An (r, `)-partition of a graph G is a partition of its vertex set into r independent sets and ` cliques. A graph is (r, `) if it admits an (r, `)-partition. A graph is well-covered if every maximal independent set is also maximum. A graph is (r, `)-well-covered if it is both (r, `) and well-covered. In this paper we consider two different decision problems. In the (r, `)-Well-Covered Graph prob...

The stability number α(G) of the graph G is the size of a maximum stable set of G. If sk denotes the number of stable sets of cardinality k in graph G, then I(G;x) = α(G)

T. S. Michael and N. Traves (2002) provided examples of wellcovered graphs whose independence polynomials are not unimodal. A. Finbow, B. Hartnell and R. J. Nowakowski (1993) proved that under certain conditions, any well-covered graph equals G∗ for some G, where G∗ is the graph obtained from G by appending a single pendant edge to each vertex of G. Y. Alavi, P. J. Malde, A. J. Schwenk and P. E...

Laser range imaging is an upcoming data source for many fields of research. Although many methods are proposed for finding surfaces and classifying points, finding the bounds of these surfaces is still a difficult problem. We propose a method for finding, in a point set of size n, a rectangle that contains a predefined subset of the points and none of the other points in O(n log n) time. Furthe...

A graph G is well-covered if every maximal independent set has the same cardinality. Let sk denote the number of independent sets of cardinality k, and define the independence polynomial of G to be S(G, z) = ∑ skz k. This paper develops a new graph theoretic operation called power magnification that preserves well-coveredness and has the effect of multiplying an independence polynomial by zc wh...

A graph is well-covered if every maximal independent set has the same cardinality, namely the vertex independence number. We answer a question of Topp and Volkmann [5] and prove that if the Cartesian product of two graphs is well-covered, then at least one of them must be well-covered.

A graph G is well covered, if any two maximal independent sets of G have the same number of vertices. A graph is called a block-cactus graph if each block is complete or a cycle. In this paper we characterize the well covered block-cactus graphs. 1. TERMINOLOGY AND INTRODUCTION In this paper we consider finite, undirected, and simple graphs G with the vertex set V(G). The degree d(x, G) of a ve...

A maximum independent set of vertices in a graph is a set of pairwise nonadjacent vertices of largest cardinality α. Plummer [14] defined a graph to be well-covered, if every independent set is contained in a maximum independent set of G. One of the most challenging problems in this area, posed in the survey of Plummer [15], is to find a good characterization of well-covered graphs of girth 4. ...

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