Maximal Independent Sets Of Vertices Research Articles

Maximal cocliques and the chromatic number of the Kneser graph on chambers of PG(3,q) $(3,q)$

AbstractLet be the graph whose vertices are the chambers of the finite projective 3‐space , with two vertices being adjacent if and only if the corresponding chambers are in general position. We show that a maximal independent set of vertices of contains , or , or at most elements. For the structure of the largest maximal independent sets is described. For the structure of the maximal independent sets of the three largest cardinalities is described. Using the cardinality of the second largest maximal independent sets, we show that the chromatic number of is .

Graph-Based Text Summarization and Its Application on COVID-19 Twitter Data

Large volumes of structured and semi-structured data are being generated every day. Processing this large amount of data and extracting important information is a challenging task. The goal of an automatic text summarization is to preserve the key information and the overall meaning of the article to be summarized. In this paper, a graph-based approach is followed to generate an extractive summary, where sentences of the article are considered as vertices, and weighted edges are introduced based on the cosine similarities among the vertices. A possible subset of maximal independent sets of vertices of the graph is identified with the assumption that adjacent vertices provide sentences with similar information. The degree centrality and clustering coefficient of the vertices are used to compute the score of each of the maximal independent sets. The set with the highest score provides the final summary of the article. The proposed method is evaluated using the benchmark BBC News data to demonstrate its effectiveness and is applied to the COVID-19 Twitter data to express its applicability in topic modeling. Both the application and comparative study with other methods illustrate the efficacy of the proposed methodology.

A Maximum Independent Set Method for Scheduling Earth-Observing Satellite Constellations

Operating Earth-observing satellites requires efficient planning methods that coordinate activities of multiple spacecraft. The satellite task-planning problem entails selecting actions that best satisfy mission objectives for autonomous execution. Task scheduling is often performed by human operators assisted by heuristic or rule-based planning tools. This approach does not efficiently scale to multiple assets as heuristics frequently fail to properly coordinate actions of multiple vehicles over long horizons. Additionally, the problem becomes more difficult to solve for large constellations as the complexity of the problem scales exponentially in the number of requested observations and linearly in the number of spacecraft. It is expected that new commercial optical- and radar-imaging constellations will require automated planning methods to meet stated responsiveness and throughput objectives. This paper introduces a new approach for solving the satellite-scheduling problem by generating an infeasibility-based graph representation of the problem and finding a maximal independent set of vertices for the graph. The approach is tested on a scenario of up to 10,000 requested imaging locations for the SkySat constellation of optical satellites as well as simulated constellations of up to 24 satellites. Performance is compared with contemporary graph-traversal and mixed-integer linear programming approaches. Empirical results demonstrate improvements in the solution time along with the number of scheduled collections beyond baseline methods. For large problems, the maximum independent set approach is able to find a feasible schedule with 8% more collections in 75% less time.

On the structure of 4-regular planar well-covered graphs

A graph G is said to be well-covered if every maximal independent set of vertices has the same cardinality. It is unknown whether membership in this class of graphs is polynomially decidable. In this paper we focus on the study of well-covered graphs which are 4-regular and planar.

Maximal independent sets on a grid graph

An independent vertex set of a graph is a set of vertices of the graph in which no two vertices are adjacent, and a maximal independent set is one that is not a proper subset of any other independent set. In this paper we count the number of maximal independent sets of vertices on a complete rectangular grid graph. More precisely, we provide a recursive matrix-relation producing the partition function with respect to the number of vertices. The asymptotic behavior of the maximal hard square entropy constant is also provided. We adapt the state matrix recursion algorithm, recently invented by the author to answer various two-dimensional regular lattice model problems in enumerative combinatorics and statistical mechanics.

Метод поиска наибольших максимальных независимых множеств вершин неориентированного графа

Метод поиска наибольших максимальных независимых множеств вершин неориентированного графа

On well-covered pentagonalizations of the plane

A graph G is said to be well-covered if every maximal independent set of vertices has the same cardinality. A planar (simple) graph in which each face is bounded by a pentagon is called a (planar) pentagonalization.In the present paper we investigate those planar pentagonalizations which are well-covered.

Well-covered triangulations: Part IV

A graph G is said to be well-covered if every maximal independent set of vertices has the same cardinality. A plane (simple) graph in which each face is a triangle is called a (plane) triangulation. In the first of a sequence of three papers the authors proved that there are no 5-connected plane well-covered triangulations. Clearly, the only plane triangulation which is exactly 2-connected is the triangle K3. Two subsequent papers culminated in the proof that there are exactly four well-covered plane triangulations which are exactly 4-connected.It is the aim of the present paper to complete the characterization of well-covered plane triangulations by characterizing the infinite family of those well-covered triangulations of the plane which are exactly 3-connected.

Pure simplicial complexes and well-covered graphs

A graph G is called well-covered if all maximal independent sets of vertices have the same cardinality. A simplicial complexis called pure if all of its facets have the same cardinality. Let G be the class of graphs with some disjoint maximal cliques covering all vertices. In this paper, we prove that for any simplicial complex or any graph, there is a corresponding graph in class G with the same well-coveredness property. Then some necessary and sufficient conditions are presented t o recognize which graphs in the class G are well-covered. This characterization has a nice algebraic interpretation according to zero-divisor elements of edge ring of graphs which is shown in this paper.

Independent dominating sets in regular graphs

Let G be a simple, regular graph of order n and degree ?. The independent domination number i(G) is defined to be the minimum cardinality among all maximal independent sets of vertices of G. We establish new upper bounds, as functions of n and ?, for the independent domination number of regular graphs with $n/6<\delta< (3-\sqrt{5})n/2$ . Our two main theorems complement recent results of Goddard et al. (Ann. Comb., 2011) for larger values of ?.

On well-covered triangulations: Part III

A graph G is said to be well-covered if every maximal independent set of vertices has the same cardinality. A planar (simple) graph in which each face is a triangle is called a triangulation. It was proved in an earlier paper Finbow et al. (2004) [3] that there are no 5-connected planar well-covered triangulations, and in Finbow et al. (submitted for publication) [4] that there are exactly four 4-connected well-covered triangulations containing two adjacent vertices of degree 4. It is the aim of the present paper to complete the characterization of 4-connected well-covered triangulations by showing that each such graph contains two adjacent vertices of degree 4.

On well-covered triangulations: Part II

A graph G is said to be well-covered if every maximal independent set of vertices has the same cardinality. A planar (simple) graph in which each face is a triangle is called a triangulation. It was proved in an earlier paper [A. Finbow, B. Hartnell, R. Nowakowski, M. Plummer, On well-covered triangulations: Part I, Discrete Appl. Math., 132, 2004, 97–108] that there are no 5-connected planar well-covered triangulations. It is the aim of the present paper to completely determine the 4-connected well-covered triangulations containing two adjacent vertices of degree 4. In a subsequent paper [A. Finbow, B. Hartnell, R. Nowakowski, M. Plummer, On well-covered triangulations: Part III (submitted for publication)], we show that every 4-connected well-covered triangulation contains two adjacent vertices of degree 4 and hence complete the task of characterizing all 4-connected well-covered planar triangulations. There turn out to be only four such graphs. This stands in stark contrast to the fact that there are infinitely many 3-connected well-covered planar triangulations.

Independent domination in triangle-free graphs

Let G be a simple graph of order n and minimum degree δ . The independent domination number i ( G ) is defined to be the minimum cardinality among all maximal independent sets of vertices of G. We establish upper bounds, as functions of n and δ ⩽ n / 2 , for the independent domination number of triangle-free graphs, and over part of the range achieve best possible results.

Upper bounds for independent domination in regular graphs

Let G be a regular graph of order n and degree δ . The independent domination number i ( G ) is defined to be the minimum cardinality among all maximal independent sets of vertices of G. We establish upper bounds, as functions of n and δ , for the sum and product of the independent domination numbers of a regular graph and its complement.

Maximal and maximum independent sets in graphs with at most r cycles

AbstractLet m(G) denote the number of maximal independent sets of vertices in a graph G and let c(n,r) be the maximum value of m(G) over all connected graphs with n vertices and at most r cycles. A theorem of Griggs, Grinstead, and Guichard gives a formula for c(n,r) when r is large relative to n, while a theorem of Goh, Koh, Sagan, and Vatter does the same when r is small relative to n. We complete the determination of c(n,r) for all n and r and characterize the extremal graphs. Problems for maximum independent sets are also completely resolved. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 283–314, 2006

Independence Sequences of Well-Covered Graphs: Non-Unimodality and the Roller-Coaster Conjecture

A graph G is well-covered provided each maximal independent set of vertices has the same cardinality. The term s k of the independence sequence (s 0,s 1,?,s ?) equals the number of independent k-sets of vertices of G. We investigate constraints on the linear orderings of the terms of the independence sequence of well-covered graphs. In particular, we provide a counterexample to the recent unimodality conjecture of Brown, Dilcher, and Nowakowski. We formulate the Roller-Coaster Conjecture to describe the possible linear orderings of terms of the independence sequence.

On well-covered triangulations: Part I

A graph G is said to be well-covered if every maximal independent set of vertices has the same cardinality. A planar (simple) graph in which each face is a triangle is called a triangulation. It is the aim of this paper to prove that there are no 5-connected planar well-covered triangulations.

On the Number of Maximal Independent Set of Vertices in Star-Like Ladders

On the Number of Maximal Independent Set of Vertices in Star-Like Ladders

Two graph algorithms derived

Much emphasis has been placed in recent years on deriving or calculating programs rather than proving them correct. Adequate calculational frameworks are needed to support such an approach. The present work explores the use of a calculus of relations to express and reason about graph properties in an algorithmic context. We first construct a generic program that computes a maximal set, over some universe, satisfying a certain given predicate. The development is presented as a step-by-step derivation, applying well-known techniques and heuristics that aid the construction of imperative programs. A calculational framework of relations is then used to obtain two instances of the generic program. These instances correspond to the computation of maximal graph components, viz. maximal independent sets of vertices and maximal sets of edges without cycles (i.e. maximal forests).

An Upper Bound for the Independent Domination Number

Let G be a simple graph of order n and minimum degree δ . The independent domination number i ( G ) is defined to be the minimum cardinality among all maximal independent sets of vertices of G . In this paper, we show that i ( G )⩽ n +2 δ −2 nδ . Thus a conjecture of Favaron is settled in the affirmative.