Independent Set Of Vertices Research Articles

Fuzzy chromatic number of union of fuzzy graphs: An algorithm, properties and its application

We focus on fuzzy graphs with crisp vertex and fuzzy edge sets. A concept of the fuzzy chromatic number of these graphs based on fuzzy independent vertex set is used in this paper. A modified algorithm called a fuzzy chromatic algorithm is developed to find the fuzzy chromatic number of union of fuzzy graphs. Running time and complexity of the algorithm are also analyzed. Furthermore, we investigate some properties of the fuzzy chromatic number of union of fuzzy graphs. Finally, an application of the fuzzy chromatic number to determine the number of phases of an integrated traffic light system is proposed. We get different phases with different degrees of safety.

New results on the [formula omitted]-matrix of connected graphs

Let G be a simple undirected connected graph. Let D(G) be the distance matrix of G and let Tr(G) be the diagonal matrix of the vertex transmissions in G. Let α∈[0,1]. In S-Y. Cui et al. (2019) [7] the matrixDα(G)=αTr(G)+(1−α)D(G) is introduced and several properties are obtained. In this paper, new properties on the Dα-matrix are derived including inequalities that involve the largest vertex transmission and the spectral radii of the distance matrix, distance signless Laplacian matrix and Dα-matrix. The necessary and sufficient condition for the equality in each of the inequalities is given. Moreover, some results on the Dα-matrix of a graph with independent sets of vertices sharing the same set of neighbors, including the case of a complete multipartite graph, are obtained. Finally, the spectrum of Dα(G) is determined when G is the H-join of regular graphs.

A family of monomial ideals with the persistence property

In this paper, we introduce a family of monomial ideals with the persistence property. Given positive integers [Formula: see text] and [Formula: see text], we consider the monomial ideal [Formula: see text] generated by all monomials [Formula: see text], where [Formula: see text] is an independent set of vertices of the path graph [Formula: see text] of size [Formula: see text], which is indeed the facet ideal of the [Formula: see text]th skeleton of the independence complex of [Formula: see text]. We describe the set of associated primes of all powers of [Formula: see text] explicitly. It turns out that any such ideal [Formula: see text] has the persistence property. Moreover, the index of stability of [Formula: see text] and the stable set of associated prime ideals of [Formula: see text] are determined.

On the Fibonacci numbers of the composition of graphs

Let G=(V,E) be a graph. A subset S of V is said to be independent if for every two vertices u,v∈S there is no edge between them. The Fibonacci number of a graph G is the total number of independent vertex sets of G. The problem of finding the Fibonacci number of a graph is an NP-complete Problem. A closely related concept is the independence polynomial of a graph, whose kth coefficient equals the number of independent sets of cardinality k in the graph. The composition of two graphs G and H is the graph obtained by replacing every vertex of G with a copy of H and all vertices of two copies are adjacent if the corresponding vertices in G are adjacent. Previous results on Fibonacci numbers of graphs, allow us to calculate the Fibonacci number of very few particular classes of graphs. In this paper, we show some new nice qualitative results, which allow us to calculate the Fibonacci numbers of infinitely many classes of graphs, obtained by the composition of graphs. In particular, we calculate the Fibonacci number of the composition of G and H in terms of the independence polynomial of G and the Fibonacci number of H.

A fast deterministic detection of small pattern graphs in graphs without large cliques

We show that for several pattern graphs on four vertices (e.g., C4), their induced copies in host graphs with n vertices and no clique on k+1 vertices can be deterministically detected in O(n2.5719k0.3176+n2k2) time for k<n0.394 and O(n2.5k0.5+n2k2) time for k≥n0.394. The aforementioned pattern graphs have a pair of non-adjacent vertices whose neighborhoods are equal. By considering dual graphs, in the same asymptotic time, we can also detect four vertex pattern graphs, that have an adjacent pair of vertices with the same neighbors among the remaining vertices (e.g., K4), in host graphs with n vertices and no independent set on k+1 vertices.By using the concept of Ramsey numbers, we can extend our method for induced subgraph isomorphism to include larger pattern graphs having a set of independent vertices with the same neighborhood and n-vertex host graphs without cliques on k+1 vertices (as well as the pattern graphs and host graphs dual to the aforementioned ones, respectively).

New bounds on the independence number of connected graphs

The independence number of a graph [Formula: see text], denoted by [Formula: see text], is the maximum cardinality of an independent set of vertices in [Formula: see text]. [Henning and Löwenstein An improved lower bound on the independence number of a graph, Discrete Applied Mathematics 179 (2014) 120–128.] proved that if a connected graph [Formula: see text] of order [Formula: see text] and size [Formula: see text] does not belong to a specific family of graphs, then [Formula: see text]. In this paper, we strengthen the above bound for connected graphs with maximum degree at least three that have a non-cut-vertex of maximum degree. We show that if a connected graph [Formula: see text] of order [Formula: see text] and size [Formula: see text] has a non-cut-vertex of maximum degree then [Formula: see text], where [Formula: see text] is the maximum degree of the vertices of [Formula: see text]. We also characterize all connected graphs [Formula: see text] of order [Formula: see text] and size [Formula: see text] that have a non-cut-vertex of maximum degree and [Formula: see text].

Maximum independent sets near the upper bound

The size of a largest independent set of vertices in a given graph G is denoted by α(G) and is called its independence number (or stability number). Given a graph G and an integer K, it is NP-complete to decide whether α(G)≥K. An upper bound for the independence number α(G) of a given graph G with n vertices and m edges is given by α(G)≤p≔⌊12+14+n2−n−2m⌋.In this paper we will consider maximum independent sets near this upper bound. Our main result is the following: There exists an algorithm with time complexity O(n2) that, given as an input a graph G with n vertices, m edges, p≔⌊12+14+n2−n−2m⌋, and an integer k≥0 with p≥2k+1, returns an induced subgraph Gp,k of G with n0≤p+2k+1 vertices such that α(G)≤p−k if and only if α(Gp,k)≤p−k. Furthermore, we will show that we can decide in time O(1.27383k+n2) whether α(Gp,k)≤p−k.

The number of independent sets in a connected graph and its complement

For a connected graph G, the total number of independent vertex sets (including the empty vertex set) is denoted by i(G). In this paper, we consider Nordhaus-Gaddum-type inequalities for the number of independent sets of a connected graph with connected complement. First we define a transformation on a graph that increases i(G) and i(G). Next, we obtain the minimum and maximum value of i(G) + i(G), where graph G is a tree T with connected complement and a unicyclic graph G with connected complement, respectively. In each case, we characterize the extremal graphs. Finally, we establish an upper bound on the i(G) in terms of the Wiener polarity index.

Independent Feedback Vertex Set for P_5-Free Graphs

The NP-complete problem Feedback Vertex Set is that of deciding whether or not it is possible, for a given integer kge 0, to delete at most k vertices from a given graph so that what remains is a forest. The variant in which the deleted vertices must form an independent set is called Independent Feedback Vertex Set and is also NP-complete. In fact, even deciding if an independent feedback vertex set exists is NP-complete and this problem is closely related to the 3-Colouring problem, or equivalently, to the problem of deciding whether or not a graph has an independent odd cycle transversal, that is, an independent set of vertices whose deletion makes the graph bipartite. We initiate a systematic study of the complexity of Independent Feedback Vertex Set for H-free graphs. We prove that it is NP-complete if H contains a claw or cycle. Tamura, Ito and Zhou proved that it is polynomial-time solvable for P_4-free graphs. We show that it remains polynomial-time solvable for P_5-free graphs. We prove analogous results for the Independent Odd Cycle Transversal problem, which asks whether or not a graph has an independent odd cycle transversal of size at most k for a given integer kge 0. Finally, in line with our underlying research aim, we compare the complexity of Independent Feedback Vertex Set for H-free graphs with the complexity of 3-Colouring, Independent Odd Cycle Transversal and other related problems.

Enumerating Some Stable Partitions Involving Stirling and r-Stirling Numbers of the Second Kind

The coefficient of the chromatic polynomial counts the number of partitions of the vertex set of a simple and finite graph G into k independent vertex sets, equivalently, it gives the number of proper colorings of G with exactly k colors subject to some constraints. In this work, we study this invariant, we establish new formulas in this context for some families of graphs and we treat some specific cases as Thorn graphs. Consequently, we derive identities for the classical Stirling numbers of the second kind, besides that, this gives rise to new explicit formulae for the r-Stirling numbers of the second kind.

On the vertex partitions of sparse graphs into an independent vertex set and a forest with bounded maximum degree

Given a graph G=(V,E), if its vertex set V(G) can be partitioned into two non-empty subsets V1 and V2 such that G[V1] is edgeless and G[V2] is a graph with maximum degree at most k, then we say that G admits an (I, Δk)-partition. A similar definition can be given for the notation (I, Fk)-partition if G[V2] is a forest with maximum degree at most k.The maximum average degree of G is defined to be mad(G)=max{2|E(H)||V(H)|:H⊆G}. Borodin and Kostochka (2014) proved that every graph G with mad(G)≤83 admits an (I, Δ2)-partition and every graph G with mad(G)≤145 admits an (I, Δ4)-partition. In this paper, we obtain a strengthening result by showing that for any k ≥ 2, every graph G with mad(G)≤2+kk+1 admits an (I, Fk)-partition. As a corollary, every planar graph with girth at least 7 admits an (I, F4)-partition and every planar graph with girth at least 8 admits an (I, F2)-partition. The later result is best possible since neither girth condition nor the class of F2 can be further improved.

Kneser graphs are like Swiss cheese

Kneser graphs are like Swiss cheese, Discrete Analysis 2018:2, 18 pp. This paper relates two very interesting areas of research in extremal combinatorics: removal lemmas, and of variables. A _removal lemma_ is a result that states that if $S$ is a set that contains only few copies of a certain structure, then one can remove just a few elements from $S$ to create a set $S'$ that has no copies of that structure. For instance, the triangle removal states that a graph with few triangles can be approximated by a graph with no triangles. Removal lemmas play a very important role in extremal and additive combinatorics. As for the of variables, the aspect that is of interest here is that there are many interesting theorems that state that if a function has a certain property, it must be because in a certain sense it depends on only a few variables. For example, a famous result of the first author shows that if a monotone graph property fails to have a sharp threshold, then the property can be approximated by one for which the minimal graphs that satisfy it are all small, with the result that the influence of individual edges on whether the graph has the property is substantial. The main result of this paper is an removal lemma for various families of graphs. It states that if $G$ is a graph in one of these families, then for every set $X$ of vertices in $G$ that spans only a few edges one can throw away just a small proportion of $X$ to obtain a subset $X'$ that is independent. Of course, a result like this is completely false in general: one has only to take a sparse random graph, and then every subset will span few edges but no large subset will be independent. But it is true, for example, for complete bipartite graphs, since a set of vertices that spans a sparse subgraph will have to be contained almost entirely in one of the two vertex sets, and then one can throw away the vertices that are in the other set. One of the families for which the authors prove an edge-removal is that of Kneser graphs. For $0<m<n/2$, the Kneser graph $K(n,m)$ has as its vertex set the set of all subsets of $\{1,2,\dots,n\}$ of size $m$, with two such subsets joined by an edge if and only if they are disjoint. Kneser graphs have large sparse sets, but the natural examples seem to depend on just a few variables: for example, one can take all sets that contain a particular element. (Note that an independent set in $K(n,m)$ is just an intersecting family of $m$-sets.) The authors wanted to prove a precise conjecture that would say that every set of vertices that spans a sparse subgraph can be approximated by a set of vertices that (i) depends on just a few variables and (ii) spans an independent set. Before this paper a weaker statement was known where instead of (ii) one had a sparse set. In this paper the authors settle the conjecture completely by showing that a sparse set of vertices can be approximated by an independent set of vertices. The other family of graphs for which the authors prove an edge removal is that of product graphs. If $H$ is a graph, then the product $G=H^n$ is the graph where $x$ and $y$ are joined if for every coordinate $i$ we have that $x_iy_i$ is an edge of $H$. (There are different definitions of product graphs, but this is the one to which the result applies: note that it makes it quite hard for two vertices to be joined.) For example, if $H$ is a complete graph on $r$ coordinates, then $G=[r]^n$ with two vertices joined if and only if they differ in every coordinate. The proof of the removal uses ideas from Jacob Fox's proof of the triangle removal lemma, which obtains the best known bound for that result. Finally, what has this to do with Swiss cheese? The rough idea is that these graphs have large and highly structured independent sets, and a part of the graph is sparse if and only if it has a big intersection with one of these independent sets. Likewise a Swiss cheese contains lots of large holes, and a region of space contained within the boundary of the cheese will contain only a small amount of cheese if and only if it is almost entirely contained in one of the holes. Article image [by G. J. Wanderer](https://www.flickr.com/photos/106561245@N07/13515457313/in/photolist-mAjfA6-f7mF8i-Tf5Ler-7i4FtY-bi74Nt-gWVXdj-7XoC4N-okRvp5-77HxiZ-7xt4hx-RSUt3B-qeB17D-as2aAc-7K64uq-oL2GMj-6trVsK-UDq452-SP1xSX-d4u57o-fATSUi-8Km2Qs-dJeLzb-qu9Z4Z-dNNvQ2-7eVY4b-6mtCks-frWJEF-aCerNx-7tyjAW-haiMTG-oysrog-9LJVj8-fzno-4VuER1-b5n85M-Jka6K5-a7vHJZ-5S7Yyt-Uwa6kC-dNU7Mm-5AWGod-YSE7a-bzagV7-8tV58z-8YW1XL-5nvECU-QAdC69-8wG84t-TSsHP4-9eoj6S) and distributed under a [CC-BY-NC-SA licence](https://creativecommons.org/licenses/by-nc-sa/2.0/).

Maximum weight independent set for [formula omitted]claw-free graphs in polynomial time

For a finite, simple, undirected graph G with a vertex weight function, the Maximum Weight Independent Set (MWIS) problem asks for an independent vertex set of G with maximum weight. MWIS is a well-known NP-hard problem of fundamental importance. For claw-free graphs, MWIS can be solved in polynomial time—in 1980, the first two such algorithms were independently found by Minty for MWIS and by Sbihi for the unweighted case. For a constant ℓ≥2, let ℓG denote the disjoint union of ℓ copies of G.In this paper, using a dynamic programming approach (inspired by Farber’s result about MWIS for 2K2-free graphs), we show that for any fixed ℓ, MWIS can be solved in polynomial time for ℓclaw-free graphs. This solves the open cases for MWIS on (P3+claw)-free graphs and on (2K2+claw)-free graphs and extends known results for claw-free graphs, ℓK2-free graphs, (K2+claw)-free graphs, and ℓP3-free graphs.

Uncertain vertex coloring problem

This paper investigates the vertex coloring problem in an uncertain graph in which all vertices are deterministic, while all edges are not deterministic and exist with some degree of belief in uncertain measures. The concept of the maximal uncertain independent vertex set of an uncertain graph is first introduced. We then present a degree of belief rule to obtain the family of maximal uncertain independent vertex sets. Based on the maximal uncertain independent vertex set, some properties of the separation degree of an uncertain graph are discussed. Following that, the concept of an uncertain chromatic set is introduced. Then, a maximum separation degree algorithm is derived to obtain the uncertain chromatic set. Finally, numerical examples are presented to demonstrate the application of the vertex coloring problem in uncertain graphs and the effectiveness of the maximum separation degree algorithm.

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.

Tight Just Chromatic Excellence In Fuzzy Graphs

Let G be a simple fuzzy graph. A family  I“a¶ = { I³1, I³2,…, I³k} of fuzzy sets on a set V is called k-fuzzy colouring of V = (V,Iƒ,µ) if i) âˆa I“a¶ = Iƒ, ii) I³i∩ I³j = Ф, iii) for every strong edge (x,y) (i.e., µ(xy) > 0) of G min{I³i(x), I³j(y)} = 0; (1 ≤ i ≤ k). The minimum number of k for which there exists a k-fuzzy colouring is called the fuzzy chromatic number of G denoted as I‡f (G). Then I“a¶  is the partition of independent sets of vertices of G in which each sets has the same colour is called the fuzzy chromatic partition. A graph G is called the just I‡f -excellent if every vertex of G appears as a singleton in exactly one _f -partition of G. A just I‡f –excellent graph of order n is called the tight just I‡f -excellent if G having exactly n, I‡f -partitions. This paper aims at the study of the new concept namely tight just Chromatic excellence in fuzzy graphs and its properties. 02000 Mathematics Subject Classification:05C72 Key words: fuzzy just chromatic excellent, tight just I‡f -excellent, fuzzy colourful vertex, fuzzy kneser graph.

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.

Positive graphs

We call a graph positive if it has a nonnegative homomorphism number into any target graph with real edge weights. The Positive Graphs Conjecture offers a structural characterization: these are exactly the graphs that can be obtained by gluing together two copies of the same graph along an independent set of vertices. In this talk I will discuss our recent results on the Positive Graphs Conjecture.

Independent [1, 2]-domination of grids via min-plus algebra

Domination of grids has been proved to be a demanding task and with the addition of independence it becomes more challenging. It is known that no grid with m,n≥5 has a perfect code, that is an independent vertex set such that each vertex not in it has exactly one neighbor in that set. So it is interesting to study the existence of an independent dominating set for grids that allows at most two neighbors, such a set is called independent [1, 2]-set. In this paper we develop a dynamic programming algorithm using min-plus algebra that computes the minimum cardinality of an independent [1, 2]-set for the grid Pm□Pn.