The strong geodetic problem is a recent variation of the geodetic problem. For a graph $G$, its strong geodetic number ${\rm sg}(G)$ is the cardinality of a smallest vertex subset $S$, such that each vertex of $G$ lies on a fixed shortest path between a pair of vertices from $S$. In this paper, the strong geodetic problem is studied on the Cartesian product of graphs. A general upper bound for ${\rm sg}(G \,\square\, H)$ is determined, as well as exact values for $K_m \,\square\, K_n$, $K_{1, k} \,\square\, P_l$, and certain prisms. Connections between the strong geodetic number of a graph and its subgraphs are also discussed.
The strong geodetic problem on a graph $G$ is to determine a smallest set of vertices such that by fixing one shortest path between each pair of its vertices, all vertices of $G$ are covered. To do this as efficiently as possible, strong geodetic cores and related numbers are introduced. Sharp upper and lower bounds on the strong geodetic core number are proved. Using the strong geodetic core number an earlier upper bound on the strong geodetic number of Cartesian products is improved. It is also proved that ${\rm sg}(G \,\square\, K_2) \geq {\rm sg}(G)$ holds for different families of graphs,…
A recent variation of the classical geodetic problem, the strong geodetic problem, is defined as follows. If $G$ is a graph, then ${\rm sg}(G)$ is the cardinality of a smallest vertex subset $S$, such that one can assign a fixed geodesic to each pair $\{x,y\}\subseteq S$ so that these $\binom{|S|}{2}$ geodesics cover all the vertices of $G$. In this paper, the strong geodesic problem is studied on Cartesian product graphs. A general upper bound is proved on the Cartesian product of a path with an arbitrary graph and showed that the bound is tight on flat grids and flat cylinders.
Let $G=(V(G),E(G))$ be a simple graph. A set $S \subseteq V(G)$ is a strong edge geodetic set if there exists an assignment of exactly one shortest path between each pair of vertices from $S$, such that these shortest paths cover all the edges $E(G)$. The cardinality of a smallest strong edge geodetic set is the strong edge geodetic number $\text{sge}(G)$ of $G$. In this paper, the strong edge geodetic problem is studied on the Cartesian product of two paths. The exact value of the strong edge geodetic number is computed for $P_n \, \square \, P_2$, $P_n \, \square \, P_3$ and $P_n \, \square …
A set S of vertices of a connected graph G is convex, if for any pair of vertices u; v 2 S, every shortest path joining u and v is contained in S . The convex hull CH(S) of a set of vertices S is defined as the smallest convex set in G containing S. The set S is geodetic, if every vertex of G lies on some shortest path joining two vertices in S, and it is said to be a hull set if its convex hull is V(G). The geodetic and the hull numbers of G are the cardinality of a minimum geodetic and a minimum hull set, respectively. In this work, we investigate the behavior of both geodetic and hull sets …
A set of vertices $S$ of a graph $G$ is a geodetic set of $G$ if every vertex $v\not\in S$ lies on a shortest path between two vertices of $S$. The minimum cardinality of a geodetic set of $G$ is the geodetic number of $G$ and it is denoted by $g(G)$. A Steiner set of $G$ is a set of vertices $W$ of $G$ such that every vertex of $G$ belongs to the set of vertices of a connected subgraph of minimum size containing the vertices of $W$. The minimum cardinality of a Steiner set of $G$ is the Steiner number of $G$ and it is denoted by $s(G)$. Let $G$ and $H$ be two graphs and let $n$ be the order o…
The Closed Geodetic Game: algorithms and strategies
Université Clermont Auvergne（克莱蒙奥弗涅大学）；CNRS（法国国家科学研究中心）；Mines de Saint-Étienne（圣艾蒂安国立高等矿业学校）；Clermont-Auvergne-INP（克莱蒙奥弗涅理工学院）；LIMOS（极限与优化实验室）；INRAE（法国国家农业、食品和环境研究所）；Laboratoire G-SCOP（G-SCOP实验室）；CNRS, Univ. Grenoble Alpes（法国国家科学研究中心，格勒诺布尔阿尔卑斯大学）；Ben-Gurion University of the Negev（内盖夫本古里安大学）；Indian Institute of Technology Dharwad（印度理工学院达瓦德分校）；Polytechnic University (Kyaing Tong)（桂开工科大学）
Recently, Bapat and Kurata [\textit{Linear Algebra Appl.}, 562(2019), 135-153] defined the Cartesian product of two square matrices $A$ and $B$ as $A\oslash B=A\otimes \J+\J\otimes B$, where $\J$ is the all one matrix of appropriate order and $\otimes$ is the Kronecker product. In this article, we find the expression for the trace of the Cartesian product of any finite number of square matrices in terms of traces of the individual matrices. Also, we establish some identities involving the Cartesian product of matrices. Finally, we apply the Cartesian product to study some graph-theoretic prope…
We study linkedness of Cartesian product of graphs and prove that the product of an $a$-linked and a $b$-linked graphs is $(a+b-1)$-linked if the graphs are sufficiently large. Further bounds in terms of connectivity are shown. We determine linkedness of product of paths and product of cycles.
We show that every nontrivial finite or infinite connected directed graph with loops and at least one vertex without a loop is uniquely representable as a Cartesian or weak Cartesian product of prime graphs. For finite graphs the factorization can be computed in linear time and space.
In this paper, we study the Cartesian product of signed graphs as defined by Germina, Hameed and Zaslavsky (2011). Here we focus on its algebraic properties and look at the chromatic number of some Cartesian products. One of our main results is the unicity of the prime factor decomposition of signed graphs. This leads us to present an algorithm to compute this decomposition in linear time based on a decomposition algorithm for oriented graphs by Imrich and Peterin (2018). We also study the chromatic number of a signed graph, that is the minimum order of a signed graph to which the input signed…
We prove upper and lower bounds on the chromatic number of the square of the cartesian product of trees. The bounds are equal if each tree has even maximum degree.
We study inheritance of path-pairability in the Cartesian product of graphs, and prove different (such as additive and multiplicative) inheritance patterns of path-pairability, depending on the size of the Cartesian product. We present path-pairable graph families, that improve the known upper bound on the minimal maximum degree of a path-pairable graph. Further results and open questions about path-pairability are also presented.
We study the structure of a minimum dominating set of $C_{2n+1}^n$, the Cartesian product of $n$ copies of the cycle of size $2n+1$, where $2n+1$ is a prime.
N. Hindman and D. Strauss had shown that, for discrete semigroups, the cartesian product of two central sets are central. They also proved that the product of J- sets and C-sets are also J-set and C-set and characterized when the infnite product of these sets are preserved. To prove these results they used the algebraic structure of Stone-Cech compactification of discrete semigroups. In this work we will give a combinatorial proof of the preserveness of those large sets under fnite cartesian product.
In 1970, Plummer defined a well-covered graph to be a graph $G$ in which all maximal independent sets are in fact maximum. Later Hartnell and Rall showed that if the Cartesian product $G \Box H$ is well-covered, then at least one of $G$ or $H$ is well-covered. In this paper, we consider the problem of classifying all well-covered Cartesian products. In particular, we show that if the Cartesian product of two nontrivial, connected graphs of girth at least $4$ is well-covered, then at least one of the graphs is $K_2$. Moreover, we show that $K_2 \Box K_2$ and $C_5 \Box K_2$ are the only well-cov…
Recently, we have introduced and modified two graph-decomposition theorems based on a new graph product, motivated by applications in the context of synchronising periodic real-time processes. This vertex-removing synchronised product (VRSP), is based on modifications of the well-known Cartesian product and is closely related to the synchronised product due to Wöhrle and Thomas. Here, we introduce a new graph-decomposition theorem based on the VRSP that provides a Cartesian decomposition of graphs.
In this ote, which has been absorbed by arXiv1702.01003, we combine a recent point-line incidence bound of Stevens and de Zeeuw with an older lemma of Bourgain, Katz and Tao to bound the number of collinear triples and quadruples in a Cartesian product in $\mathbb{F}_p^2$.
We determine the values of s and t for which there is a coloring of the edges of the complete bipartite graph K_{s,t} which admits only the identity automorphism. In particular this allows us to determine the distinguishing number of the Cartesian product of complete graphs.
An identifying code in a graph $G$ is a dominating set $C$ such that the closed neighborhood of each vertex in $G$ has a distinct intersection with $C$. In 2008, Gravier et al. determined the minimum cardinality of an identifying code of the Cartesian product of two cliques with the same size. In this note, we generalize this result to the Cartesian product of any two nontrivial cliques.
Betweenness centrality is a widely-used measure in the analysis of large complex networks. It measures the potential or power of a vertex to control the communication over the network under the assumption that information primarily flows over the shortest paths between them. In this paper we prove several results on betweenness centrality of Cartesian product of graphs.
An Italian dominating function on a (di)graph $G$ with vertex set $V(G)$ is a function $f: V(G) \to \{0, 1, 2\}$ such that every vertex $v \in V(G)$ such that $f(v) = 0$ has an (in)neighbour assigned 2 or two (in)neighbours assigned 1. We complete the investigation of the Italian domination numbers of Cartesian products of directed cycles.
We prove bounds on the number of incidences between a set of algebraic curves in $\mathbb{C}^2$ and a Cartesian product $A\times B$ with finite sets $A,B\subset \mathbb{C}$. Similar bounds are known under various conditions, but we show that the Cartesian product assumption leads to a simpler proof. This assumption holds in a number of interesting applications, and with our bound these applications can be extended from $\mathbb{R}$ to $\mathbb{C}$. The proof is a new application of the polynomial partitioning technique introduced by Guth and Katz.
In this paper we study the weak Roman domination number and the secure domination number of a graph. In particular, we obtain general bounds on these two parameters and, as a consequence of the study, we derive new inequalities of Nordhaus-Gaddum type involving secure domination and weak Roman domination. Furthermore, the particular case of Cartesian product graphs is considered.
A proper vertex coloring of a graph $G$ is called a star coloring if every two color classes induce a forest whose each component is a star, which means there is no bicolored $P_4$ in $G$. In this paper, we show that the Cartesian product of any two cycles, except $C_3 \square C_3$ and $C_3 \square C_5$, has a $5$-star coloring.
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 \cite{HaRi} conjectured that every simple connected graph, but $K_2$, is antimagic. In this article, we prove that a new class of Cartesian product graphs are antimagic. In addition, by combining this res…
The generalized $k$-connectivity $κ_{k}(G)$ of a graph $G$, which was introduced by Chartrand et al.(1984) is a generalization of the concept of vertex connectivity. Let $G$ and $H$ be nontrivial connected graphs. Recently, Li et al. gave a lower bound for the generalized $3$-connectivity of the Cartesian product graph $G \square H$ and proposed a conjecture for the case that $H$ is $3$-connected. In this paper, we give two different forms of lower bounds for the generalized $3$-connectivity of Cartesian product graphs. The first lower bound is stronger than theirs, and the second confirms the…
In this paper, we prove a problem proposed by Brešar: for any graphs $G$ and $H$, $Γ(G\square H)\geΓ(G)Γ(H)+ \min\{|V(G)|-Γ(G),|V(H)|-Γ(H)\}$, where $Γ(G)$ denotes the upper domination number of $G$.
we define the sunlet graph is the graph obtained by taking one copy of the cycle and joined every vertex of the cycle to an exactly one pendant vertex such that the degree of each vertex in the cycle is three. In this paper, we establish necessary and sufficient conditions for the existence of decomposition of the Cartesian product of complete graphs into sunlet graph of order eight.
We extend the definition of the Cartesian product to graphs with loops and show that the Sabidussi-Vizing unique factorization theorem for connected finite simple graphs still holds in this context for all connected finite graphs with at least one unlooped vertex. We also prove that this factorization can be computed in O(m) time, where m is the number of edges of the given graph.
Koh and Tay proved a fundamental classification of $G$ vertex-multiplications into three classes $\mathscr{C}_0, \mathscr{C}_1$ and $\mathscr{C}_2$. In this paper, we prove that vertex-multiplications of cartesian products of graphs $G\times H$ lie in $\mathscr{C}_0$ ($\mathscr{C}_0\cup \mathscr{C}_1$ resp.) if $G^{(2)}\in \mathscr{C}_0$ ($\mathscr{C}_1$ resp.), $d(G)\ge 2$ and $d(G\times H)\ge 4$. We also focus on cartesian products involving trees, paths and cycles and show that most of them lie in $\mathscr{C}_0$.
The exact crossing number is only known for a small number of families of graphs. Many of the families for which crossing numbers have been determined correspond to cartesian products of two graphs. Here, the cartesian product of the Sunlet graph, denoted $\mathcal{S}_n$, and the Star graph, denoted $K_{1,m}$, is considered for the first time. It is proved that the crossing number of $\mathcal{S}_n \Box K_{1,2}$ is $n$, and the crossing number of $\mathcal{S}_n \Box K_{1,3}$ is $3n$. An upper bound for the crossing number of $\mathcal{S}_n \Box K_{1,m}$ is also given.
There has been significant research dedicated towards computing the crossing numbers of families of graphs resulting from the Cartesian products of small graphs with arbitrarily large paths, cycles and stars. For graphs with four or fewer vertices, these have all been computed, but there are still various gaps for graphs with five or more vertices. We contribute to this field by determining the crossing numbers for fifteen such families.
The acyclic chromatic index, denoted by $a'(G)$, of a graph $G$ is the minimum number of colours used in any proper edge colouring of $G$ such that the union of any two colour classes does not contain a cycle, that is, forms a forest. We show that $a'(G\Box H)\le a'(G) + a'(H)$ for any two graphs $G$ and $H$ such that $max\{a'(G), a'(H)\} > 1$. Here, $G \Box H$ denotes the cartesian product of $G$ and $H$. This extends a recent result of [15] where tight and constructive bounds on $a'(G)$ were obtained for a class of grid-like graphs which can be expressed as the cartesian product of a number …
If each minimal dominating set in a graph is a minimum dominating set, then the graph is called well-dominated. Since the seminal paper on well-dominated graphs appeared in 1988, the structure of well-dominated graphs from several restricted classes have been studied. In this paper we give a complete characterization of nontrivial direct products that are well-dominated. We prove that if a strong product is well-dominated, then both of its factors are well-dominated. When one of the factors of a strong product is a complete graph, the other factor being well-dominated is also a sufficient cond…
A b-coloring of a graph $G$ is a coloring of its vertices such that every color class contains a vertex that has neighbors in all other classes. The b-chromatic number of $G$ is the largest integer $k$ such that $G$ has a b-coloring with $k$ colors. Javadi and Omoomi ("On b-coloring of cartesian product of graphs", Ars Combinatoria 107 (2012) 521-536) proved that the b-chromatic number of $K_n \times K_n$ (the Cartesian product of two complete graphs on $n$ vertices) is in the set $\{2n-3, 2n-2\}$ and conjectured that the exact value is $2n-3$ for all $n \ge 5$. We give counterexamples to this…
This paper is concerned with the linkedness of Cartesian products of complete graphs. A graph with at least $2k$ vertices is {\it $k$-linked} if, for every set of $2k$ distinct vertices organised in arbitrary $k$ pairs of vertices, there are $k$ vertex-disjoint paths joining the vertices in the pairs. We show that the Cartesian product $K^{d_{1}+1}\times K^{d_{2}+1}$ of complete graphs $K^{d_{1}+1}$ and $K^{d_{2}+1}$ is $\floor{(d_{1}+d_{2})/2}$-linked for $d_{1},d_{2}\ge 2$, and this is best possible. %A polytope is said to be {\it $k$-linked} if its graph is $k$-linked. This result is connec…
A star edge coloring of a graph $G$ is a proper edge coloring of $G$ such that every path and cycle of length four in $G$ uses at least three different colors. The star chromatic index of a graph $G$, is the smallest integer $k$ for which $G$ admits a star edge coloring with $k$ colors. In this paper, we first obtain some upper bounds for the star chromatic index of the Cartesian product of two graphs. We then determine the exact value of the star chromatic index of $2$-dimensional grids. We also obtain some upper bounds on the star chromatic index of the Cartesian product of a path with a cyc…
Let $p ,r $ and $n $ be positive integers. Then the O-Fibonacci $(p,r)$-cube $OΓ^{(p,r)}_{n}$ is the subgraph of $Q_{n}$ induced on the binary words in which there is at least $p-1$ zeros between any two $1$s and there is at most $r$ consecutive $10^{p-1}$. These cubes include a wide range of cubes as their special cases, such as hypercubes, Fibonacci cubes, and postal networks. In this note it is proved that $OΓ^{(p,r)}_{n}$ is a non-trivial Cartesian product if and only if $p=1$ and $r\geq n\geq2$.
Graph embedding is a powerful method in parallel computing that maps a guest network $G$ into a host network $H$. The performance of an embedding can be evaluated by certain parameters, such as the dilation, the edge congestion and the wirelength. In this manuscript, we obtain the wirelength (exact and minimum) of embedding complete multi-partite graphs into Cartesian product of paths and cycles, which include $n$-cube, $n$-dimensional mesh (grid), $n$-dimensional cylinder and $n$-dimensional torus, etc., as the subfamilies.
Let $γ(C_m\Box C_n)$ be the domination number of the Cartesian product of directed cycles $C_m$ and $C_n$ for $m,n\geq2$. Shaheen [] and Liu and al.[ ], [ ] determined the value of $γ(C_m\Box C_n)$ when $m \leq 6$ and when both $m$ and $n$ $\equiv 0$ $(mod\: 3)$. In this article we give, in general, the value of $γ(C_m\Box C_n)$ when $m\equiv 2$ $(mod\: 3)$ and improve the known lower bound for most of the remaining cases. We also disprove the conjectured formula for the case $m$ $\equiv 0$ $(mod\: 3)$ appearing in \cite{}
Ho proved in [A note on the total domination number, Util.Math. 77 (2008) 97--100] that the total domination number of the Cartesian product of any two graphs with no isolated vertices is at least one half of the product of their total domination numbers. We extend a result of Lu and Hou from [Total domination in the Cartesian product of a graph and $K_2$ or $C_n$, Util. Math. 83 (2010) 313--322] by characterizing the pairs of graphs $G$ and $H$ for which $γ_t(G\Box H)=\frac{1}{2}γ_t(G) γ_t(H)\,$, whenever $γ_t(H)=2$. In addition, we present an infinite family of graphs $G_n$ with $γ_t(G_n)=2n…
