Independent Set Of Vertices Research Articles

Some sufficient conditions for the existence of kernels in infinite 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 . If every induced subdigraph of D has a kernel, D is said to be a kernel perfect digraph. D is called a critical kernel imperfect digraph when D has no kernel but every proper induced subdigraph of D has a kernel. If F is a set of arcs of D , a semikernel modulo F of D is an independent set of vertices S of D such that for every z ∈ V ( D ) − S for which there exists an ( S , z ) -arc of D − F , there also exists an ( z , S ) -arc in D . In this work we show sufficient conditions for an infinite digraph to be a kernel perfect digraph, in terms of semikernel modulo F . As a consequence it is proved that symmetric infinite digraphs and bipartite infinite digraphs are kernel perfect digraphs. Also we give sufficient conditions for the following classes of infinite digraphs to be kernel perfect digraphs: transitive digraphs, quasi-transitive digraphs, right (or left)-pretransitive digraphs, the union of two right (or left)-pretransitive digraphs, the union of a right-pretransitive digraph with a left-pretransitive digraph, the union of two transitive digraphs, locally semicomplete digraphs and outward locally finite digraphs.

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.

Kernel perfect and critical kernel imperfect digraphs structure

Abstract 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 . If every induced subdigraph of D has a kernel, D is said to be a kernel perfect digraph. Minimal non-kernel perfect digraph are called critical kernel imperfect digraph. If F is a set of arcs of D , a semikernel modulo F , S of D is an independent set of vertices of D such that for every z ∈ V ( D ) − S for which there exists an Sz− arc of D − F , there also exists an zS− arc in D . In this talk some structural results concerning critical kernel imperfect and sufficient conditions for a digraph to be a critical kernel imperfect digraph are presented.

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.

Poly-free constructions for right-angled Artin groups

We show that every right-angled Artin group AΓ defined by a graph Γ of finite chromatic number is poly-free with poly-free length bounded between the clique number and the chromatic number of Γ. Further, a characterization of all right-angled Artin groups of poly-free length 2 is given, namely the group AΓ has poly-free length 2 if and only if there exists an independent set of vertices D in Γ such that every cycle in Γ meets D at least twice. Finally, it is shown that AΓ is a semidirect product of two free groups of finite rank if and only if Γ is a finite tree or a finite complete bipartite graph. All of the proofs of the existence of polyfree structures are constructive.

The neighborhood union of independent sets and hamiltonicity of graphs

Let G be a graph, N ( u ) the neighborhood of u for each u ∈ V ( G ) , and N ( U ) = ⋃ u ∈ U N ( u ) for each U ⊆ V ( G ) . For any two positive integers s and t, we prove that there exists a least positive integer N ( s , t ) such that every ( s + t ) -connected graph G of order n ⩾ N ( s , t ) is hamiltonian if | N ( S ) | + | N ( T ) | ⩾ n for every two disjoint independent vertex sets S, T with | S | = s and | T | = t .

Maximum weight independent set of circular-arc graph and its application

In this paper, an algorithm is designed to find a maximum weight independent set of a circular-arc graph withn vertices. The weights considered here are all non-negative real numbers and associated with each of the vertex of the graph. The proposed algorithm runs in timeO(n2). Here we shown that the program slots of television channels during 24 hours can be modeled as a circular-arc graph. Each program represents a vertex and number of viewers of that program represents the weight of the corresponding vertex. Two vertices are connected by an edge iff the corresponding program slots have a common program time, i.e., ifIi andIj are the program slots of two programsi andj then the corresponding verticesi andj are connected by an edge iffIi ∩ Ij 6⊋ Φ. We also shown that the non-overlapping program slots with maximum number of viewers can be selected by computing maximum weight independent set on the corresponding circular-arc graph.

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

Hamiltonian graphs involving neighborhood unions

AbstractDirac proved that a graph G is hamiltonian if the minimum degree $\delta(G) \geq n/2$, where n is the order of G. Let G be a graph and $A \subseteq V(G)$. The neighborhood of A is $N(A)=\{ b: ab \in E(G)$ for some $a \in A\}$. For any positive integer k, we show that every (2k − 1)‐connected graph of order n ≥ 16k3 is hamiltonian if |N(A)| ≥ n/2 for every independent vertex set A of k vertices. The result contains a few known results as special cases. The case of k = 1 is the classic result of Dirac when n is large and the case of k = 2 is a result of Broersma, Van den Heuvel, and Veldman when n is large. For general k, this result improves a result of Chen and Liu. The lower bound 2k − 1 on connectivity is best possible in general while the lower bound 16k3 for n is conjectured to be unnecessary. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 83–100, 2006

Well-covered graphs and factors

A maximum independent set of vertices in a graph is a set of pairwise nonadjacent vertices of largest cardinality α . Plummer [Some covering concepts in graphs, J. Combin. Theory 8 (1970) 91–98] defined a graph to be well-covered, if every independent set is contained in a maximum independent set of G. Every well-covered graph G without isolated vertices has a perfect [ 1 , 2 ] -factor F G , i.e. a spanning subgraph such that each component is 1-regular or 2-regular. Here, we characterize all well-covered graphs G satisfying α ( G ) = α ( F G ) for some perfect [ 1 , 2 ] -factor F G . This class contains all well-covered graphs G without isolated vertices of order n with α ⩾ ( n - 1 ) / 2 , and in particular all very well-covered graphs.

New sufficient conditions for the existence of kernels in digraphs

Abstract 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. If every induced subdigraph of D has a kernel, D is said to be a kernel perfect digraph. Minimal non-kernel perfect digraph are called critical kernel imperfect digraph. If F is a set of arcs of D, a semikernel modulo F, S of D is an independent set of vertices of D such that for every z ∈ V ( D ) − S for which there exists an S z − arc of D − F , there also exists an z S − arc in D. In this work new sufficient conditions for a digraph to be a critical kernel imperfect digraph, in terms of semikernel modulo F, are presented.

Graphs with the Erdős–Ko–Rado property

For a graph G vertex v of G and integer r ⩾ 1 , we denote the family of independent r-sets of V ( G ) by I ( r ) ( G ) and the subfamily { A ∈ I ( r ) ( G ) : v ∈ A } by I v ( r ) ( G ) ; such a subfamily is called a star. Then, G is said to be r-EKR if no intersecting subfamily of I ( r ) ( G ) is larger than the largest star in I ( r ) ( G ) . If every intersecting subfamily of I v ( r ) ( G ) of maximum size is a star, then G is said to be strictly r- EKR. We show that if a graph G is r -EKR then its lexicographic product with any complete graph is r-EKR. For any graph G , we define μ ( G ) to be the minimum size of a maximal independent vertex set. We conjecture that, if 1 ⩽ r ⩽ 1 2 μ ( G ) , then G is r-EKR, and if r < 1 2 μ ( G ) , then G is strictly r-EKR. This is known to be true when G is an empty graph, a cycle, a path or the disjoint union of complete graphs. We show that it is also true when G is the disjoint union of a pair of complete multipartite graphs.

Neighborhood union of independent sets and hamiltonicity of claw-free graphs

Let G be a graph, for any u∈V(G), let N(u) denote the neighborhood of u and d(u)=|N(u)| be the degree of u. For any U⊆V(G), let N(U)=∪ u eU N(u), and d(U)=|N(U)|. A graph G is called claw-free if it has no induced subgraph isomorphic to K 1,3. One of the fundamental results concerning cycles in claw-free graphs is due to Tian Feng, et al.: Let G be a 2-connected claw-free graph of order n, and d(u)+d(v)+d(w)≥n−2 for every independent vertex set {u,v,w} of G, then G is Hamiltonian.

Kernel perfect and critical kernel imperfect digraphs structure

A kernel $N$ of a digraph $D$ is an independent set of vertices of $D$ such that for every $w \in V(D)-N$ there exists an arc from $w$ to $N$. If every induced subdigraph of $D$ has a kernel, $D$ is said to be a kernel perfect digraph. Minimal non-kernel perfect digraph are called critical kernel imperfect digraph. If $F$ is a set of arcs of $D$, a semikernel modulo $F$, $S$ of $D$ is an independent set of vertices of $D$ such that for every $z \in V(D)- S$ for which there exists an $Sz-$arc of $D-F$, there also exists an $zS-$arc in $D$. In this talk some structural results concerning critical kernel imperfect and sufficient conditions for a digraph to be a critical kernel imperfect digraph are presented.

Average independence polynomials

The independence polynomial of a graph G is the function i ( G , x ) = ∑ k ⩾ 0 i k x k , where i k is the number of independent sets of vertices in G of cardinality k. We investigate here the average independence polynomial, where the average is taken over all independence polynomials of (labeled) graphs of order n. We prove that while almost every independence polynomial has a nonreal root, the average independence polynomials always have all real, simple roots.

The structure of strong arc-locally semicomplete digraphs

Arc-locally semicomplete digraphs were introduced in (Preprint, No. 10, 1993, Department of Mathematics and Computer Science, University of Southern Denmark) as a common generalization of semicomplete digraphs and semicomplete bipartite digraphs. In this note, we prove that the structure of these digraphs is very closely related to that of semicomplete and semicomplete bipartite digraphs. In fact, we show that if a strong arc-locally semicomplete digraph is neither semicomplete nor semicomplete bipartite, then it is obtained from a directed cycle by substituting independent sets of vertices for each vertex of the cycle. We also identify a new class of digraphs for which the hamiltonian cycle problem seems tractable and non-trivial. As it is the case for arc-locally semicomplete digraphs, this new class of digraphs contains all semicomplete digraphs and all semicomplete bipartite digraphs.

On the Location of Roots of Independence Polynomials

The independence polynomial of a graph G is the function i(G, x) = ? k?0 i k x k, where i k is the number of independent sets of vertices in G of cardinality k. We prove that real roots of independence polynomials are dense in (??, 0], while complex roots are dense in ?, even when restricting to well covered or comparability graphs. Throughout, we exploit the fact that independence polynomials are essentially closed under graph composition.

The structure of strong arc-locally semicomplete digraphs

Arc-locally semicomplete digraphs were introduced in (Preprint, No. 10, 1993, Department of Mathematics and Computer Science, University of Southern Denmark) as a common generalization of semicomplete digraphs and semicomplete bipartite digraphs. In this note, we prove that the structure of these digraphs is very closely related to that of semicomplete and semicomplete bipartite digraphs. In fact, we show that if a strong arc-locally semicomplete digraph is neither semicomplete nor semicomplete bipartite, then it is obtained from a directed cycle by substituting independent sets of vertices for each vertex of the cycle. We also identify a new class of digraphs for which the hamiltonian cycle problem seems tractable and non-trivial. As it is the case for arc-locally semicomplete digraphs, this new class of digraphs contains all semicomplete digraphs and all semicomplete bipartite digraphs.

Independent sets and 2‐factors in edge‐chromatic‐critical graphs

AbstractIn 1968, Vizing made the following two conjectures for graphs which are critical with respect to the chromatic index: (1) every critical graph has a 2‐factor, and (2) every independent vertex set in a critical graph contains at most half of the vertices. We prove both conjectures for critical graphs with many edges, and determine upper bounds for the size of independent vertex sets in those graphs. © 2003 Wiley Periodicals, Inc. J Graph Theory 45: 113–118, 2004

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.