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...

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. ...

The independence polynomial i(G, x) of a graph G is the generating function of the numbers of independent sets of each size. A graph of order n is very well-covered if every maximal independent set has size n/2. Levit and Mandrescu conjectured that the independence polynomial of every very well-covered graph is unimodal (that is, the sequence of coefficients is nondecreasing, then nonincreasing...

The Hamiltonian cycle problem in digraph is mapped into a matching cover bipartite graph. Based on this mapping, it is proved that determining existence a Hamiltonian cycle in graph is O(n). Abstract. Hamiltonian Cycle, Z-mapping graph, complexity, decision, matching covered, optimization Hamiltonian Cycle, Z-mapping graph, complexity, decision, matching covered, optimization

A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function w is defined on its vertices. Then G is w-well-covered if all maximal independent sets are of the same weight. For every graph G, the set of weight functions w such that G is w-well-covered is a vector space. Given an input claw-free graph G, we present an O (

Motivated by the definition of the edge elimination polynomial of a graph we define the covered components polynomial counting spanning subgraphs with respect to their number of components, edges and covered components. We prove a recurrence relation, which shows that both graph polynomials are substitution instances of each other. We give some properties of the covered components polynomial an...

A graph is well-covered if every maximal independent set has the same cardinality. The recognition problem of well-covered graphs is known to be co-NP-complete. Let w be a weight function defined on the vertices of G. Then G is w-well-covered if all maximal independent sets of G are of the same weight. The set of weight functions w for which a graph is w-well-covered is a vector space. We prove...

A graph is well-covered if every maximal independent set has the same cardinality. The recognition problem of well-covered graphs is known to be co-NP-complete. Let w be a weight function defined on the vertices of G. Then G is w-well-covered if all maximal independent sets of G are of the same weight. The set of weight functions w for which a graph is w-well-covered is a vector space. We prove...

A snake in a graph G is defined to be a closed path in G without proper chords. Let Kd n be the product of d copies of the complete graph Kn. Wojciechowski [13] proved that for any d ≥ 2 the hypercube Kd 2 can be vertex covered with at most 16 disjoint snakes. Alsardary [6] proved that for any odd integer n ≥ 3,d ≥ 2 the graph Kd n can be vertex covered with 2n 3 snakes. We show that for any ev...

A graph G is a Zm-well-covered graph if |I1| ≡ |I2| (modm) for all maximal independent sets I1 and I2 in V (G) [3]. The recognition problem of Zm-well-covered graphs is a Co-NP-Complete problem. We give a characterization of Zm-well-covered graphs for chordal, simplicial and circular arc graphs. c © 2001 Elsevier Science B.V. All rights reserved.