply this te…
In this series of papers, the primary goal is to enumerate Hamiltonian cycles (HC's) on the grid cylinder graphs $P_{m+1}\times C_n$, where $n$ is allowed to grow whilst $m$ is fixed. In Part~I, we studied the so-called non-contractible HC's. Here, in Part~II, we proceed further on to the contractible case. We propose two different novel characterizations of contractible HC's, from which we construct digraphs for enumerating the contractible HC's. Given the impression which the computational data for $m \leq 9$ convey, we conjecture that the asymptotic domination of the contractible HC's versu…
We say that a vertex-coloring of a graph is a proper k-distance domatic coloring if for each color, every vertex is within distance k from a vertex receiving that color. The maximum number of colors for which such a coloring exists is called the k-distance domatic number of the graph. The problem of determining the k-distance domatic number is motivated by questions about multi-agent networks including arrangements of sensors and robotics. Here, we find the exact k-distance domatic numbers for all grid graphs formed from the Cartesian product of two sufficiently long paths.
This paper deals with three graph characteristics related to graph covering named the (vertex, edge, and total, resp.) H-irregularity strength of a graph G admitting H-covering. Those are the minimum values of positive integer k such that G has an H-irregular (vertex, edge, and total, resp.) k-labeling. The exact values of these three characteristics are determined for grid graph admitting grid-covering
An \emph{antimagic labeling} of a finite undirected simple graph with $m$ edges and $n$ vertices is a bijection from the set of edges to the integers $1,...,m$ such that all $n$ vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called \emph{antimagic} if it has an antimagic labeling. In 1990, Hartsfield and Ringel conjectured that every connected graph, but $K_2$, is antimagic. In 2004, N. Alon et al showed that this conjecture is true for $n$-vertex graphs with minimum degree $Ω(\log n)$. They also proved that com…
The domination number of a finite graph $G$ with vertex set $V$ is the cardinality of the smallest set $S\subseteq V$ such that for every vertex $v\in V$ either $v\in S$ or $v$ is adjacent to a vertex in $S$. A set $S$ satisfying these conditions is called a dominating set. In 2015 Blessing, Insko, Johnson, and Mauretour introduced $(t,r)$ broadcast domination, a generalization of graph domination parameterized by the nonnegative integers $t$ and $r$. In this setting, we say that the signal a vertex $v\in V$ receives from a tower of strength $t$ located at vertex $T$ is defined by $sig(v,T)=ma…
We consider a classic model known as bootstrap percolation on the $n \times n$ square grid. To each vertex of the grid we assign an initial state, infected or healthy, and then in consecutive rounds we infect every healthy vertex that has at least $2$ already infected neighbours. We say that percolation occurs if the whole grid is eventually infected. In this paper, contributing to a recent series of extremal results in this field, we prove that the maximum time a bootstrap percolation process can take to eventually infect the entire vertex set of the grid is $13n^2/18+O(n)$.
A pebbling move on a graph consists of removing $2$ pebbles from a vertex and adding $1$ pebble to one of the neighbouring vertices. A vertex is called reachable if we can put $1$ pebble on it after a sequence of moves. The optimal pebbling number of a graph is the minimum number $m$ such that there exists a distribution of $m$ pebbles so that each vertex is reachable. For the case of a square grid $n \times m$, Győri, Katona and Papp recently showed that its optimal pebbling number is at least $\frac{2}{13}nm \approx 0.1538nm$ and at most $\frac{2}{7}nm +O(n+m) \approx 0.2857nm$. We improve t…
College of Saint Benedict and Saint John’s University（圣本笃学院与圣约翰大学）；Northern Arizona University（北亚利桑那大学）；Lewis University（路易斯大学）；University of Colorado Boulder（科罗拉多大学博尔德分校）
We introduce new methods to describe admissible states of the six-vertex and the eight-vertex lattice models of statistical mechanics. For the six-vertex model, we view the admissible states as differential forms on a grid graph. This yields a new proof of the correspondence between admissible states and 3-colorings of a rectangular grid. For the eight-vertex model, we interpret the set of admissible states as an $\mathbb{F}_2$-vector space. This viewpoint lets us enumerate the set of admissible states. Finally, we find necessary conditions for a Yang-Baxter equation to hold for the general ei…
We consider some problems concerning the maximum number of (strong) dominating sets in a regular graph, and their weighted analogues. Our primary tool is Shearer's entropy lemma. These techniques extend to a reasonably broad class of graph parameters enumerating vertex colorings satisfying conditions on the multiset of colors appearing in (closed) neighborhoods. We also generalize further to enumeration problems for what we call existence homomorphisms. Here our results are substantially less complete, though we do solve some natural problems.
A vertex k-ranking is a labeling of the vertices of a graph with integers from 1 to k so any path connecting two vertices with the same label will pass through a vertex with a greater label. The rank number of a graph is defined to be the minimum possible k for which a k-ranking exists for that graph. For mxn grid graphs, the rank number has been found only for m<4. In this paper, we determine its for m=4 and improve its upper bound for general grids. Furthermore, we improve lower bounds on the rank numbers for square and triangle grid graphs from logarithmic to linear. These new lower bounds …
This paper gives lower bounds on the spectral radius of vertex-transitive graphs, based on the number of ``prime cycles'' at a vertex. The bounds are obtained by constructing circuits in the graph that resemble ``cactus trees'', and enumerating them. Counting these circuits gives a coefficient-wise underestimation of the Green function of the graph, and hence and underestimation of its spectral radius. The bounds obtained are very good for the Cayley graph of surface groups of genus g>=2, with standard generators (these graphs are the 1-skeletons of tessellations of hyperbolic plane by 4g-gons…
In this paper, we continue the study of locating-paired-dominating set, abbreviated LPDS, in graphs introduced by McCoy and Henning. Given a finite or infinite graph $G=(V,E)$, a set $S\subset V$ is paired-dominating if the induced subgraph $G[S]$ has a perfect matching and every vertex in $V$ is adjacent to a vertex in $S$. The other condition for LPDS requires that for any distinct vertices $u,v \in V\backslash S$, we have $N(u)\cap S\neq N(v)\cap S$. Motivated by the conjecture of Kinawi, Hussain and Niepel, we prove the minimal density of LPDS in the king grid is between $8/37$ and $2/9$, …
Given a graph $G$, an identifying code $C \subseteq V(G)$ is a vertex set such that for any two distinct vertices $v_1,v_2\in V(G)$, the sets $N[v_1]\cap C$ and $N[v_2]\cap C$ are distinct and nonempty (here $N[v]$ denotes a vertex $v$ and its neighbors). We study the case when $G$ is the infinite hexagonal grid $H$. Cohen et.al. constructed two identifying codes for $H$ with density $3/7$ and proved that any identifying code for $H$ must have density at least $16/39\approx0.410256$. Both their upper and lower bounds were best known until now. Here we prove a lower bound of $12/29\approx0.4137…
In Zombies and Survivors, a set of zombies attempts to eat a lone survivor loose on a given graph. The zombies randomly choose their initial location, and during the course of the game, move directly toward the survivor. At each round, they move to the neighbouring vertex that minimizes the distance to the survivor; if there is more than one such vertex, then they choose one uniformly at random. The survivor attempts to escape from the zombies by moving to a neighbouring vertex or staying on his current vertex. The zombies win if eventually one of them eats the survivor by landing on their ver…
For a simple graph $G=(V,E)$ and for a pair of vertices $u,v \in V$, we say that a vertex $w \in V$ resolves $u$ and $v$ if the shortest path from $w$ to $u$ is of a different length than the shortest path from $w$ to $v$. A set of vertices ${R \subseteq V}$ is a resolving set if for every pair of vertices $u$ and $v$ in $G$, there exists a vertex $w \in R$ that resolves $u$ and $v$. The minimum weight resolving set problem is to find a resolving set $M$ for a weighted graph $G$ such that$\sum_{v \in M} w(v)$ is minimum, where $w(v)$ is the weight of vertex $v$. In this paper, we explore the p…
Domination of grids has been proved to be a demanding task and with the addition of independence it becomes more challenging. It is known that no grid with $m,n \geq 5$ has an efficient dominating set, also called perfect code, that is, an independent vertex set such that each vertex not in it has exactly one neighbor in that set. So it is interesting to study the existence of independent dominating sets for grids that allow at most two neighbors, such sets are called independent $[1,2]$-sets. In this paper we prove that every grid has an independent $[1,2]$-set, and we develop a dynamic progr…
We study the $(p,q)$-Maker Breaker Crossing game introduced by Day and Falgas Ravry in 'Maker-Breaker percolation games I: crossing grids'. The game described in their paper involves two players Maker and Breaker who take turns claiming p and q as yet unclaimed edges of the graph respectively. Maker aims to make a horizontal path from a leftmost vertex to a rightmost vertex and Breaker aims to prevent this. The game is a version of the more general Shannon switching game and is played on a square grid graph. We consider the same game played on the triangular grid graph $Δ_{(m,n)}$ (m vertices …
A dominating set $S$ in a graph $G$ is said to be perfect if every vertex of $G$ not in $S$ is adjacent to just one vertex of $S$. Given a vertex subset $S'$ of a side $P_m$ of an $m\times n$ grid graph $G$, the perfect dominating sets $S$ in $G$ with $S'=S\cap V(P_m)$ can be determined via an exhaustive algorithm $Θ$ of running time $O(2^{m+n})$. Extending $Θ$ to infinite grid graphs of width $m-1$, periodicity makes the binary decision tree of $Θ$ prunable into a finite threaded tree, a closed walk of which yields all such sets $S$. The graphs induced by the complements of such sets $S$ can …
In this paper, we consider the \emph{firefighter problem} on a graph $G=(V,E)$ that is either finite or infinite. Suppose that a fire breaks out at a given vertex $v \in V$. In each subsequent time unit, a firefighter protects one vertex which is not yet on fire, and then the fire spreads to all unprotected neighbors of the vertices on fire. The objective of the firefighter is to save as many vertices as possible (if $G$ is finite) or to stop the fire from spreading (for an infinite case). The surviving rate $ρ(G)$ of a finite graph $G$ is defined as the expected percentage of vertices that ca…
Suppose each vertex of a graph is originally occupied by contamination, except for those vertices occupied by lions. As the lions wander on the graph, they clear the contamination from each vertex they visit. However, the contamination simultaneously spreads to any adjacent vertex not occupied by a lion. How many lions are required in order to clear the graph of contamination? We give a lower bound on the number of lions needed in terms of the Cheeger constant of the graph. Furthermore, the lion and contamination problem has been studied in detail on square grid graphs by Brass et al. and Berg…
A longest sequence $S$ of distinct vertices of a graph $G$ such that each vertex of $S$ dominates some vertex that is not dominated by its preceding vertices, is called a Grundy dominating sequence; the length of $S$ is the Grundy domination number of $G$. In this paper we study the Grundy domination number in the four standard graph products: the Cartesian, the lexicographic, the direct, and the strong product. For each of the products we present a lower bound for the Grundy domination number which turns out to be exact for the lexicographic product and is conjectured to be exact for the stro…
Let $G=(V,E)$ be a graph and $t,r$ be positive integers. The \emph{signal} that a tower vertex $T$ of signal strength $t$ supplies to a vertex $v$ is defined as $sig(T,v)=max(t-dist(T,v),0),$ where $dist(T,v)$ denotes the distance between the vertices $v$ and $T$. In 2015 Blessing, Insko, Johnson, and Mauretour defined a \emph{$(t,r)$ broadcast dominating set}, or simply a \emph{$(t,r)$ broadcast}, on $G$ as a set $\mathbb{T}\subseteq V$ such that the sum of all signals received at each vertex $v \in V$ from the set of towers $\mathbb{T}$ is at least $r$. The $(t,r)$ broadcast domination numbe…
An open-locating-dominating set (OLD-set) is a subset of vertices of a graph such that every vertex in the graph has at least one neighbor in the set and no two vertices in the graph have the same set of neighbors in the set. This is an analogue to the well-studied identifying code in the literature. In this paper, we prove that the optimal density of the OLD-set for the infinite triangular grid is $4/13$.
An $r$-identifying code on a graph $G$ is a set $C\subset V(G)$ such that for every vertex in $V(G)$, the intersection of the radius-$r$ closed neighborhood with $C$ is nonempty and unique. On a finite graph, the density of a code is $|C|/|V(G)|$, which naturally extends to a definition of density in certain infinite graphs which are locally finite. We present new lower bounds for densities of codes for some small values of $r$ in both the square and hexagonal grids.
Let $G=(V,E)$ be a graph and $t,r$ be positive integers. The signal that a vertex $v$ receives from a tower of signal strength $t$ located at vertex $T$ is defined as $sig(v,T)=max(t-dist(v,T),0)$, where $dist(v,T)$ denotes the distance between the vertices $v$ and $T$. In 2015 Blessing, Insko, Johnson, and Mauretour defined a $(t,r)$ broadcast dominating set, or simply a $(t,r)$ broadcast, on $G$ as a set $\mathbb{T}\subseteq V$ such that the sum of all signal received at each vertex $v \in V$ is at least $r$. We say that $\mathbb{T}$ is optimal if $|\mathbb{T}|$ is minimal among all such set…
The firefighter problem with $k$ firefighters on an infinite graph $G$ is an iterative graph process, defined as follows: Suppose a fire breaks out at a given vertex $v\in V(G)$ on Turn 1. On each subsequent even turn, $k$ firefighters protect $k$ vertices that are not on fire, and on each subsequent odd turn, any vertex that is on fire spreads the fire to all adjacent unprotected vertices. The firefighters' goal is to eventually stop the spread of the fire. If there exists a strategy for $k$ firefighters to eventually stop the spread of the fire, then we say $G$ is $k$-containable. We conside…
Given an undirected simple graph, a subset of the vertices of the graph is a {\em dominating set} if every vertex not in the subset is adjacent to at least one vertex in the subset. A subset of the vertices of the graph is a {\em connected dominating set} if the subset is a dominating set and the subgraph induced by the subset is connected. In this paper, we determine the minimum cardinality of a connected dominating set, called the {\em connected domination number}, of an $m \times n$ grid graph for any $m$ and $n$.
An induced matching $M$ in a graph $G$ is a matching in $G$ that is also the edge set of an induced subgraph of $G$. That is, any edge not in $M$ must have no more than one incident vertex saturated by $M$. The maximum size $|M|$ of an induced matching $M$ of $G$ is maximum induced matching number of $G$, which is denoted by $\textrm{Max}(G)$. In this article, we obtain upper bounds for $\textrm{Max}(G)$, for $G=G_{n,m}$, grids with $n,m \geq 9$, $m\equiv 1 \mod 4$ and $nm$ odd.
We answer some questions concerning the so called sigma-game of Sutner. It is played on a graph where each vertex has a lamp, the light of which is toggled by pressing any vertex with an edge directed to the lamp. For example, we show that every configuration of lamps can be lit if and only if the number of complete matchings in the graph is odd. In the special case of an orthogonal grid one gets a criterion for whether the number of monomer-dimer tilings of an m times n grid is odd or even.
