A graph is well-covered if every independent set can be extended to a maximum independent set. We show that it is co-NP-complete to determine whether an arbitrary graph is well-covered, even when restricted to the family of circulant graphs. Despite the intractability of characterizing the complete set of well-covered circulant graphs, we apply the theory of independence polynomials to show tha...

The $k$-token graph $T_k(G)$ is the whose vertices are $k$-subsets of a $G$, with two adjacent if their symmetric difference an edge $G$. We explore when well-covered graph, that is, all its maximal independent sets have same cardinality. For bipartite graphs we classify well-covered. arbitrary show $T_2(G)$ well-covered, then girth $G$ at most four. include upper and lower bounds on independen...

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...

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 inflation $G_{I}$ of a graph $G$ with $n(G)$ vertices and $m(G)$ edges is obtained from $G$ by replacing every vertex of degree $d$ of $G$ by a clique, which is isomorph to the complete graph $K_{d}$, and each edge $(x_{i},x_{j})$ of $G$ is replaced by an edge $(u,v)$ in such a way that $uin X_{i}$, $vin X_{j}$, and two different edges of $G$ are replaced by non-adjacent edges of $G_{I}$. T...

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...