Intersection Graph Research Articles

Percolation on High‐Dimensional Product Graphs

ABSTRACTWe consider percolation on high‐dimensional product graphs, where the base graphs are regular and of bounded order. In the subcritical regime, we show that typically the largest component is of order logarithmic in the number of vertices. In the supercritical regime, our main result recovers the sharp asymptotic of the order of the largest component and shows that all the other components are typically of order logarithmic in the number of vertices. In particular, we show that this phase transition is quantitatively similar to the one of the binomial random graph. This generalizes the results of Ajtai, Komlós, and Szemerédi [1] and of Bollobás, Kohayakawa, and Łuczak [5] who showed that the ‐dimensional hypercube, which is the ‐fold Cartesian product of an edge, undergoes a phase transition quantitatively similar to the one of the binomial random graph.

Bootstrap Percolation in Strong Products of Graphs

Given a graph $G$ and assuming that some vertices of $G$ are infected, the $r$-neighbor bootstrap percolation rule makes an uninfected vertex $v$ infected if $v$ has at least $r$ infected neighbors. The $r$-percolation number, $m(G,r)$, of $G$ is the minimum cardinality of a set of initially infected vertices in $G$ such that after continuously performing the $r$-neighbor bootstrap percolation rule each vertex of $G$ eventually becomes infected. In this paper, we consider percolation numbers of strong products of graphs. If $G$ is the strong product $G_1\boxtimes \cdots \boxtimes G_k$ of $k$ connected graphs, we prove that $m(G,r)=r$ as soon as $r\le 2^{k-1}$ and $|V(G)|\ge r$. As a dichotomy, we present a family of strong products of $k$ connected graphs with the $(2^{k-1}+1)$-percolation number arbitrarily large. We refine these results for strong products of graphs in which at least two factors have at least three vertices. In addition, when all factors $G_i$ have at least three vertices we prove that $m(G_1 \boxtimes \dots \boxtimes G_k,r)\leq 3^{k-1} -k$ for all $r\leq 2^k-1$, and we again get a dichotomy, since there exist families of strong products of $k$ graphs such that their $2^{k}$-percolation numbers are arbitrarily large. While $m(G\boxtimes H,3)=3$ if both $G$ and $H$ have at least three vertices, we also characterize the strong prisms $G\boxtimes K_2$ for which this equality holds. Some of the results naturally extend to infinite graphs, and we briefly consider percolation numbers of strong products of two-way infinite paths.

Geodetic numbers of tensor product and lexicographic product of graphs

A shortest u - v path between two vertices u and v of a graph G is a u - v geodesic of G. Let I[u, v] denote the set of all internal vertices lying on some u - v geodesic of G. For a nonempty subset S of V ( G ) , let I ( S ) = ∪ u , v ∈ S I [ u , v ] . If I ( S ) = V ( G ) , then S is a geodetic set of G. The cardinality of a minimum geodetic set of G is the geodetic number of G and it is denoted by g ( G ) . In this paper, the exact geodetic numbers of the product graphs T × K m and T ° K ¯ m are obtained, where T is a tree, K ¯ m denotes the complement of the complete graph K m and, × and ° denote the tensor product and lexicographic product $($also called the wreath product$)$ of graphs, respectively.

The algorithm and complexity of co-secure domination in geometric intersection graphs

The algorithm and complexity of co-secure domination in geometric intersection graphs

Analyzing Single-Valued Neutrosophic Fuzzy Graphs Through Matroid Perspectives

We hope to present this paper on the emergence of a novel category of matroids derived from single-valued neutrosophic (SVN) fuzzy-graphs. The findings of this study make a substantial contribution to both matroid theory and the field of neutrosophic systems. In this study, we aim to introduce the concept of SVN-matroids and conduct a thorough investigation into their fundamental properties. Additionally, we explore the essential requirements for establishing duality within the realm of matroids formed from neutrosophic fuzzy-networks. This investigation encompasses co-weak isomorphism, weak isomorphism, and isomorphism between two matroids. Furthermore, our research proposes the expansion of other graph products and the creation of a SVN-graph to utilize the Greedy and Kruskal algorithms for determining optimal productivity in an information flow of a single-source network within a SVN-network. The application of neutronosophic graph theory offers a paradigm for addressing optimization problems in chemical graph theory, such as identifying the maximal independent set and a minimum spanning tree. When faced with uncertainty, neutrosophic optimization provides more reliable solutions by considering the uncertainties associated with the objective function or constraints.

Algorithms and Turing Kernels for Detecting and Counting Small Patterns in Unit Disk Graphs

In this paper we investigate the parameterized complexity of counting and detecting small patterns in unit disk graphs: Given an n-vertex unit disk graph G with an embedding of ply p (i.e. G is an intersection graph of closed unit disks, and each point is contained in at most p disks) and a k-vertex unit disk graph P, count the number of (induced) copies of P in G. For general patterns P, we give an 2O(pk/log⁡k)nO(1) time algorithm for counting pattern occurrences. We show this is tight, even for ply p=2: any 2o(n/log⁡n)nO(1) time algorithm violates the Exponential Time Hypothesis (ETH). Our approach combines tools developed for planar subgraph isomorphism such as ‘efficient inclusion-exclusion’ from [Nederlof STOC'20], and ‘isomorphisms checks’ from [Bodlaender et al. ICALP'16] with a different separator hierarchy and a new bound on the number of non-isomorphic separations tailored for unit disk graphs.

Lost in the Shuffle: Testing Power in the Presence of Errorful Network Vertex Labels

Two-sample network hypothesis testing is an important inference task with applications across diverse fields such as medicine, neuroscience, and sociology. Many of these testing methodologies operate under the implicit assumption that the vertex correspondence across networks is a priori known. This assumption is often untrue, and the power of the subsequent test can degrade when there are misaligned/label-shuffled vertices across networks. This power loss due to shuffling is theoretically explored in the context of random dot product and stochastic block model networks for a pair of hypothesis tests based on Frobenius norm differences between estimated edge probability matrices or between adjacency matrices. The loss in testing power is further reinforced by numerous simulations and experiments, both in the stochastic block model and in the random dot product graph model, where the power loss across multiple recently proposed tests in the literature is considered. Lastly, the impact that shuffling can have in real-data testing is demonstrated in a pair of examples from neuroscience and from social network analysis.

A framework for generalizing toric inequalities for holographic entanglement entropy

We conjecture a multi-parameter generalization of the toric inequalities of [1]. We then extend their proof methods for the generalized toric inequalities in two ways. The first extension constructs the graph corresponding to the toric inequalities and the generalized toric conjectures by tiling the Euclidean space. An entanglement wedge nesting relation then determines the geometric structure of the tiles. In the second extension, we exploit the cyclic nature of the inequalities and conjectures to construct cycle graphs. Then, the graph can be obtained using graph Cartesian products of cycle graphs. In addition, we define a set of knots on the graph by following [1]. These graphs with knots then imply the validity of their associated inequality. We study the case where the graph can be decomposed into disjoint unions of torii. Under the specific case, we explore and prove the conjectures for some ranges of parameters. We also discuss ways to explore the conjectured inequalities whose corresponding geometries are d-dimensional torii (d > 2).

ZONAL LABELING OF EDGE COMB PRODUCT OF GRAPHS

Given a plane graph $G=(V,E)$. A zonal labeling of graph $G$ is defined as an assignment of the two nonzero elements of the ring $\mathbb{Z}_3$, which are $1$ and $2$, to the vertices of $G$ such that the sum of the labels of the vertices on the border of each region of the graph is $0\in\mathbb{Z}_3$. A graph $G$ that possess such a labeling is termed as zonal graph. This paper will characterize edge comb product graphs that are zonal. The results show that $P_m\trianglerighteq_eC_n$, $C_n\trianglerighteq_e C_r$, $S_p\trianglerighteq_e C_n$, and $S_p\trianglerighteq_e F_t$ are zonal in some cases, but not in others.

Intersection graph and writhe polynomial

We prove that two virtual knots have equivalent intersection graphs if and only if they have the same writhe polynomial.

Graph and wreath products of cellular automata

Graph and wreath products of cellular automata

Fault-tolerant metric dimension of the intersection graph of gamma sets in the zero-divisor graph

Fault-tolerant metric dimension of the intersection graph of gamma sets in the zero-divisor graph

Connected power domination number of product graphs

Connected power domination number of product graphs

Dynamic Random Intersection Graph: Dynamic Local Convergence and Giant Structure

ABSTRACTRandom intersection graphs containing an underlying community structure are a popular choice for modeling real‐world networks. Given the group memberships, the classical random intersection graph is obtained by connecting individuals when they share at least one group. We extend this approach and make the communities dynamic by letting them alternate between an active and inactive phase. We analyse the new model, delivering results on degree distribution, local convergence, largest connected component, and maximum group size, paying particular attention to the dynamic description of these properties. We also describe the connection between our model and the bipartite configuration model, which is of independent interest.

Online coloring of disk graphs

In this paper, we give a family of online algorithms for the classical coloring problem of intersection graphs of disks with bounded diameter. Our algorithms make use of a geometric representation of such graphs and are inspired by an algorithm of Fiala et al., but have better competitive ratios. The improvement comes from using two techniques of partitioning the set of vertices before coloring them. One of them is an application of a b-fold coloring of the plane. The method is more general and we show how it can be applied to coloring other shapes on the plane as well as adjust it for online L(2,1)-labeling.

Corona Products and Rooted Products of Soft Graphs

Corona Products and Rooted Products of Soft Graphs

Maker-Breaker domination game played on corona products of graphs

In the Maker-Breaker domination game, Dominator and Staller play on a graph G by taking turns in which each player selects a not yet played vertex of G. Dominator’s goal is to select all the vertices in a dominating set, while Staller aims to prevent this from happening. In this paper, the game is investigated on corona products of graphs. Its outcome is determined as a function of the outcome of the game on the second factor. Staller-Maker-Breaker domination numbers are determined for arbitrary corona products, while Maker-Breaker domination numbers of corona products are bounded from both sides. All the bounds presented are demonstrated to be sharp. Corona products as well as general graphs with small (Staller-)Maker-Breaker domination numbers are described.

On Study of Some Bounds for Fault-Tolerant Metric Dimension and Adjacency Fault-Tolerant Resolving Set of Corona Product Graphs

On Study of Some Bounds for Fault-Tolerant Metric Dimension and Adjacency Fault-Tolerant Resolving Set of Corona Product Graphs

Characterizations and Clique Coloring of Edge Intersection Graphs on a Triangular Grid

Characterizations and Clique Coloring of Edge Intersection Graphs on a Triangular Grid

Transitive path decompositions of Cartesian products of complete graphs

Transitive path decompositions of Cartesian products of complete graphs