K-independent Set Research Articles

Cyclically k-partite digraphs and k-kernels

Let D be a digraph, V (D) and A(D) will denote the sets of vertices and arcs of D, respectively. A (k, l)-kernel N of D is a k-independent set of vertices (if u, v ∈ N then d(u, v), d(v, u) ≥ k) and l-absorbent (if u ∈ V (D) − N then there exists v ∈ N such that d(u, v) ≤ l). A k-kernel is a (k, k − 1)-kernel. A digraph D is cyclically k-partite if there exists a partition {Vi} i=0 of V (D) such that every arc in D is a ViVi+1-arc (mod k). We give a characterization for an unilateral digraph to be cyclically k-partite through the lengths of directed cycles and directed cycles with one obstruction, in addition we prove that such digraphs always have a k-kernel. A study of some structural properties of cyclically k-partite digraphs is made which bring interesting consequences, e.g., sufficient conditions for a digraph to have k-kernel; a generalization of the well known and important theorem that states if every cycle of a graph G has even length, then G is bipartite (cyclically 2-partite), we prove that if every cycle of a graph G has length ≡ 0 (mod k) then G is cyclically k-partite; and a generalization of another well known result about bipartite digraphs, a strong digraph D is bipartite if and only if every directed cycle has even length, we prove that an unilateral digraphD is bipartite if and only if every directed cycle with at most one obstruction has even length. keywords: digraph, kernel, (k, l)-kernel, k-kernel, cyclically k-partite. AMS Subject Classification: 05C20.

On the existence of k-kernels in digraphs and in weighted digraphs

Let D be a digraph, V(D) and A(D) will denote the sets of vertices and arcs of D, respectively.A (k, l)-kernel N of D is a k-independent set of vertices (if u, v ∊ N then d(u, v), d(v, u) ≥ k) and l-absorbent (if u ∊ V(D) \\ N then exists v ∊ N such that d(u, v) ≤ l). A k-kernel is a (k, k - 1)-kernel. We propose an extension of the definition of (k, l)-kernel to (arc-) weighted digraphs, verifying which of the existing results for k-kernels are valid in this extension. If D is a digraph and w: A(D) → Z is a weight function for the arcs of D, we can restate the problem of finding a k-kernel in the following way. If 풞 is a walk in D, the weight of 풞 is defined as . A subset S ⊆ V(D) is (k, w)-independent if, for every u, v ∊ S there does not exist an uv-directed path of weight less than k. A subset S ⊆ V(D) will be (l, w)-absorbent if, for every u ⊆ V(D) \\ S, there exists an uS-directed path of weight less than or equal to l. A subset N ⊆ V(D) is a (k, l, w)-kernel if it is (k, w)-independent and (l, w)-absorbent. We prove, among other results, that every transitive digraph has a (k, k - l, w)-kernel for every k, that if T is a tournament and for every a ⊆ A(T), then T has a (k, w)-kernel and that if every directed cycle in a quasi-transitive digraph D has weight ≤ , then D has a (k, w)-kernel.Also, we let the weight function to have an arbitrary group as codomain and propose another variation of the concept of k-kernel.

K-independence stable graphs upon edge removal

Let k be a positive integer and G = (V (G); E(G)) a graph. A subset S of V (G) is a k-independent set of G if the subgraph induced by the vertices of S has maximum degree at most k 1. The maximum cardinality of a k-independent set of G is the k-independence number k(G). A graph G is called k -stable if k(G e) = k(G) for every edge e of E(G). First we give a necessary and sucien t condition for k -stable graphs. Then we establish four equivalent conditions for k -stable trees.

Maximal k-independent sets in graphs

A subset of vertices of a graph G is k-independent if it induces in G a subgraph of maximum degree less than k. The minimum and maximum cardinalities of a maximal k-independent set are respectively denoted ik(G) and k(G). We give some relations between k(G) and j(G) and between ik(G) and ij(G) for j 6 k. We study two families of extremal graphs for the inequality i2(G) i(G) + (G). Finally we give an upper bound on i2(G) and a lower bound when G is a cactus.

Extremal double hexagonal chains with respect to k-matchings and k-independent sets

“Double hexagonal chains” can be considered as benzenoids constructed by successive fusions of successive naphthalenes along a zig-zag sequence of triples of edges as appear on opposite sides of each naphthalene unit. In this paper, we discuss the numbers of k-matchings and k-independent sets of double hexagonal chains, as well as Hosoya indices and Merrifield–Simmons indices, and obtain some extremal results: among all the double hexagonal chains with the same number of naphthalene units, (a) the double linear hexagonal chain has minimal k-matching number and maximal k-independent set number and (b) the double zig-zag hexagonal chain has maximal k-matching number and minimal k-independent set number, which are extensions to hexagonal chains [L. Zhang and F. Zhang, Extremal hexagonal chains concerning k-matchings and k-independent sets, J. Math. Chem. 27 (2000) 319–329].

The Cardinality of Sets of k-Independent Vectors over Finite Fields

A set of vectors is k-independent if all its subsets with no more than k elements are linearly independent. We obtain a result concerning the maximal possible cardinality Ind q (n, k) of a k-independent set of vectors in the n-dimensional vector space F q n over the finite field F q of order q. Namely, we give a necessary and sufficient condition for Ind q (n, k) = n + 1. We conclude with some pertinent remarks re applications of our results to codes, graphs and hypercubes.

Large k-independent sets of regular graphs

Large k-independent sets of regular graphs

Maximum weight k-independent set problem on permutation graphs

Based on a Directed Acyclic Graph approach, an O(kn 2) time sequential algorithm is presented to solve the maximum weight k-independent set problem on weighted-permutation graphs. The weights considered here are all non-negative and associated with each of the n vertices of the graph. This problem has many applications in practical problems like k-machines job scheduling problem, k-colourable subgraph problem, VLSI design and routing problem.

Powers of geometric intersection graphs and dispersion algorithms

We study powers of certain geometric intersection graphs: interval graphs, m-trapezoid graphs and circular-arc graphs. We define the pseudo-product, (G,G′)→G∗G′, of two graphs G and G′ on the same set of vertices, and show that G∗G′ is contained in one of the three classes of graphs mentioned here above, if both G and G′ are also in that class and fulfill certain conditions. This gives a new proof of the fact that these classes are closed under taking power; more importantly, we get efficient methods for computing the representation for G k if k⩾1 is an integer and G belongs to one of these classes, with a given representation sorted by endpoints. We then use these results to give efficient algorithms for the k-independent set, dispersion and weighted dispersion problem on these classes of graphs, provided that their geometric representations are given.

An Efficient Algorithm for Finding a Maximum Weight k-Independent Set on Trapezoid Graphs

The maximum weight k-independent set problem has applications in many practical problems like k-machines job scheduling problem, k-colourable subgraph problem, VLSI design layout and routing problem. Based on DAG (Directed Acyclic Graph) approach, an O(kn2) time sequential algorithm is designed in this paper to solve the maximum weight k-independent set problem on weighted trapezoid graphs. The weights considered here are all non-negative and associated with each of the n vertices of the graph.

αk- and γk-stable graphs

A set I of vertices of a graph G is k-independent if the distance between every two vertices of I is at least k+1. The k-independence number, α k ( G), is the cardinality of a maximum k-independent set of G. A set D of vertices of G is k-dominating if every vertex in V( G)− D is at distance at most k from some vertex in D. The k-domination number, γ k ( G), is the cardinality of a minimum k-dominating set of G. A graph G is α k -stable ( γ k -stable) if α k ( G− e)= α k ( G) ( γ k ( G− e)= γ k ( G)) for every edge e of G. We establish conditions under which a graph is α k - and γ k -stable. In particular, we give constructive characterizations of α k - and γ k -stable trees.

Extremal hexagonal chains concerning k-matchings and k-independent sets

Denote by ℬ n the set of the hexagonal chains with n hexagons. For any B n ∈ℬ n , let m k (B n ) and i k (B n ) be the numbers of k-matchings and k-independent sets of B n , respectively. In the paper, we show that for any hexagonal chain B n ∈ℬ n and for any k≥0, m k (L n )≤m k (B n )≤m k (Z n ) and i k (L n )≥i k (B n )≥i k (Z n ), with left equalities holding for all k only if B n =L n , and the right equalities holding for all k only if B n =Z n , where L n and Z n are the linear chain and the zig-zag chain, respectively. These generalize some related results known before.

A sequential algorithm for finding a maximum weightK-independent set on interval graphs

In this paper an O(kn√logc+γ) time algorithm is presented to solve the maximum weight k-independent set problem on an interval graph with n vertices and non-negative integer weights, where c is the weight of the longest path of the interval graph and γ is the total size of all maximal cliques, given its interval representation. If the intervals are not sorted then considering the time for sorting the time complexity is of O(nlogn+kn √logc+γ). Using this algorithm the maximum weight 2-independent set problem for an interval graph with n vertices can be solved in O(n √logc + γ) time. The best known previous algorithm for 2-independent set problem requires O(n 2) time.

On unique independent sets in graphs

For a nonnegative integer k, a subset I of the vertex set V( G) of a simple graph G is said to be k-independent if I is independent and every independent subset I′ of G with | I′|⩾| I|−( k−1) is a subset of I. A set I of vertices is called a strong k-independent set of G if I is k-independent and the set V( G)− I is independent in G. First we give several characterizations of k-independent sets for some classes of graphs. Then we characterize trees which have (strong) k-independent sets. Finally, we obtain lower bounds on the number of edges in graphs which have k-independent sets.

An efficient algorithm for finding a maximum weight 2-independent set on interval graphs

In this paper, we introduce an O( n) time algorithm to solve the maximum weight independent set problem on an interval graph with n vertices given its interval representation with sorted endpoints list. Based on this linear algorithm, we design an O( n 2) time algorithm using O( n 2) space to solve the maximum weight 2-independent set problem on an interval graph with n vertices. With a slight extension and modification of our algorithm, the maximum weight k-independent set problem on an interval graph with n vertices can be solved in O( n k ) time using O( n k ) space.

A polynomial algorithm for constructing families of k-independent sets

The present paper describes an algorithm for constructing families of k-independent subsets F k of {1, 2,…, n} with | F k |⩾2 c kn , where c k = d (k−1)2 k and d is a certain constant. The algorithm has a polynomial complexity with respect to the size of the family constructed.

Explicit construction of exponential sized families of k-independent sets

Error correcting codes are used to describe explicit collections F k of subsets of {1,2,...; n}, with | F k | > 2 c k · n ( C k > 0), such that for any selections A, B of k 2 and k 1 of members of F k with k 1 + k 2 = k, there are elements in all the members of A and not in the members of B. This settles a problem of Kleitman and Spencer and a similar problem of Kleimman, Shearer and Sturtevant.