Independent Set Of Vertices Research Articles

Bounds on the independence number of a graph in terms of order, size and maximum degree

The independence number of a graph G is the maximum cardinality of an independent set of vertices in G. In this paper we obtain several new lower bounds for the independence number of a graph in terms of its order, size and maximum degree, and characterize graphs achieving equalities for these bounds. Our bounds improve previous bounds for graphs with large maximum degree.

Enumerating independent vertex sets in grid graphs

A set of vertices in a graph is called independent if no two vertices of the set are connected by an edge. In this paper we use the state matrix recursion algorithm, developed by Oh, to enumerate independent vertex sets in a grid graph and even further to provide the generating function with respect to the number of vertices. We also enumerate bipartite independent vertex sets in a grid graph. The asymptotic behavior of their growth rates is presented.

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.

The Fibonacci numbers of certain subgraphs of circulant graphs

The Fibonacci number ℱ(G) of a graph G with vertex set V(G), is the total number of independent vertex sets S⊂V(G); recall that a set S⊂V(G) is said to be independent whenever for every two different vertices u,v∈S there is no edge between them. In general, the problem to find the Fibonacci number of a graph is NP-complete. Prodinger and Tichy proved that the Fibonacci number of Pn, the path of order n is Fn+2, the n+2-Fibonacci number; and the Fibonacci number of Cn, the cycle of order n, is Ln, the n-Lucas number.A circulant graph Cn(m1,m2,…,mr) is a graph of order n with vertex set V={v1,v2,…,vn} and edge set E={vivi+mj(modn):i∈{1,2,…,n},j∈{1,2,…,r}}, where r∈Z+. The values mj are the jump sizes. In this paper we only deal with the circulant graphs of order n with r consecutive jumps 1,2,…,r. Cn(1,2,…,r) is denoted by Cn[r].We prove that the total number of independent vertex sets of the family of graphs Cn[r] for all n≥r+1, and for several subgraphs of this family is completely determined by some sequences which are constructed recursively like the Fibonacci and Lucas sequences, even more, these new sequences generalize the Fibonacci and Lucas sequences.

Regular independent sets

The regular independence number, introduced by Albertson and Boutin in 1990, is the size of a largest set of independent vertices with the same degree. Lower bounds were proven for this invariant, in terms of the order, for trees and planar graphs. In this article, we generalize and extend these results to find lower bounds for the regular k-independence number for trees, forests, planar graphs, k-trees and k-degenerate graphs.

Some results on the structure of kernel-perfect and critical kernel-imperfect digraphs

A kernel N of a digraph D is an independent set of vertices of D such that for every w∈V(D)∖N there exists an arc from w to N. The digraph D is said to be a kernel-perfect digraph when every induced subdigraph of D has a kernel. Minimal non kernel-perfect digraphs are called critical kernel-imperfect digraphs. The broader sufficient condition for the existence of kernels in digraphs known so far is that states: (1) If D is a digraph such that every odd cycle has two consecutive poles, then D is kernel-perfect.In this paper is studied the structure of critical kernel-imperfect digraphs which belong to a very large special classes of digraphs and many structural properties are obtained. As a consequence (1) is widely generalized in this class of digraphs, where the condition of the poles is requested only for odd cycles whose edges alternate in a set of arcs. As consequence, some classic results of kernel-perfect and finite critical kernel-imperfect digraphs are generalized for these classes of digraphs.

FrogCOL and FrogMIS: new decentralized algorithms for finding large independent sets in graphs

Finding large (and generally maximal) independent sets of vertices in a given graph is a fundamental problem in distributed computing. Applications include, for example, facility location and backbone formation in wireless ad hoc networks. In this paper, we study a decentralized (or distributed) algorithm inspired by the calling behavior of male Japanese tree frogs, originally introduced for the graph-coloring problem, for its potential usefulness in the context of finding large independent sets. Moreover, we adapt this algorithm to directly produce maximal independent sets without the necessity of first producing a graph-coloring solution. Both algorithms are compared to a wide range of decentralized algorithms from the literature on a diverse set of benchmark instances for the maximal independent set problem. The results show that both algorithms compare very favorably to their competitors.

On a minimization problem for a set of Boolean functions

We study the set of Boolean functions that consist of a single connected component, have minimal complexes of faces which are not shortest, and do not satisfy the sufficient minimality conditions based on the notion of an independent set of vertices. For the minimization of functions with the indicated properties, the available efficient methods such as the independent minimization for the connected components and the fulfillment of sufficientminimality conditions are inapplicable. For this set of functions, we obtain some lower bounds for the capacity and the maximal number of complexes of faces minimal under additive measures of the linear and polynomial complexity.

Formula omitted]-kernels and [formula omitted]-obstructions in [formula omitted]-colored digraphs

Let D be a digraph. V(D) and A(D) will denote the vertex and arc sets of D respectively. A kernel K of a digraph D is an independent set of vertices of D such that for every vertex w in V(D)-K there exists an arc from w to a vertex in K. Let H be a digraph possibly with loops and D a digraph without loops whose arcs are colored with the vertices of H (D is said to be an H-colored digraph). A directed path W in D is said to be an H-path if and only if the consecutive colors encountered on W form a directed walk in H. A generalization of the concept of kernel is the concept of H-kernel, where an H-kernel N of an H-colored digraph D is a set of vertices of D such that for every pair of different vertices in N there is no H-path between them, and for every vertex u in V(D)-N there exists an H-path in D from u to N. A classical result in kernel theory establishes that if D is a digraph without cycles of odd length, then D has a kernel; this result is known as Richardson’s theorem and in this paper we will show an extension of this theorem which is given by the main result.Let D be an H-colored digraph. For an arc (z1, z2) of D we will denote its color by c(z1, z2). We introduce the concept of obstruction in an H-colored digraph as follows. Let W=(v0,v1,…,vn−1,v0) be a closed directed walk in D. We will say that there is an obstruction on vi if (c(vi−1, vi), c(vi, vi+1)) is not an arc of A(H) (indices modulo n). The main result establishes that if D is an H-colored digraph such that the number of obstructions in every closed directed trail of D is even, then D has an H-kernel. Previous interesting results are generalized, as for example Sands, Sauer and Woodrow’s theorem.

Computing unique maximum matchings in $$O(m)$$ O ( m ) time for König–Egerváry graphs and unicyclic graphs

Let $$\alpha \left( G\right) $$źG denote the maximum size of an independent set of vertices and $$\mu \left( G\right) $$μG be the cardinality of a maximum matching in a graph $$G$$G. A matching saturating all the vertices is a perfect matching. If $$\alpha \left( G\right) +\mu \left( G\right) =\left| V(G)\right| $$źG+μG=V(G), then $$G$$G is called a Konig---Egervary graph. A graph is unicyclic if it is connected and has a unique cycle. It is known that a maximum matching can be found in $$O(m\cdot \sqrt{n})$$O(m·n) time for a graph with $$n$$n vertices and $$m$$m edges. Bartha (Proceedings of the 8th joint conference on mathematics and computer science, Komarno, Slovakia, 2010) conjectured that a unique perfect matching, if it exists, can be found in $$O(m)$$O(m) time. In this paper we validate this conjecture for Konig---Egervary graphs and unicylic graphs. We propose a variation of Karp---Sipser leaf-removal algorithm (Karp and Sipser in Proceedings of the 22nd annual IEEE symposium on foundations of computer science, 364---375, 1981) , which ends with an empty graph if and only if the original graph is a Konig---Egervary graph with a unique perfect matching (obtained as an output as well). We also show that a unicyclic non-bipartite graph $$G$$G may have at most one perfect matching, and this is the case where $$G$$G is a Konig---Egervary graph.

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.

On choosability with separation of planar graphs with lists of different sizes

A (k,d)-list assignment L of a graph G is a mapping that assigns to each vertex v a list L(v) of at least k colors and for any adjacent pair xy, the lists L(x) and L(y) share at most d colors. A graph G is (k,d)-choosable if there exists an L-coloring of G for every (k,d)-list assignment L. This concept is also known as choosability with separation.It is known that planar graphs are (4, 1)-choosable but it is not known if planar graphs are (3, 1)-choosable. We strengthen the result that planar graphs are (4, 1)-choosable by allowing an independent set of vertices to have lists of size 3 instead of 4.Our strengthening is motivated by the observation that in (4, 1)-list assignment, vertices of an edge have together at least 7 colors, while in (3, 1)-list assignment, they have only at least 5. Our setting gives at least 6 colors.

Algorithms for the Independent Feedback Vertex Set Problem

Algorithms for the Independent Feedback Vertex Set Problem

On the asymptotics of a sequence of lacunary binomial-type polynomials

We examine the asymptotics of a sequence of lacunary binomial-type polynomials $\wp_n(z)$ as $n\rightarrow\infty$ that have arisen in the problem of the expected number of independent sets of vertices of finite simple graphs. We extend the recent analysis of Gawronski and Neuschel by employing the method of steepest descents applied to an integral representation. The case of complex $z$ with $|z|<1$ is also considered. Numerical results are presented to illustrate the accuracy of the resulting expansions.

An improved lower bound on the independence number of a graph

The independence number of a graph G, denoted α(G), is the maximum cardinality of an independent set of vertices in G. Our main result is the strengthening of a lower bound for the independence number of a graph due to Löwenstein et al. (2011) who proved that if G is a connected graph of order n and size m, then α(G)≥23n−14m−13. We show that if G does not belong to a specific family of graphs, then α(G)>23n−14m. Further, we characterize the graphs G for which α(G)≤23n−14m.

Kernels in circulant digraphs

A kernel $J$ of a digraph $D$ is an independent set of vertices of $D$ such that for every vertex $w,in,V(D),setminus,J$ there exists an arc from $w$ to a vertex in $J.$‎ ‎In this paper‎, ‎among other results‎, ‎a characterization of $2$-regular circulant digraph having a kernel is obtained‎. ‎This characterization is a partial solution to the following problem‎: ‎Characterize circulant digraphs which have kernels; it appeared in the book Digraphs‎ - ‎theory‎, ‎algorithms and applications‎, ‎Second Edition‎, ‎Springer-Verlag‎, ‎2009‎, ‎by J‎. ‎Bang-Jensen and G‎. ‎Gutin‎.

A note on graphs contraction-critical with respect to independence number

Let α(G) denote the independence number of graph G, i.e., the size of any maximum independent set of vertices. A graph G is contraction-critical (with respect to α) if α(G/xŷ)<α(G), for every edge xy∈E(G), where G/xŷ denotes the graph obtained from G by shrinking the edge xy to a single vertex xŷ and deleting the loop thus formed. Let us denote the class of all such contraction-critical graphs by CCR.First it is shown that all graphs in CCR must be bipartite.Let G=(A,B) be a bipartite graph with bipartition A∪B. It is then shown that (a) if |A|<|B|, then G∈CCR if and only if B is the unique maximum independent set in G if and only if G is 1-expanding, while (b) if |A|=|B|, then G∈CCR if and only if A and B are the only two maximum independent sets in G if and only if G is 1-extendable.It follows that CCR graphs can be recognized in polynomial time.

A New Lower Bound on the Independence Number of a Graph and Applications

The independence number of a graph $G$, denoted $\alpha(G)$, is the maximum cardinality of an independent set of vertices in $G$. The independence number is one of the most fundamental and well-studied graph parameters. In this paper, we strengthen a result of Fajtlowicz [Combinatorica 4 (1984), 35-38] on the independence of a graph given its maximum degree and maximum clique size. As a consequence of our result we give bounds on the independence number and transversal number of $6$-uniform hypergraphs with maximum degree three. This gives support for a conjecture due to Tuza and Vestergaard [Discussiones Math. Graph Theory 22 (2002), 199-210] that if $H$ is a $3$-regular $6$-uniform hypergraph of order $n$, then $\tau(H) \le n/4$.

Cycle Double Covers in Cubic Graphs having Special Structures

In the first part of this article, we employ Thomason's Lollipop Lemma 25 to prove that bridgeless cubic graphs containing a spanning lollipop admit a cycle double cover (CDC) containing the circuit in the lollipop; this implies, in particular, that bridgeless cubic graphs with a 2-factor F having two components admit CDCs containing any of the components in the 2-factor, although it need not have a CDC containing all of F. As another example consider a cubic bridgeless graph containing a 2-factor with three components, all induced circuits. In this case, two of the components may separately be used to start a CDC although it is uncertain whether the third component may be part of some CDC. Numerous other corollaries shall be given as well. In the second part of the article, we consider special types of bridgeless cubic graphs for which a prominent circuit can be shown to be included in a CDC. The interest here is the proof technique and therefore we only give the simplest case of the theorem. Notably, we show that a cubic graph that consists of an induced 2k-circuit C together with an induced 4k-circuit T and an independent set of 2k vertices, each joined by one edge to C and two edges to T, has a CDC starting with T.

Heuristic algorithm for finding the maximum independent set

A heuristic algorithm is developed for finding the maximum independent set of vertices in an undirected graph. To this end, the technique of finite partially ordered sets is used, in particular, the technique of partitioning such a set into a minimum number of chains. A special digraph is constructed and a solution algorithm is proposed on the basis of a hypothesis about its properties. Some experimental data are presented for well-known examples.