Outer-connected open packing sets in graphs

A set [Formula: see text] of a graph [Formula: see text] is an [Formula: see text] [Formula: see text] [Formula: see text] of [Formula: see text] if no two vertices of [Formula: see text] have a common neighbor in [Formula: see text]. An open packing set [Formula: see text] is called an outer-connected open packing set (ocop-set) if either [Formula: see text] or [Formula: see text] is connected. The minimum and maximum cardinalities of an ocop-set are called the lower outer-connected open packing number and the outer-connected open packing number, respectively, and are denoted by [Formula: see text] and [Formula: see text], respectively. In this paper, we initiate a study on these parameters.

OPEN PACKING NUMBER FOR SOME CLASSES OF PERFECT GRAPHS

Let \(G\) be a graph with the vertex set \(V(G)\). A subset \(S\) of \(V(G)\) is an open packing set of \(G\) if every pair of vertices in \(S\) has no common neighbor in \(G.\) The maximum cardinality of an open packing set of \(G\) is the open packing number of \(G\) and it is denoted by \(\rho^o(G)\). In this paper, the exact values of the open packing numbers for some classes of perfect graphs, such as split graphs, \(\{P_4, C_4\}\)-free graphs, the complement of a bipartite graph, the trestled graph of a perfect graph are obtained.

On the Open Packing Number of a Graph

A non-empty set of a graph G is an open packing set of G if no two vertices of S have a common neighbor in G. The maximum cardinality of an open packing set is the open packing number of G and is denoted by . An open packing set of cardinality is a -set of G. In this paper, the classes of trees and unicyclic graphs for which the value of is either 2 or 3 are characterized. Moreover, the exact values of the open packing number for some special classes of graphs have been found.

DOMINATION, TOTAL DOMINATION AND OPEN PACKING OF THE CORCOR DOMAIN OF GRAPHENE

A dominating set of a graph G = (V,E) is a subset D of V such that everyvertex not in D is adjacent to at least one vertex in D. A dominating set D is a totaldominating set, if every vertex in V is adjacent to at least one vertex in D. The set Pis said to be an open packing set if no two vertices of P have a common neighbor inG. In this paper, we obtain domination number, total domination number and openpacking number of the molecular graph of a new type of graphene named CorCor thatis a 2-dimensional carbon network.

Open packing bondage number of a graph

In a graph [Formula: see text], a set [Formula: see text] is said to be an open packing set if no two vertices of [Formula: see text] have a common neighbor in [Formula: see text] The maximum cardinality of an open packing set is called the open packing number and is denoted by [Formula: see text]. The open packing bondage number of a graph [Formula: see text], denoted by [Formula: see text], is the cardinality of the smallest set of edges [Formula: see text] such that [Formula: see text]. In this paper, we initiate a study on this parameter.

A Robust, Porous and Solution-Processable Molecular Crystal Containing Multiple Donor-Acceptor Units

We report a curious porous molecular crystal that is devoid of the common traits of related systems. Namely, the molecule does not rely on directional hydrogen bonds to enforce open packing; and it offers neither large concave faces (i.e., high internal free volume) to frustrate close packing, nor any inherently built-in cavity like in the class of organic cages. Instead, the permanent porosity (as unveiled by the X-ray crystal structure and CO₂ sorption studies) arises from the strong push-pull units built into a Sierpinski-like molecule that features four symmetrically backfolded (<b>SBF</b>) side arms. Each side arm consists of the 1,1,4,4-tetracyanobuta-1,3-diene acceptor (TCBD) coupled with the dimethylaminophenyl donor, which is conveniently installed by a cycloaddition-retroelectrocyclization (CA-RE) reaction. Unlike the poor/fragile crystalline order of many porous molecular solids, the molecule here readily crystallizes and the crystalline phase can be easily deposited into thin films from solutions. Moreover, both the bulk sample and thin film exhibit excellent thermal stability with the porous crystalline order maintained even at 200 °C. The intermolecular forces underlying this robust porous molecular crystal likely include the strong dipole interactions and the multiple C···N and C···O short contacts afforded by the strongly donating and accepting groups integrated within the rigid molecular scaffold.

A Robust, Porous and Solution-Processable Molecular Crystal Containing Multiple Donor-Acceptor Units

We report a curious porous molecular crystal that is devoid of the common traits of related systems. Namely, the molecule does not rely on directional hydrogen bonds to enforce open packing; and it offers neither large concave faces (i.e., high internal free volume) to frustrate close packing, nor any inherently built-in cavity like in the class of organic cages. Instead, the permanent porosity (as unveiled by the X-ray crystal structure and CO₂ sorption studies) arises from the strong push-pull units built into a Sierpinski-like molecule that features four symmetrically backfolded (<b>SBF</b>) side arms. Each side arm consists of the 1,1,4,4-tetracyanobuta-1,3-diene acceptor (TCBD) coupled with the dimethylaminophenyl donor, which is conveniently installed by a cycloaddition-retroelectrocyclization (CA-RE) reaction. Unlike the poor/fragile crystalline order of many porous molecular solids, the molecule here readily crystallizes and the crystalline phase can be easily deposited into thin films from solutions. Moreover, both the bulk sample and thin film exhibit excellent thermal stability with the porous crystalline order maintained even at 200 °C. The intermolecular forces underlying this robust porous molecular crystal likely include the strong dipole interactions and the multiple C···N and C···O short contacts afforded by the strongly donating and accepting groups integrated within the rigid molecular scaffold.