K-independent Set Research Articles

Finding maximum independent set based on multi-stage simulated quantum adiabatic evolution

The ubiquitous nature of the maximum independent set (MIS) problem across diverse fields, including biology and chip design, presents a significant challenge. Recently, quantum algorithms have been recognized for their increased potential in addressing combinatorial optimization problems, including the MIS problem. Several quantum systems, including optical quantum circuits and Rydberg atomic systems, have been designed specifically for the MIS problem. However, elucidating the precise nature of instances that pose challenges for quantum algorithms constitutes a formidable task. In an effort to comprehend this issue, we endeavor to delineate the conditions under which instances become hard to solve by analyzing the Hamiltonian of the k-independent set (k-IS). Furthermore, we present two multi-stage strategies grounded in the simulated quantum adiabatic evolution (SQAE) of Kerr-nonlinear parametric oscillator (KPO) system for identifying the MIS in instances deemed hard to solve. More specifically, we present the algorithm leveraging SQAE of the KPO system for approximating the MIS. Our investigation of the SQAE algorithm reveals that optimal solutions are unattainable under two distinct conditions: (1) when the number of solutions is exponentially smaller to the problem’s scale, and (2) when the counts of non-optimal independent sets (IS) with m overlaps between the optimal k-IS (k>m) is exponentially smaller relative to the problem’s scale. With these two distinctive features, we propose two multi-stage SQAE strategies tailored for hard-to-solve instances. In the first approach, the SQAE algorithm is executed multiple times to identify a set of candidate vertices for the MIS. For each vertex or combinations of vertices in this set, we extract the subgraphs induced by the vertices not adjacent to them. Subsequently, the SQAE algorithm is applied to the subgraph to obtain the (k−1)-IS. In the second multi-stage strategy, we consider each vertex vi in the given graph, generating the subgraph induced by the vertices not adjacent to vi. The SQAE algorithm is then employed to find the (k−1)-IS of the subgraph. Experimental results demonstrate that both multi-stage strategies significantly enhance the performance of the SQAE algorithm for hard-to-solve instances.

A note on the k-restriction problem

Consider a set of demands, each taking length-k strings as input. The k-restriction problem is to construct a small set of length-m strings, such that given any k positions and any demand, there exists a string in the set satisfying the demand at these positions. The k-restriction problem relates to many problems, such as k-independent sets, covering arrays, and many other combinatorial applications. By considering the VC-dimension of demands, we prove bounds independent of the number of demands with the Lovász Local Lemma. As a result, we can prove better bounds for demands with finite VC-dimension.

Complexity results on k-independence in some graph products

For a positive integer k, a subset S of vertices of a graph G is k-independent if each vertex in S has at most k − 1 neighbors in S. We consider k-independent sets in two graph products: Cartesian and complementary prism. We show that the problem of determining k-independence remains NP-complete even for Cartesian products and complementary prisms. Furthermore, we present results on k-independence in the Cartesian product of two paths, known as grid graphs.

Nordhaus–Gaddum-Type Results for the k-Independent Number of Graphs

The concept of [Formula: see text]-independent set, introduced by Fink and Jacobson in 1986, is a natural generalization of classical independence set. A [Formula: see text]-independent set is a set of vertices whose induced subgraph has maximum degree at most [Formula: see text]. The [Formula: see text]-independence number of [Formula: see text], denoted by [Formula: see text], is defined as the maximum cardinality of a [Formula: see text]-independent set of [Formula: see text]. As a natural counterpart of the [Formula: see text]-independence number, we introduced the concept of [Formula: see text]-edge-independence number. An edge set [Formula: see text] in [Formula: see text] is called [Formula: see text]-edge-independent if the maximum degree of the subgraph induced by the edges in [Formula: see text] is less or equal to [Formula: see text]. The [Formula: see text]-edge-independence number, denoted [Formula: see text], is defined as the maximum cardinality of a [Formula: see text]-edge-independent set. In this paper, we study the Nordhaus–Gaddum-type results for the parameter [Formula: see text] and [Formula: see text]. We obtain sharp upper and lower bounds of [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] for a graph [Formula: see text] of order [Formula: see text]. Some graph classes attaining these bounds are also given.

Independence numbers of polyhedral graphs

A polyhedral graph is a 3-connected planar graph. We find the least possible order p(k,a) of a polyhedral graph containing a k-independent set of size a for all positive integers k and a. In the case k=1 and a even, we prove that the extremal graphs are exactly the vertex-face (radial) graphs of maximal planar graphs.

The optimal bound on the 3-independence number obtainable from a polynomial-type method

A k-independent set in a connected graph is a set of vertices such that any two vertices in the set are at distance greater than k in the graph. The k-independence number of a graph, denoted αk, is the size of a largest k-independent set in the graph. Recent results have made use of polynomials that depend on the spectrum of the graph to bound the k-independence number. They are optimized for the cases k=1,2. There are polynomials that give good (and sometimes) optimal results for general k, including case k=3. In this paper, we provide the best possible bound that can be obtained by choosing a polynomial for case k=3 and apply this bound to well-known families of graphs including the Hamming graph.

DAG-Pathwidth: Graph Algorithmic Analyses of DAG-Type Blockchain Networks

This paper analyzes a blockchain network forming a directed acyclic graph (DAG), called a DAG-type blockchain, from the viewpoint of graph algorithm theory. To use a DAG-type blockchain, NP-hard graph optimization problems on the DAG are required to be solved. Although various problems for undirected and directed graphs can be efficiently solved by using the notions of graph parameters, these currently known parameters are meaningless for DAGs, which implies that it is hopeless to design efficient algorithms based on the parameters for such problems. In this work, we propose a novel graph parameter for directed graphs called a DAG-pathwidth, which represents the closeness to a directed path. This is an extension of the pathwidth, a well-known graph parameter for undirected graphs. We analyze the features of the DAG-pathwidth and prove that computing the DAG-pathwidth of a DAG (directed graph in general) is NP-complete. Finally, we propose an efficient algorithm for a variant of the maximum k-independent set problem for the DAG-type blockchain when the DAG-pathwidth of the input graph is small.

The [formula omitted]-independence number of [formula omitted]-connected graphs

A set S⊆V(G) is a k-independent set of a graph G if the distance between every two vertices of S is greater than k. The k-independence number αk(G) of G is the maximum cardinality over all k-independent set in G. In this note, we show that if G is a t-connected graph of order n, then α2k+1(G)≤max{1,n−ttk+1} and α2k(G)≤max{1,ntk+1} for any integer k≥0. Both bounds are sharp. For any integer s≥2,p(G,s)=max{min1≤i≤j≤sd(vi,vj):{v1,…,vs}⊆V(G)},is a generalization of the diameter of G. We determine the exact value of the maximum p(G,s) among all t-connected graphs of order n. This includes the main results of a recent paper by Knor and Škrekovski (On the minimum distance in a k-vertex set in a graph, Appl. Math. Comput. 356 (2019) 99–104) and solve an open problem therein.

Optical Filter-Less WDM for Visible Light Communications Using Defocused MIMO

This paper experimentally investigates, for the first time, a new wavelength-division multiplexing-based visible light communications link based on a defocused non-imaging multiple-input multiple-output (MIMO), which removes the need for tuned optical bandpass filters paired with each receiver. The proposed system is based on using the natural diversity of the individual light emitting diodes (LEDs) within a single light source to generate an H-matrix, which is independent of spatial diversity. We show that, by transmitting K-independent sets of non-return to zero on-and-off keying signals on separate wavelengths, the received superposed symbols can be demultiplexed. The non-imaging MIMO diversity is achieved by considering the power-current characteristics of the light emitting diode, the responsivity of the photodetector array, and the defocused beam spot. The system is empirically verified for K = 3 using red, green, and blue LEDs with Q-factors of 7.66, 7.69, and 4.75 dB, respectively.

Remarks on k-Clique, k-Independent Set and 2-Contamination in Complementary Prisms

Complementary prism graphs arise from the disjoint union of a graph [Formula: see text] and its complement [Formula: see text] by adding the edges of a perfect matching joining pairs of corresponding vertices of [Formula: see text] and [Formula: see text]. Classical graph problems such as Clique and Independent Set were proved to be NP-complete on such a class of graphs. In this work, we study the complexity of both problems on complementary prism graphs from the parameterized complexity point of view. First, we prove that both problems admit a kernel and therefore are fixed-parameter tractable (FPT) when parameterized by the size of the solution, [Formula: see text]. Then, we show that [Formula: see text]-Clique and [Formula: see text]-Independent Set on complementary prisms do not admit polynomial kernel when parameterized by [Formula: see text], unless [Formula: see text]. Furthermore, we address the [Formula: see text]-Contamination problem in the context of complementary prisms. This problem consists in completely contaminating a given graph [Formula: see text] using a minimum set of initially infected vertices. For a vertex to be contaminated, it is enough that at least two of its neighbors are contaminated. The propagation of the contamination follows this rule until no more vertex can be contaminated. It is known that the minimum set of initially contaminated vertices necessary to contaminate a complementary prism of connected graphs [Formula: see text] and [Formula: see text] has cardinality at most [Formula: see text]. In this paper, we show that the tight upper bound for this invariant on complementary prisms is [Formula: see text], improving a result of Duarte et al. (2017).

MAX for [formula omitted]-independence in multigraphs

For a fixed positive integer k, a set S of vertices of a graph or multigraph is called a k-independent set if the subgraph induced by S has maximum degree less than k. The well-known algorithm MAX finds a maximal k-independent set in a graph or multigraph by iteratively removing vertices of maximum degree until what remains has maximum degree less than k. We give an efficient procedure that determines, for a given degree sequence D, the smallest cardinality b(D) of a k-independent set that can result from any application of MAX to any loopless multigraph with degree sequence D. This analysis of the worst case is sharp for each degree sequence D in that there exists a multigraph G with degree sequence D such that some application of MAX to G will result in a k-independent set of cardinality exactly b(D).

Local Algorithms, Regular Graphs of Large Girth, and Random Regular Graphs

We introduce a general class of algorithms and analyse their application to regular graphs of large girth. In particular, we can transfer several results proved for random regular graphs into (deterministic) results about all regular graphs with sufficiently large girth. This reverses the usual direction, which is from the deterministic setting to the random one. In particular, this approach enables, for the first time, the achievement of results equivalent to those obtained on random regular graphs by a powerful class of algorithms which contain prioritised actions. As a result, we obtain new upper or lower bounds on the size of maximum independent sets, minimum dominating sets, maximum k-independent sets, minimum k-dominating sets and maximum k-separated matchings in r-regular graphs with large girth.

The k-independence number of graph products

The concept of k-independent number is a natural generalization of classical independence number. A k-independent set is a set of vertices whose induced subgraph has maximum degree at most k. The k-independence number of G, denoted by αk(G), is defined as the maximum cardinality of a k-independent set of G. In this paper, we study the k-independence number on the lexicographical, strong, Cartesian and direct product and present several upper and lower bounds for these products of graphs.

Extremal Polygonal Chains Concerning Merrified-Simmons Index

ABSTRACTDenote by the set of h-polygonal chains with n congruent regular h-polygons (h ⩾ 6). For any , let ik(An) be the numbers of k-independent sets of An. In this article, we show that ik(Z2n) ⩾ ik(An) ⩾ ik(Z1n) for any k ⩾ 0, with the equalities on the left holding for all k only if An = Z2n, and the equalities on the right holding for all k only if An = Z1n, where Z1n and Z2n are extremal chains of type one and type two (their definitions are given in the main text), respectively. Thus, we extend the main results of (7) to a more general case.

Lower bounds for the independence and [formula omitted]-independence number of graphs using the concept of degenerate degrees

Let G be a graph and v be any vertex of G. We define the degenerate degree of v, denoted by ζ(v) as ζ(v)=maxH:v∈Hδ(H), where the maximum is taken over all subgraphs of G containing the vertex v. We show that the degenerate degree sequence of any graph can be determined by an efficient algorithm. A k-independent set in G is any set S of vertices such that Δ(G[S])≤k. The largest cardinality of any k-independent set is denoted by αk(G). For k∈{1,2,3}, we prove that αk−1(G)≥∑v∈Gmin{1,1/(ζ(v)+(1/k))}. Using the concept of cheap vertices we strengthen our bound for the independence number. The resulting lower bounds improve greatly the famous Caro–Wei bound and also the best known bounds for α1(G) and α2(G) for some families of graphs. We show that the equality in our bound for the independence number happens for a large class of graphs. Our bounds are achieved by Cheap-Greedy algorithms for αk(G) which are designed by the concept of cheap sets. At the end, a bound for αk(G) is presented, where G is a forest and k an arbitrary non-negative integer.

Linearly connected sequences and spectrally optimal dual frames for erasures

In the case that a frame is prescribed for applications and erasures occur in the process of data transmissions, we examine optimal dual frames for the recovery from single erasures. In contrast to earlier papers, we consider the spectral radius of the error operator instead of its operator norm as a measure of optimality. This notion of optimality is natural when the Neumann series is used to recover the original data in an iterative manner. We obtain a complete characterization of spectrally one-erasure optimal dual frames in terms of the redundancy distribution of the prescribed frame. Our characterization relies on the connection between erasure optimal frames and the linear connectivity property of the frame. We prove that the linear connectivity property is equivalent to the intersection dependent property, and is also closely related to the well-known concept of a k-independent set. Additionally, we also establish several necessary and sufficient conditions for the existence of an alternate dual frame to make the iterative reconstruction work.

A characterization of trees for a new lower bound on the k-independence number

Let k be a positive integer and G = (V, E) a graph of order n. A subset S of V is a k-independent set of G if the maximum degree of the subgraph induced by the vertices of S is less or equal to k − 1. The maximum cardinality of a k-independent set of G is the k-independence number βk(G). In this paper, we show that for every graph G, βk(G) ≥ l n + (χ(G) − 1) P v2S(G) min(|Lv| , k − 1) /χ(G) m , where χ(G), s(G) and Lv are the chromatic number, the number of supports vertices and the number of leaves neighbors of v, in the graph G, respectively. Moreover, we characterize extremal trees attaining these bounds.

Some identities for the generalized Fibonacci numbers and the generalized Lucas numbers

In this paper we study some properties of the generalized Fibonacci numbers and the generalized Lucas numbers. These numbers are equal to the total numbers of k-independent sets in special graphs. We give some identities for the generalized Fibonacci numbers and the generalized Lucas numbers, which can be useful also in problems of counting of k-independent sets in graphs.

K-kernels in generalizations of transitive 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 , u 6= v, 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. Quasi-transitive, right-pretransitive and left-pretransitive digraphs are generalizations of transitive digraphs. In this paper the following results are proved: Let D be a right-(left-) pretransitive strong digraph such that every directed triangle of D is symmetrical, then D has a k-kernel for every integer k ≥ 3; the result is also valid for non-strong digraphs in the right-pretransitive case. We also give a proof of the fact that every quasi-transitive digraph has a (k, l)-kernel for every integers k > l ≥ 3 or k = 3 and l = 2. keywords: digraph, kernel, (k, l)-kernel, k-kernel, transitive digraph, quasi-transitive digraph, rightpretransitive digraph, left-pretransitive digraph, pretransitive digraph. AMS Subject Classification: 05C20.

K-Kernels in Multipartite Tournaments

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. An m-partite tournament is an orientation of an m-partite complete graph. In this paper we use a tool for finding sufficient conditions for a digraph to have a k-kernel, the k-closure of a digraph, which is employed to prove that every m-partite tournament has a k-kernel for every k ≥ 4, m ≥ 2, and to give a simple proof of the fact that for k ≥ 2 it is NP-complete to decide if a digraph D has a k-kernel, or not. We also characterize the m-partite tournaments that have a 3-kernel for m ≥ 2 as those m-partite tournaments having a 2-absorbent vertex for the directed cycles of length four.