Vertex Set Research Articles

On Sharp Bounds of Local Fractional Metric Dimension for Certain Symmetrical Algebraic Structure Graphs

The smallest set of vertices needed to differentiate or categorize every other vertex in a graph is referred to as the graph’s metric dimension. Finding the class of graphs for a particular given metric dimension is an NP-hard problem. This concept has applications in many different domains, including graph theory, network architecture, and facility location problems. A graph G with order n is known as a Toeplitz graph over the subset S of consecutive collections of integers from one to n, and two vertices will be adjacent to each other if their absolute difference is a member of S. A graph G(Zn) is called a zero-divisor graph over the zero divisors of a commutative ring Zn, in which two vertices will be adjacent to each other if their product will leave the remainder zero under modulo n. Since the local fractional metric dimension problem is NP-hard, it is computationally difficult to identify an optimal solution or to precisely determine the minimal size of a local resolving set; in the worst case, the process takes exponential time. Different upper bound sequences of local fractional metric dimension are suggested in this article, along with a comparison analysis for certain families of Toeplitz and zero-divisor graphs. Furthermore, we note that the analyzed local fractional metric dimension upper bounds fall into three metric families: constant, limited, and unbounded.

On the super graphs and reduced super graphs of some finite groups

For a finite group G, let B be an equivalence (equality, conjugacy or order) relation on G and let A be a simple (power, enhanced power or commuting) graph with vertex set G. The B super A graph is a simple graph with vertex set G and two distinct vertices are adjacent if either they are in the same B-equivalence class or there are elements in their B-equivalence classes that are adjacent in the original A graph. The graph obtained by deleting the dominant vertices (adjacent to all other vertices) from a B super A graph is called the reduced B super A graph. In this article, for some pairs of B super A graphs, we characterize the finite groups for which a pair of graphs are equal. We also characterize the dominant vertices for the order super commuting graph Δo(G) of G and prove that for n≥4 the identity element is the only dominant vertex of Δo(Sn) and Δo(An). We characterize the values of n for which the reduced order super commuting graph Δo(Sn)⁎ of Sn and the reduced order super commuting graph Δo(An)⁎ of An are connected. We also prove that if Δo(Sn)⁎ (or Δo(An)⁎) is connected then the diameter is at most 3 and show that the diameter is 3 for many values of n.

Brooks' theorem with forbidden colors

AbstractWe consider extensions of Brooks' classic theorem on vertex coloring where some colors cannot be used on certain vertices. In particular we prove that if is a connected graph with maximum degree that is not a complete graph and is a set of vertices where either at most colors are forbidden for every vertex in , and any two vertices of are at distance at least 4, or at most colors are forbidden for every vertex in , and any two vertices of are at distance at least 3,then there is a proper ‐coloring of respecting these constraints. In fact, we shall prove that these results hold in the more general setting of list colorings. These results are sharp.

Spectrum of the Cozero-Divisor Graph Associated to Ring Zn

Let R be a commutative ring with identity 1≠0 and let Z(R)′ be the set of all non-unit and non-zero elements of ring R. Γ′(R) denotes the cozero-divisor graph of R and is an undirected graph with vertex set Z(R)′, w∉zR, and z∉wR if and only if two distinct vertices w and z are adjacent, where qR is the ideal generated by the element q in R. In this article, we investigate the signless Laplacian eigenvalues of the graphs Γ′(Zn). We also show that the cozero-divisor graph Γ′(Zp1p2) is a signless Laplacian integral.

Ad-Hoc Lanzhou Index

This paper initiates the study of the mathematical aspects of the ad-hoc Lanzhou index. If G is a graph with the vertex set {x1,…,xn}, then the ad-hoc Lanzhou index of G is defined by Lz˜(G)=∑i=1ndi(n−1−di)2, where di represents the degree of the vertex xi. Several identities for the ad-hoc Lanzhou index, involving some existing topological indices, are established. The problems of finding graphs with the extremum values of the ad-hoc Lanzhou index from the following sets of graphs are also attacked: (i) the set of all connected ξ-cyclic graphs of a fixed order, (ii) the set of all connected molecular ξ-cyclic graphs of a fixed order, (iii) the set of all graphs of a fixed order, and (iv) the set of all connected molecular graphs of a fixed order.

Metric Dimension Parameterized by Feedback Vertex Set and Other Structural Parameters

Metric Dimension Parameterized by Feedback Vertex Set and Other Structural Parameters

On the price of independence for vertex cover, feedback vertex set and odd cycle transversal

Let vc(G), fvs(G) and oct(G), respectively, denote the size of a minimum vertex cover, minimum feedback vertex set and minimum odd cycle transversal in a graph G. One can ask, when looking for these sets in a graph, how much bigger might they be if we require that they are independent; that is, what is the price of independence? If G has a vertex cover, feedback vertex set or odd cycle transversal that is an independent set, then we let ivc(G), ifvs(G) or ioct(G), respectively, denote the minimum size of such a set. Similar to a recent study on the price of connectivity (Hartinger et al. EuJC 2016), we investigate for which graphs H the values of ivc(G), ifvs(G) and ioct(G) are bounded in terms of vc(G), fvs(G) and oct(G), respectively, when the graph G belongs to the class of H-free graphs. We find complete classifications for vertex cover and feedback vertex set and an almost complete classification for odd cycle transversal (subject to three non-equivalent open cases). We also investigate for which graphs H the values of ivc(G), ifvs(G) and ioct(G) are equal to vc(G), fvs(G) and oct(G), respectively, when the graph G belongs to the class of H-free graphs. We find a complete classification for vertex cover and almost complete classifications for feedback vertex set (subject to one open case) and odd cycle transversal (subject to three open cases).

Cohen-Macaulay binomial edge ideals of small graphs

A combinatorial property that characterizes Cohen-Macaulay binomial edge ideals has long been elusive. A recent conjecture ties the Cohen-Macaulayness of a binomial edge ideal JG to special disconnecting sets of vertices of its underlying graph G, called cut sets. More precisely, the conjecture states that JG is Cohen-Macaulay if and only if JG is unmixed and the collection of the cut sets of G is an accessible set system. In this paper we prove the conjecture theoretically for all graphs with up to 12 vertices and develop an algorithm that allows to computationally check the conjecture for all graphs with up to 15 vertices and all blocks with whiskers where the block has at most 11 vertices. This significantly extends previous computational results.

Color-avoiding percolation of random graphs: Between the subcritical and the intermediate regime

Fix a graph G in which every edge is colored in some of k≥2 colors. Two vertices u and v are CA-connected if u and v may be connected using any subset of k−1 colors. CA-connectivity is an equivalence relation dividing the vertex set into classes called CA-components.In two recent papers, Ráth, Varga, Fekete, and Molontay, and Lichev and Schapira studied the size of the largest CA-component in a randomly colored random graph. The second of these works distinguished and studied three regimes (supercritical, intermediate, and subcritical) in which the largest CA-component has respectively linear, logarithmic, and bounded size. In this short note, we describe the phase transition between the intermediate and the subcritical regime.

On the order of antipodal covers

AbstractA noncomplete graph of diameter is called an antipodal ‐cover if its vertex set can be partitioned into the subsets (also called fibres) of vertices each, in such a way that two vertices of are at distance if and only if they belong to the same fibre. We say that is symmetric if for every , there exist such that , where . In this paper, we prove that, for , an antipodal ‐cover which is not a cycle, has at least vertices provided , and at least vertices provided it is symmetric. Our results extend those of Göbel and Veldman.

Many Hamiltonian subsets in large graphs with given density

AbstractA set of vertices in a graph is a Hamiltonian subset if it induces a subgraph containing a Hamiltonian cycle. Kim, Liu, Sharifzadeh, and Staden proved that for large $d$ , among all graphs with minimum degree $d$ , $K_{d+1}$ minimises the number of Hamiltonian subsets. We prove a near optimal lower bound that takes also the order and the structure of a graph into account. For many natural graph classes, it provides a much better bound than the extremal one ( $\approx 2^{d+1}$ ). Among others, our bound implies that an $n$ -vertex $C_4$ -free graph with minimum degree $d$ contains at least $n2^{d^{2-o(1)}}$ Hamiltonian subsets.

METHODOLOGY FOR PLANNING THE LOCATION OF SWITCHING NODES OF IP-TELEPHONY NETWORKS

The relevance of the topic is due to the high pace of development and implementation of new systems and processes in information structures and the lack of a methodology for determining the effectiveness of using new infocommunication technologies in them. Currently, there are no unified scientifically based methods for assessing the feasibility and effectiveness of using IP telephony technology. This is due to insufficient experience in the operation of telecommunication systems based on IP telephony technology and their significant difference from traditional telephone systems. It is indicated that for optimal planning of the location of switching nodes of IP-telephony networks, a methodology has been developed, consisting of the development of a mathematical basis for constructing the projected IP-telephony network, algorithmic implementation and estimation of the complexity of the implementation of this algorithm, which can be used in the design of real IP-telephony networks. The algorithmic implementation of the developed technique uses three arrays from among the nodes of the network, i.e. the first array contains labels with two values, the second array contains the distances - the current shortest distances from the specified vertex to the corresponding vertex, and the third array contains the vertex numbers. In turn, the algorithmic implementation depends on how the source vertex is found, how the set of unmarked vertices is stored, and how labels are updated. Keywords: Telecommunications network, IP-telephony technology, packet switching of flows, planning of the location of switching nodes, algorithmic implementation of the planning methodology, subscriber network, parameters of information systems, network parameters.

Partitioning P4-tidy graphs into a stable set and a forest

Given a graph G, a near-bipartition of G is a partition of V(G) into S and F, where S is a stable set, and F induces a forest. Given a graph G and a vertex set P inducing a P4 in G, a vertex v is said partner of P if v∈V(G)∖P and G[P∪{v}] has at least two P4’s. A graph G is P4-tidy if any P4 in G has at most one partner. In this paper, we study the property of having a near-bipartition on P4-tidy graphs. We also consider a variant of that property that requires F being a tree. While the problem of determining whether an input graph has a near-bipartition is NP-complete even for graphs with maximum degree four, perfect graphs, graphs with diameter three, line graphs, and planar graphs, we show that the class of near-bipartite P4-tidy graphs admit finite forbidden induced subgraph characterization. In addition, we also present a characterization for P4-tidy graphs admitting a partition of its vertex set into a stable set and a tree.

Wiener index and vulnerability parameters of graphs

Let G be a finite simple graph with vertex set V(G). We define the Wiener index of G as W(G)=∑{u,v}∈V(G)d(u,v), where d(u,v) is the distance between u and v. This topological index was defined by Wiener in 1947 for studying the structural graphs of molecules (Wiener, 1947 [17]) (although, Wiener’s definition included a 1/2 multiple in front of the sum). Since that time, many theoretical results have been derived. The results we present concern the graph parameters integrity, toughness, tenacity and binding number. Lower bounds on these parameters have been used extensively to determine properties of graphs. For each of the four parameters, we provide a best possible upper bound on the Wiener index of G that guarantees a specified lower bound on that parameter. This provides a continuation of results provided in Feng et al. (2017).

Solution to a conjecture on resistance diameter of lexicographic product of paths

The resistance distance R[u,v] between two vertices u and v of a graph G is defined as the net effective resistance between them in the electric network constructed from G by replacing each edge with a unit resistor. The resistance diameter Dr(G) of G is the maximum resistance distance among all pairs of vertices of G. Given two path graphs Pn=a1a2…an and Pm=b1b2…bm, let Pn[Pm] be the lexicographic product of Pn and Pm with vertex set {(ai,bj)|i=1,…,n;j=1,…,m}. In [J. Appl. Math. Comput. 68 (2022) 1743–1755], Li et al. proved that for n>10, Dr(Pn[Pm])=R[(a1,b1),(an,bm)]=R[(a1,b1),(an,b1)]=R[(a1,bm),(an,b1)]=R[(a1,bm),(an,bm)]. In addition, they found that the result is not true for n=2. For 3≤n≤10 and enough small m, they checked by computer that the result is still true. Based on their observation, they conjectured that the result is true for 3≤n≤10. In this paper, by combinatorial and electrical network approaches, we confirm the conjecture.

Fractional matching preclusion numbers of Cartesian product graphs

The Cartesian product of two simple graphs G and H is the graph G□H whose vertex set is V(G)×V(H) and whose edge set is the set of all pairs (u1,v1)(u2,v2) such that either u1u2∈E(G) and v1=v2, or v1v2∈E(H) and u1=u2. The fractional matching preclusion number of a graph G, denoted by fmp(G), is the minimum number of edges whose deletion results in a graph with no fractional perfect matching. In this paper, we determine fmp(G□H) when H is a cycle or a path of even order; Moreover, given any integers a,b with a≥1 and 0≤b≤a+1, we construct a graph G such that δ(G)=a and fmp(G□H)=b when H is a path of odd order.

[1,2]-dimension of graphs

Let G be a connected graph with vertex set V, where the distance between two vertices is the length of a shortest path between them. A set S⊆V is [1,2]-resolving if each vertex of G is at most distance-two away from a vertex in S and, given a pair of distinct vertices not in S, either there is a vertex in S adjacent to exactly one member of the given pair, or there are two vertices in S each of which is distance-two from exactly one member of the given pair. The [1,2]-dimension of G is the minimum cardinality of a [1,2]-resolving set of G. In this paper, we study the [1,2]-dimension of graphs by proving that the [1,2]-dimension problem is an NP-complete problem, and determine the [1,2]-dimension of some classes of graphs, such as paths, cycles, and full k-ary trees. We also introduce a generalization of metric dimension of which the (original) metric dimension and the [1,2]-dimension, as well as other metric dimension variants in the literature, are special instances.

A simple [formula omitted]-approximation algorithm for Split Vertex Deletion

A split graph is a graph whose vertex set can be partitioned into a clique and a stable set. Given a graph G and weight function w:V(G)→Q≥0, the Split Vertex Deletion (SVD) problem asks to find a minimum weight set of vertices X such that G−X is a split graph. It is easy to show that a graph is a split graph if and only if it does not contain a 4-cycle, 5-cycle, or a two edge matching as an induced subgraph. Therefore, SVD admits an easy 5-approximation algorithm. On the other hand, for every δ>0, SVD does not admit a (2−δ)-approximation algorithm, unless P=NP or the Unique Games Conjecture fails.For every ϵ>0, Lokshtanov, Misra, Panolan, Philip, and Saurabh (Lokshtanov et al., 2020) recently gave a randomized(2+ϵ)-approximation algorithm for SVD. In this work we give an extremely simple deterministic (2+ϵ)-approximation algorithm for SVD.

On sum of powers of normalized Laplacian eigenvalues and resistance distances of graphs

Let G=(V,E) be a simple connected graph with vertex set V={1,2,…,n} and edge set E. The normalized Laplacian matrix of G is L=D−1/2(D−A)D−1/2, where D is the degree-diagonal matrix of G and A is the adjacency matrix of G. Let λ1≥λ2≥⋯≥λn−1≥λn=0 be the eigenvalues of L, for any real number α, the topological index sα∗(G) of G is defined as sα∗(G)=∑i=1n−1λiα.Let Ωij denote the resistance distance between vertices i and j, the degree-Kirchhoff index Kf∗(G) of G is defined as Kf∗(G)=∑i<jdidjΩij,where di is the degree of the vertex i. In this paper, we first express sα∗(G) in terms of Ωij’s explicitly for any real number α. Then as an application, we give a new proof for the classical Foster’s kth formula of resistance distances. Finally, we generalize the well-known relation 2|E|s−1∗(G)=Kf∗(G)to any integer k≥−1, which provides some upper bounds for sk∗(G).

Complete characterizations of the 2-domination and P3-hull number polytopes

Given a graph G=(V,E), a subset S⊆V is 2-dominating if every vertex in S¯ has at least two neighbors in S. The minimum cardinality of such a set is called the 2-domination number of G. Consider a process in discrete time that, starting with an initial set of marked vertices S, at each step marks all unmarked vertices having two marked neighbors. In such a process, the minimum number of initial vertices in S such that eventually all vertices are marked is called the P3-hull number of G. In this work, we explore a polyhedral relation between these two parameters and, in addition, we provide new families of valid inequalities for the associated polytopes. Finally, we give explicit descriptions of the polytopes associated to these problems when G is a path, a cycle, a complete graph, or a tree.