Intersection Graph Research Articles

Identifying Secure Domination in the Cartesian and Lexicographic Products of Graphs

Let G be a connected simple graph. A subset S of V(G) is a dominating set of G if for every v∈V(G)\S, there exists x∈S such that xv∈E(G). An identifying code of a graph G is a dominating set C⊆V(G) such that for every v∈V(G),N_G [v]∩C is distinct. An identifying code of a graph G is an identifying secure dominating set if for each u∈V(G)\C, there exists v∈C such that uv∈E(G) and the set (C\{v})∪{u} is a dominating set of G. The minimum cardinality of an identifying secure dominating set of G, denoted by γ_s^ID, is called the identifying secure domination number of G. In this paper, the researchers initiate the study of the concept and give some important results. In particular, the researchers show some properties of the identifying secure dominating sets in the Cartesian product and lexicographic product of two connected graphs.

Recognizing geometric intersection graphs stabbed by a line

In this paper, we determine the computational complexity of recognizing two graph classes, groundedL-graphs and stabbable grid intersection graphs. An L-shape is made by joining the bottom end-point of a vertical (|) segment to the left end-point of a horizontal (−) segment. The top end-point of the vertical segment is known as the anchor of the L-shape. Grounded L-graphs are the intersection graphs of L-shapes such that all the L-shapes' anchors lie on the same horizontal line. We show that recognizing grounded L-graphs is NP-complete. This answers an open question asked by Jelínek & Töpfer (Electron. J. Comb., 2019).Grid intersection graphs are the intersection graphs of axis-parallel line segments in which two vertical (similarly, two horizontal) segments cannot intersect. We say that a (not necessarily axis-parallel) straight line ℓ stabs a segment s, if s intersects ℓ. A graph G is a stabbable grid intersection graph (▪) if there is a grid intersection representation of G in which the same line stabs all its segments. We show that recognizing ▪ graphs is NP-complete, even on a restricted class of graphs. This answers an open question asked by Chaplick et al. (Order, 2018).

On the total chromatic number of the direct product of cycles and complete graphs

A k-total coloring of a graph G is an assignment of k colors to the elements (vertices and edges) of G so that adjacent or incident elements have different colors. The total chromatic number is the smallest integer k for which G has a k-total coloring. The well known Total Coloring Conjecture states that the total chromatic number of a graph is either Δ(G) + 1 (called Type 1) or Δ(G) + 2 (called Type 2), where Δ(G) is the maximum degree of G. We consider the direct product of complete graphs Km × Kn. It is known that if at least one of the numbers m or n is even, then Km × Kn is Type 1, except for K2 × K2. We prove that the graph Km × Kn is Type 1 when both m and n are odd numbers, by using that the conformable condition is sufficient for the graph Km × Kn to be Type 1 when both m and n are large enough, and by constructing the target total colorings by using Hamiltonian decompositions and a specific color class, called guiding color. We additionally apply our technique to the direct product Cm × Kn of a cycle with a complete graph. Interestingly, we are able to find a Type 2 infinite family Cm × Kn, when m is not a multiple of 3 and n = 2. We provide evidence to conjecture that all other Cm × Kn are Type 1.

Brouwer's Conjecture for the cartesian product of graphs

Let G be a simple, undirected graph with n vertices and m edges and Sk(G) the sum of its k largest Laplacian eigenvalues. Brouwer conjectured that Sk(G)≤m+(k+12) for any 1≤k≤n. We show that if G and H are simple undirected graphs satisfying Brouwer's Conjecture, then the cartesian product G□H also satisfies Brouwer's Conjecture.

Crossing Numbers of the Cartesian Product of the Double Triangular Snake Graphs With Path Pm.

The crossing number Cr(G) of a graph G is the least number of edge crossings in all possible good drawings of G in the plane. Join and Cartesian products of graphs have many interesting graph-theoretical properties. In this paper, we evaluate the crossing number of the Cartesian product of double triangular snake graph DT2 with the path Pm. In this paper, we proved Cr(DT2 × Pm) = 6(m − 2), form ≥ 2 where Cr denotes the crossing number

Closed formulas for 2-domination and 2-outer-independent domination numbers of rooted product graphs

Closed formulas for 2-domination and 2-outer-independent domination numbers of rooted product graphs

Homology Growth, Hyperbolization, and Fibering

We introduce a hyperbolic reflection group trick which builds closed aspherical manifolds out of compact ones and preserves hyperbolicity, residual finiteness, and—for almost all primes p—\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{F} _{p}$\\end{document}-homology growth above the middle dimension. We use this trick, embedding theory and manifold topology to construct Gromov hyperbolic 7-manifolds that do not virtually fiber over a circle out of graph products of large finite groups.

A generalization of the Davis–Moussong complex for Dyer groups

A common feature of Coxeter groups and right-angled Artin groups is their solution to the word problem. Matthew Dyer introduced a class of groups, which we call Dyer groups, sharing this feature. This class includes, but is not limited to, Coxeter groups, right-angled Artin groups, and graph products of cyclic groups. We introduce Dyer groups by giving their standard presentation and show that they are finite-index subgroups of Coxeter groups. We then introduce a piecewise Euclidean cell complex \Sigma which generalizes the Davis–Moussong complex and the Salvetti complex. The construction of \Sigma uses simple categories without loops and complexes of groups. We conclude by proving that the cell complex \Sigma is \mathrm{CAT}(0) .

Aut-Invariant Quasimorphisms on Graph Products of Abelian Groups

Aut-Invariant Quasimorphisms on Graph Products of Abelian Groups

Integer flows on triangularly connected signed graphs

AbstractA triangle‐path in a graph is a sequence of distinct triangles in such that for any with , and if . A connected graph is triangularly connected if for any two nonparallel edges and there is a triangle‐path such that and . For ordinary graphs, Fan et al. characterize all triangularly connected graphs that admit nowhere‐zero 3‐flows or 4‐flows. Corollaries of this result include the integer flow of some families of ordinary graphs, such as locally connected graphs due to Lai and some types of products of graphs due to Imrich et al. In this paper, Fan's result for triangularly connected graphs is further extended to signed graphs. We proved that a flow‐admissible triangularly connected signed graph admits a nowhere‐zero 4‐flow if and only if it is not the wheel associated with a specific signature. Moreover, this result is sharp since there are infinitely many unbalanced triangularly connected signed graphs admitting a nowhere‐zero 4‐flow but no 3‐flow.

The CCAP for graph products of operator algebras

For a simple graph Γ and for unital C*-algebras with GNS-faithful states (Av,φv) for v∈VΓ, we consider the reduced graph product (A,φ)=⁎v,Γ(Av,φv), and show that if every C*-algebra Av has the completely contractive approximation property (CCAP) and satisfies some additional condition, then the graph product has the CCAP as well. The additional condition imposed is satisfied in natural cases, for example for the reduced group C*-algebra of a discrete group G that possesses the CCAP.Our result is an extension of the result of Ricard and Xu in [28, Proposition 4.11] where they prove this result under the same conditions for free products. Moreover, our result also extends the result of Reckwerdt in [27, Theorem 5.5], where he proved for groups that weak amenability with Cowling-Haagerup constant 1 is preserved under graph products. Our result further covers many new cases coming from Hecke-algebras and discrete quantum groups.

G-Small Intersection Graph of a Module

Let be a commutative ring with identity, and be a left -module. The g-small intersection graph of non-trivial submodules of , indicated by , is a simple undirected graph whose vertices are in one-to-one correspondence with all non-trivial submodules of and two distinct vertices are adjacent if and only if the intersection of the corresponding submodules is a g-small submodule of . In this article, the interplay among the algebraic properties of , and the graph properties of are studied. Properties of such as connectedness, and completeness are considered. Besides, the girth and the diameter of are determined, as well as presenting a formula to compute the clique and domination numbers of . The graph is complete if, is a generalized hollow module or is a direct sum of two simple modules, is proved.

A Survey on Variant Domination Problems in Geometric Intersection Graphs

Given a graph [Formula: see text] with vertex set [Formula: see text], a subset [Formula: see text] of [Formula: see text] is called a dominating set of [Formula: see text] if every vertex is either in [Formula: see text] or is adjacent to a vertex in [Formula: see text]. There are a lot of variants for dominating sets, such as connected dominating sets, total dominating sets, total restricted dominating sets, secure (connected, total) dominating sets, etc. Geometric intersection graphs are graphs for which there is a bijection [Formula: see text] between the vertices and a set [Formula: see text] of geometric objects (for example, disks, rectangles, etc.) such that there is an edge between two vertices [Formula: see text] and [Formula: see text] if and only if the objects [Formula: see text] and [Formula: see text] intersect. In this paper, we offer a survey about complexity and algorithmic results on variant domination problems in geometric intersection graphs.

Online coloring of short intervals

We study the online graph coloring problem restricted to the intersection graphs of intervals with lengths in [1,σ]. For σ=1 it is the class of unit interval graphs and for σ=∞ the class of all interval graphs. Our focus is on intermediary classes. We present a (1+σ)-competitive algorithm, which beats the state of the art for 1<σ<2, and proves that the problem we study can be strictly easier than online coloring of general interval graphs. On the lower bound side, we prove that no algorithm is better than 5/3-competitive for any σ>1, nor better than 7/4-competitive for any σ>2, and that no algorithm beats the 5/2 asymptotic competitive ratio for all, arbitrarily large, values of σ. That last result shows that the problem we study can be strictly harder than unit interval coloring. Our main technical contribution is a recursive composition of strategies, which seems essential to prove any lower bound higher than 2. Moreover, we prove that the natural algorithm FirstFit is no better than 5/2 for any σ>3, no better than 11/4 for any σ>12, and finally for every ɛ>0 FirstFit is no better than (5−ɛ)-competitive for any σ>σɛ (where σɛ is some finite length depending on ɛ). We also give some upper bounds for FirstFit for σ∈[5,13].

Valued-inverse dombi fuzzy graphs

Fuzzy graphs as labeled graphs (fuzzy vertex labeling and fuzzy edge labeling) have many applications in real life such as complex networks, coding theory, medical sciences, communication networks, and management sciences. Also, triangular norms as a special class of functions, have many applications in fuzzy set theory, probability and statistics, and other areas. This paper considers the notations of an inverse fuzzy graph and triangular norms to introduce the new type of graphs as valued-inverse Dombi fuzzy graphs. The valued-inverse Dombi fuzzy graphs are a generalization of inverse fuzzy graphs and are dual to Dombi fuzzy graphs. For any given greater than or equal to one real number, we construct a type of Dombi inverse fuzzy graph and investigate some conditions such that the product and union of Dombi inverse fuzzy graphs be a Dombi inverse fuzzy graph.

The Subset-Strong Product of Graphs

Abstract In this paper, we introduce the subset-strong product of graphs and give a method for calculating the adjacency spectrum of this product. In addition, exact expressions for the first and second Zagreb indices of the subset-strong products of two graphs are reported. Examples are provided to illustrate the applications of this product in some growing graphs and complex networks.

On Sombor indices of generalized tensor product of graph families

Graph invariants are extensively being used in different fields for analysing the structural properties of different compounds. Several topological indices, their properties and relations between different indices have been explored. Ivan Gutman recently introduced a new topological index, based on the vertex degree, termed as Sombor indices in the field of Chemical Graph theory. It is based on the Eulerian (distance) metric. Sombor indices and its properties have been explored on a wide range of graphs. However, the existing literature of graph theory bears fewer exploration in the analysis of topological indices of product graphs, a significant operation on graphs. Studying the topological features of molecular paths and cycles is vital for understanding chemical compounds, offering key insights into their structures and behaviours, benefiting diverse sectors such as drug discovery, material science, and chemical engineering. With this motivation, the Sombor indices of tensor product and 2-tensor product of certain families of graphs are established in this paper. Some of these product graphs are analysed by means of SO based graphical entropy.

Monitoring-edge-geodetic sets in product networks

Let G be a graph with vertex set V ( G ) and edge set E ( G ) . For any e ∈ E ( G ) and u , v ∈ V ( G ) , the edge e is monitored by two vertices u and v in graph G if d G ( u , v ) ≠ d G − e ( u , v ) . A set M of vertices of G is a monitoring-edge-geodetic set of G if for any edge e ∈ E ( G ) there exists a pair u , v ∈ M such that e is monitored by u, v. The monitoring-edge-geodetic number meg ⁡ ( G ) is the cardinality of the minimum MEG-set in G. In this paper, we obtain the exact values or bounds for the MEG numbers of graph products, including join, corona, cluster, lexicographic products and direct products.

Functionality of Box Intersection Graphs

Functionality is a graph complexity measure that extends a variety of parameters, such as vertex degree, degeneracy, clique-width, or twin-width. In the present paper, we show that functionality is bounded for box intersection graphs in R1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^1$$\\end{document}, i.e. for interval graphs, and unbounded for box intersection graphs in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document}. We also study a parameter known as symmetric difference, which is intermediate between twin-width and functionality, and show that this parameter is unbounded both for interval graphs and for unit box intersection graphs in R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^2$$\\end{document}.

Pangenome comparison via ED strings.

An elastic-degenerate (ED) string is a sequence of sets of strings. It can also be seen as a directed acyclic graph whose edges are labeled by strings. The notion of ED strings was introduced as a simple alternative to variation and sequence graphs for representing a pangenome, that is, a collection of genomic sequences to be analyzed jointly or to be used as a reference. In this study, we define notions of matching statistics of two ED strings as similarity measures between pangenomes and, consequently infer a corresponding distance measure. We then show that both measures can be computed efficiently, in both theory and practice, by employing the intersection graph of two ED strings. We also implemented our methods as a software tool for pangenome comparison and evaluated their efficiency and effectiveness using both synthetic and real datasets. As for efficiency, we compare the runtime of the intersection graph method against the classic product automaton construction showing that the intersection graph is faster by up to one order of magnitude. For showing effectiveness, we used real SARS-CoV-2 datasets and our matching statistics similarity measure to reproduce a well-established clade classification of SARS-CoV-2, thus demonstrating that the classification obtained by our method is in accordance with the existing one.