Universal Vertex Research Articles

Edge open packing: Complexity, algorithmic aspects, and bounds

Given a graph G, two edges e1,e2∈E(G) are said to have a common edge e≠e1,e2 if e joins an endvertex of e1 to an endvertex of e2. A subset B⊆E(G) is an edge open packing set in G if no two edges of B have a common edge in G, and the maximum cardinality of such a set in G is called the edge open packing number, ρeo(G), of G. In this paper, we prove that the decision version of the edge open packing number is NP-complete even when restricted to graphs with universal vertices, Eulerian bipartite graphs, and planar graphs with maximum degree 4, respectively. In contrast, we present a linear-time algorithm that computes the edge open packing number of a tree. We also resolve two problems posed in the seminal paper (Chelladurai et al. (2022) [5]). Notably, we characterize the graphs G that attain the upper bound ρeo(G)≤|E(G)|/δ(G), and provide lower and upper bounds for the edge-deleted subgraph of a graph and establish the corresponding realization result.

A weak box-perfect graph theorem

A graph G is called perfect if ω(H)=χ(H) for every induced subgraph H of G, where ω(H) is the clique number of H and χ(H) its chromatic number. The Weak Perfect Graph Theorem of Lovász states that a graph G is perfect if and only if its complement G‾ is perfect. This does not hold for box-perfect graphs, which are the perfect graphs whose stable set polytope is box-totally dual integral.We prove that both G and G‾ are box-perfect if and only if G‾+ is box-perfect, where G+ is obtained by adding a universal vertex to G. Consequently, G+ is box-perfect if and only if G‾+ is box-perfect. As a corollary, we characterize when the complete join of two graphs is box-perfect.

Strictly chordal graphs: Structural properties and integer Laplacian eigenvalues

In this paper, we establish the relation between classic invariants and integer Laplacian eigenvalues of strictly chordal graphs, pointing out how their computation can be efficiently implemented. We review results concerning general graphs showing that the number of universal vertices and the degrees of twins provide integer Laplacian eigenvalues and their respective multiplicities. For strictly chordal graphs, we show new results about integer Laplacian eigenvalues which are directly related to particular vertex sets of the graph.

Combinatorial relations among relations for level 2 standard Cn(1)-modules

For an affine Lie algebra ĝ, the coefficients of certain vertex operators that annihilate level k standard ĝ-modules are the defining relations for level k standard modules. In this paper, we study a combinatorial structure of the leading terms of these relations for level k = 2 standard ĝ-modules for affine Lie algebras of type Cn(1) and the main result is the construction of combinatorially parameterized relations among the coefficients of annihilating fields. It is believed that the constructed relations among relations will play a key role in the construction of Gröbner-like basis of the maximal ideal of the universal vertex operator algebra Vgk for k = 2.

Bounds and Extremal Graphs for Total Dominating Identifying Codes

An identifying code $C$ of a graph $G$ is a dominating set of $G$ such that any two distinct vertices of $G$ have distinct closed neighbourhoods within $C$. The smallest size of an identifying code of $G$ is denoted $\gamma^{\text{ID}}(G)$. When every vertex of $G$ also has a neighbour in $C$, it is said to be a total dominating identifying code of $G$, and the smallest size of a total dominating identifying code of $G$ is denoted by $\gamma_t^{\text{ID}}(G)$.
 Extending similar characterizations for identifying codes from the literature, we characterize those graphs $G$ of order $n$ with $\gamma_t^{\text{ID}}(G)=n$ (the only such connected graph is $P_3$) and $\gamma_t^{\text{ID}}(G)=n-1$ (such graphs either satisfy $\gamma^{\text{ID}}(G)=n-1$ or are built from certain such graphs by adding a set of universal vertices, to each of which a private leaf is attached).
 Then, using bounds from the literature, we remark that any (open and closed) twin-free tree of order $n$ has a total dominating identifying code of size at most $\frac{3n}{4}$. This bound is tight, and we characterize the trees reaching it. Moreover, by a new proof, we show that this upper bound actually holds for the larger class of all twin-free graphs of girth at least 5. The cycle $C_8$ also attains the upper bound. We also provide a generalized bound for all graphs of girth at least 5 (possibly with twins).
 Finally, we relate $\gamma_t^{\text{ID}}(G)$ to the similar parameter $\gamma^{\text{ID}}(G)$ as well as to the location-domination number of $G$ and its variants, providing bounds that are either tight or almost tight.

Unimodality and monotonic portions of certain domination polynomials

Given a graph G on n vertices, a subset of vertices U⊆V(G) is dominating if every vertex of V(G) is either in U or adjacent to a vertex of U. The domination polynomial of G is the generating function whose coefficients are the number of dominating sets of a given size. In 2014, Alikhani and Peng conjectured that the domination polynomial is unimodal, i.e., its coefficients are non-decreasing and then non-increasing. We prove unimodality for spider graphs with at most 400 legs (of arbitrary length), lollipop graphs, arbitrary direct products of complete graphs, and Cartesian products of two complete graphs. We show that for every graph, a portion of the coefficients are non-increasing, and, for certain graphs with low upper domination number, this is sufficient to prove unimodality. Furthermore, we show that for graphs with m universal vertices, i.e., vertices of degree n−1, the last 2−m−1(2m−1)n coefficients of their domination polynomial are non-increasing.

Spanning trees in graphs of high minimum degree with a universal vertex I: An asymptotic result

AbstractIn this paper and a companion paper, we prove that, if is sufficiently large, every graph on vertices that has a universal vertex and minimum degree at least contains each tree with edges as a subgraph. Our result confirms, for large , an important special case of a recent conjecture by Havet, Reed, Stein and Wood. The present paper already contains an approximate version of the result.

Spanning trees in graphs of high minimum degree with a universal vertex II: A tight result

AbstractWe prove that, if is sufficiently large, every graph on vertices that has a universal vertex and minimum degree at least contains each tree with edges as a subgraph. Our result confirms, for large , an important special case of a conjecture by Havet, Reed, Stein, and Wood. The present paper builds on the results of a companion paper in which we proved the statement for all trees having a vertex that is adjacent to many leaves.

Leap eccentric connectivity index in graphs with universal vertices

For a graph X, the leap eccentric connectivity index (LECI) is ∑x∈V(X)d2(x,X)ε(x,X), where d2(x,X) is the 2-distance degree and ε(x,X) the eccentricity of x. We establish a lower and an upper bound for the LECI of X in terms of its order and the number of universal vertices, and identify the extremal graphs. We prove an upper bound on the index for trees of a given order and diameter, and determine the extremal trees. We also determine trees with maximum LECI among all trees of a given order.

On the double-critical graph conjecture

A connected graph [Formula: see text] is double-critical if the chromatic number of [Formula: see text] is two less than that of [Formula: see text] whenever [Formula: see text] and [Formula: see text] are two adjacent vertices of [Formula: see text]. The double-critical graph conjecture states that the complete graphs are the only double-critical graphs. We give a proof of this conjecture for any double-critical graph that contains at least one universal vertex. Using this result we prove the conjecture for all double-critical graphs.

On co-maximal subgroup graph of a group

The co-maximal subgroup graph \(\Gamma (G)\) of a group G is a graph whose vertices are non-trivial proper subgroups of G and two vertices H and K are adjacent if \(HK=G\). In this paper, we continue the study of \(\Gamma (G)\), especially when \(\Gamma (G)\) has isolated vertices. We define a new graph \(\Gamma ^*(G)\), which is obtained by removing isolated vertices from \(\Gamma (G)\). We characterize when \(\Gamma ^*(G)\) is connected, a complete graph, star graph, has an universal vertex etc. We also find various graph parameters like diameter, girth, bipartiteness etc. in terms of properties of G.

On the connectivity of the non-generating graph

Given a 2-generated finite group G, the non-generating graph of G has as vertices the elements of G and two vertices are adjacent if and only if they are distinct and do not generate G. We consider the graph Sigma (G) obtained from the non-generating graph of G by deleting the universal vertices. We prove that if the derived subgroup of G is not nilpotent, then this graph is connected, with diameter at most 5. Moreover, we give a complete classification of the finite groups G such that Sigma (G) is disconnected.

Multiplicative version of eccentric connectivity index

The eccentric connectivity index is defined for a connected graph G as the summation of the terms ɛG(u)+ɛG(v) over all edges uv of G, in which ɛG(u) denotes the eccentricity of the vertex u in G. In this paper, we study a multiplicative version of this graph invariant under the name multiplicative eccentric connectivity index. We present a sharp lower bound on the multiplicative eccentric connectivity index of n-vertex trees with fixed diameter and a sharp upper bound on the multiplicative eccentric connectivity index of n-vertex trees with given number of pendent vertices. Using these results, the trees on n vertices with the first three minimum and maximum values of the multiplicative eccentric connectivity index are characterized. Moreover, we present sharp bounds on the multiplicative eccentric connectivity index of connected graphs in terms of certain graph parameters including the order, size, radius, diameter, and number of universal vertices, and study the relationship between this invariant and several other vertex-eccentricity-based invariants. In particular, the minimal graphs with respect to the multiplicative eccentric connectivity index in the set of all connected graphs of given order, connected graphs of given size, and unicyclic and bicyclic graphs of given order are characterized and a Nordhaus–Gaddum-type result for this invariant is given.

Minimum Spanning Tree Cycle Intersection problem

Consider a connected graph G and let T be a spanning tree of G. Every edge e∈G−T induces a cycle in T∪{e}. The intersection of two distinct such cycles is the set of edges of T that belong to both cycles. We consider the problem of finding a spanning tree that has the least number of such non-empty intersections. In this article we analyze the particular case of complete graphs, and formulate a conjecture for graphs that have a universal vertex.

Zero Forcing Number of Some Families of Graphs

The work is devoted to the study of the zero forcing number of some families of graphs. The concept of zero forcing is a relatively new research topic in discrete mathematics, which already has some practical applications, in particular, is used in studies of the minimum rank of the matrices of adjacent graphs. The zero forcing process is an example of the spreading process on graphs. Such processes are interesting not only in terms of mathematical and computer research, but also interesting and are used to model technical or social processes in other areas: statistical mechanics, physics, analysis of social networks, and so on. Let the vertices of the graph G be considered white, except for a certain set of S black vertices. We will repaint the vertices of the graph from white to black, using a certain rule.Colour change rule: A white vertex turns black if it is the only white vertex adjacent to the black vertex.[5] The zero forcing number Z(G) of the graph G is the minimum cardinality of the set of black vertices S required to convert all vertices of the graph G to black in a finite number of steps using the ”colour change rule”.It is known [10] that for any graph G, its zero forcing number cannot be less than the minimum degree of its vertices. Such and other already known facts became the basis for finding the zero forcing number for two given below families of graphs:A gear graph, denoted W2,n is a graph obtained by inserting an extra vertex between each pair of adjacent vertices on the perimeter of a wheel graph Wn. Thus, W2,n has 2n + 1 vertices and 3n edges.A prism graph, denoted Yn, or in general case Ym,n, and sometimes also called a circular ladder graph, is a graph corresponding to the skeleton of an n-prism.A wheel graph, denoted Wn is a graph formed by connecting a single universal vertex to all vertices of a cycle of length n.In this article some known results are reviewed, there is also a definition, proof and some examples of the zero forcing number and the zero forcing process of gear graphs and prism graphs.

Some Results on Palette Index of Cartesian Product Graphs

Given a proper edge coloring α of a graph G, we define the palette SG(ν, α) of a vertex ν ∈ V (G) as the set of all colors appearing on edges incident to ν. The palette index š(G) of G is the minimum number of distinct palettes occurring in a proper edge coloring of G. The windmill graph Wd(n, k) is an undirected graph constructed for k ≥ 2 and n ≥ 2 by joining n copies of the complete graph Kk at a shared universal vertex. In this paper, we determine the bound on the palette index of Cartesian products of complete graphs and simple paths. We also consider the problem of determining the palette index of windmill graphs. In particular, we show that for any positive integers n, k ≥ 2, š(Wd(n, 2k)) = n + 1.

Injective coloring of complementary prism and generalized complementary prism graphs

The complementary prism [Formula: see text] of a graph [Formula: see text] is the graph obtained by drawing edges between the corresponding vertices of a graph [Formula: see text] and its complement [Formula: see text]. In this paper, we generalize the concept of complementary prisms of graphs and determine the injective chromatic number of generalized complementary prisms of graphs. We prove that for any simple graph [Formula: see text] of order [Formula: see text], [Formula: see text] and if [Formula: see text] is a graph with a universal vertex, then [Formula: see text].

A preferential attachment process approaching the Rado graph

AbstractWe consider a simple preferential attachment graph process, which begins with a finite graph and in which a new (t + 1)st vertex is added at each subsequent time step t that is connected to each previous vertex u ≤ t with probability du(t)/t, where du(t) is the degree of u at time t. We analyse the graph obtained as the infinite limit of this process, and we show that, as long as the initial finite graph is neither edgeless nor complete, with probability 1 the outcome will be a copy of the Rado graph augmented with a finite number of either isolated or universal vertices.

T-Pebbling in k-connected graphs with a universal vertex

t-Pebbling in k-connected graphs with a universal vertex

The Identifying Code, the Locating-dominating, the Open Locating-dominating and the Locating Total-dominating Problems Under Some Graph Operations

The problems of determining minimum identifying, locating-dominating, open locating-dominating or locating total-dominating codes in a graph G are variations of the classical minimum dominating set problem in G and are all known to be hard for general graphs. A typical line of attack is therefore to determine the cardinality of minimum such codes in special graphs. In this work we study the change of minimum such codes under three operations in graphs: adding a universal vertex, taking the generalized corona of a graph, and taking the square of a graph. We apply these operations to paths and cycles which allows us to provide minimum codes in most of the resulting graph classes.