Maximal Independent Sets Of Vertices Research Articles

Local Structure When All Maximal Independent Sets Have Equal Weight

In many combinatorial situations there is a notion of independence of a set of points. Maximal independent sets can be easily constructed by a greedy algorithm, and it is of interest to determine, for example, if they all have the same size or the same parity. Both of these questions may be formulated by weighting the points with elements of an abelian group, and asking whether all maximal independent sets have equal weight. If a set is independent precisely when its elements are pairwise independent, a graph can be used as a model. The question then becomes whether a graph, with its vertices weighted by elements of an abelian group, is well-covered, i.e., has all maximal independent sets of vertices with equal weight. This problem is known to be co-NP-complete in general. We show that whether a graph is well-covered or not depends on its local structure. Based on this, we develop an algorithm to recognize well-covered graphs. For graphs with n vertices and maximum degree $\Delta$, it runs in linear time if $\Delta$ is bounded by a constant, and in polynomial time if $\Delta = O(\root 3 \of {\log n})$. We mention various applications to areas including hypergraph matchings and radius k independent sets. We extend our results to the problem of determining whether a graph has a weighting which makes it well-covered.

Independent domination in regular graphs

Let G be a simple graph of order n. The independent domination numberi(G) is defined to be the minimum cardinality among all maximal independent sets of vertices of G. Motivated by work of Cockayne et al. (1991) and Cockayne and Mynhardt (1989), we investigate the maximum value of the product of the independent domination numbers of a graph and its complement, as a function of n. In particular we prove that if G is regular then i(G) · i(G) < (n + 14)2/12.68.

Recent examples in the theory of partition graphs

A partition graph is an intersection graph for a collection of subsets of auniversal set S with the property that every maximal independent set of vertices corresponds to a partition of S. Two questions which arose in the study of partition graphs are answered by recently discovered examples. An enumeration of the partition graphs on ten or fewer vertices is provided.

PCBN: a high-performance partitionable circular bus network for distributed systems

The authors present a dynamically partitionable circular bus network (PCBN) and efficient algorithms for maximizing its utilization. In their approach, a distributed network is transformed into a graph, in which a vertex represents a communication request and an edge denotes the conflict between a pair of communication requests. A graph traversal algorithm is applied to the graph to identify some maximal independent sets of vertices. The communication requests corresponding to the vertices of a maximum independent set call proceed in parallel. By computing the expected size of the maximal independent sets of a graph, the improvement ratio of the network can be obtained. The network control and synchronization techniques of PCBN are described in detail. The idling problem in the execution of nonconflicting requests is also discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Combinatorial properties of some macrocyclic polyhex systems

Explicit formulae are given for the number of Clar structures, ζ (which represent maximal independent sets of vertices), of several classes of corona-condensed polyhex systems (primitive coronoids). There is a striking equivalence between expressions for ζ and K, the Kekulé counts for the classes reported by Bergan, Cyvin and Cyvin.

On minimum maximal independent sets of a graph

For a simple graph G, the independent domination number i(G) is defined to be the minimum cardinality among all maximal independent sets of vertices of G. If G has order n and minimum degree δ⩽ n/2, we give upper bounds for i(G) as functions of n and δ, and over part of the range achieve best possible results. In particular, we extend work of Favaron (1988).

Graphs associated with triangulations of lattice polygons

AbstractTwo graphs, the edge crossing graph E and the triangle graph T are associated with a simple lattice polygon. The maximal independent sets of vertices of E and T are derived including a formula for the size of the fundamental triangles. Properties of E and T are derived including a formula for the size of the maximal independent sets in E and T. It is shown that T is a factor graph of edge-disjoint 4-cycles, which gives corresponding geometric information, and is a partition graph as recently defined by the authors and F. Harary.

The complexity of parallel search

This paper studies parallel search algorithms within the framework of independence systems. It is motivated by earlier work on parallel algorithms for concrete problems such as the determination of a maximal independent set of vertices or a maximum matching in a graph and by the general question of determining the parallel complexity of a search problem when an oracle is available to solve the associated decision problem. Our results provide a parallel analog of the self-reducibility process that is so useful in sequential computation. An abstract independence system is specified by a ground set E and a family of subsets of E called the independent sets; it is required that every subset of an independent set be independent. We investigate parallel algorithms for determining a maximal independent set through oracle queries of the form “Is the set A independent?,” as well as parallel algorithms for determining a maximum independent set through queries to a more powerful oracle called a rank oracle. We also study these problems for three special types of independence systems: matroids, graphic matroids, and partition matroids. We derive lower and upper bounds on the deterministic and randomized complexity of these problems. These bounds are sharp enough to give a clear picture of the processor-time trade-offs that are possible, to establish that randomized parallel algorithms are much more powerful than deterministic ones, and to show that even randomized algorithms cannot make effective use of extremely large numbers of processors.

On the Characterization of Fibonacci Numbers as Maximal Independent Sets of Vertices of Certain Trees

Abstract Fibonacci numbers are identified for the first time as maximal independent sets of vertices of certain caterpillar trees. Their relation to king patterns of certain classes of polyomino graphs as well as polyhex graphs is illustrated.

The Number of Maximal Independent Sets in a Tree

We find the largest number of maximal independent sets of vertices that any tree of n vertices can have.

Disjoint cliques and disjoint maximal independent sets of vertices in graphs

In this paper, we find lower bounds for the maximum and minimum numbers of cliques in maximal sets of pairwise disjoint cliques in a graph. By complementation, these yield lower bounds for the maximum and minimum numbers of independent sets in maximal sets of pairwise disjoint maximal independent sets of vertices in a graph. In the latter context, we show by examples that one of our bounds is best possible.

Contributions to the theory of domination, independence and irredundance in graphs

A vertex x in a subset X of vertices of an undirected graph is redundant if its closed neighbourhood is contained in the union of closed neighbourhoods of vertices of X−{ x}. In the context of a communications network, this means that any vertex which may receive communications from X may also be informed from X−{ x}. The lower and upper irredundance numbers ir( G) and IR( G) are respectively the minimum and maximum cardinalities taken over all maximal sets of vertices having no redundancies. The domination number γ(G) and upper domination number Γ( G) are respectively the minimum and maximum cardinalities taken over all minimal dominating sets of G. The independent domination number i( G) and the independence number β( G) are respectively the minimum and maximum cardinalities taken over all maximal independent sets of vertices of G. A variety of inequalities involving these quantities are established and sufficient conditions for the equality of the three upper parameters are given. In particular a conjecture of Hoyler and Cockayne [9], namely i+β⩽2p + 2δ - 2 2pδ , is proved.

On maximal independent sets of vertices in claw-free graphs

A graph is claw-free if: whenever three (distinct) vertices are joined to a single vertex, those three vertices are a nonindependent (nonstable) set. Given a finite claw-free graph with real numbers (weights) assigned to the vertices, we exhibit an algorithm for producing an independent set of vertices of maximum total weight. This algorithm is “efficient” in the sense of J. Edmonds, that is to say, the number of computational steps required is of polynomial (not exponential or factorial) order in n, the number of vertices of the graph. This problem was solved earlier by Edmonds for the special case of “edge-graphs”; our solution is by reducing the more general problem to the earlier-solved special case. Separate attention is given to the case in which all weights are (+1) and thus an independent set is sought which is maximal in the sense of its cardinality.

Graph‐theoretic parameters concerning domination, independence, and irredundance

AbstractA vertex x in a subset X of vertices of an undericted graph is redundant if its closed neighbourhood is contained in the union of closed neighborhoods of vertices of X – {x}. In the context of a communications network, this means that any vertex that may receive communications from X may also be informed from X – {x}. The irredundance number ir (G) is the minimum cardinality taken over all maximal sets of vertices having no redundancies. The domination number γ(G) is the minimum cardinality taken over all dominating sets of G, and the independent domination number i(G) is the minimum cardinality taken over all maximal independent sets of vertices of G. The paper contians results that relate these parameters. For example, we prove that for any graph G, ir (G) &gt; γ(G)/2 and for any grpah Gwith p vertices and no isolated vertices, i(G) ≤ p‐γ(G) + 1 ‐ ⌈(p ‐ γ(G))/γ(G)⌉.

A new method to find out the chromatic partition of a symmetric graph

In this paper an algorithm to find out the chromatic partition of a symmetric graph is described. The algorithm is developed in two stages. In the first stage, all possible maximal independent sets of vertices of the graph ore determined starting from adjacency matrix description of the given graph. Several important results are arrived at about the lower bound on the chromatic number of the graph. Lastly, in the second phase of the algorithm, a minimal cover of the vertices with the maximal independent seta is enumerated by introducing the classical concept of prime implicant covering to obtain the desired chromatic partition of the graph. The proposed algorithm is simple and straightforward and may be programmed on a general purpose digital computer.

Disjoint cliques in regular graphs of degree seven and eight

Received October 15, 1975 1. INTRODUCTION This paper is motivated by Berge’s conjecture that every undirectefl regular graph, except the complement of a complete graph, has at least two disjoint maximal independent sets of vertices. By complementation this is equivalent to the conjecture that every incomplete regular graph has two or more disjoint cliques. In [l], we established this form of the conjecture for regular graphs of degree

Lattice distortion, specific heat, and antiferromagnetic ordering in AuMn

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 a triangle is called a triangulation. It was proved in an earlier paper Finbow et al. (2004) [3] that there are no 5-connected planar well-covered triangulations, and in Finbow et al. (submitted for publication) [4] that there are exactly four 4-connected well-covered triangulations containing two adjacent vertices of degree 4. It is the aim of the present paper to complete the characterization of 4-connected well-covered triangulations by showing that each such graph contains two adjacent vertices of degree 4.