We introduce the concept of combed graphs and present an ear decomposition theorem for this class of graphs. This theorem includes the well known ear decomposition theorem for matching covered graphs proved by Lovász and Plummer. Then we use the ear decomposition theorem to show that any two edges of a 2-connected combed graph lie in a balanced circuit of an equivalent combed graph. This result...

We provide a family of examples of graph manifolds which admit taut foliations, but no R-covered foliations. We also show that, with very few exceptions, R-covered foliations are taut.

A graph is called matching covered if for its every edge there is a maximum matching containing it. It is shown that minimal matching covered graphs contain a perfect matching.

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 all its maximal stable sets have the same size, denoted by α(G) (M. D. Plummer, 1970). If sk denotes the number of stable sets of cardinality k in graph G, and α(G) is the size of a maximum stable set, then I(G;x) = α(G) ∑ k=0 skx k is the independence polynomial of G (I. Gutman and F. Harary, 1983). J. I. Brown, K. Dilcher and R. J. Nowakowski (2000) conjectured th...

A graph G is well-covered if every maximal independent set has the same cardinality q. Let ik(G) denote the number of independent sets of cardinality k in G. Brown, Dilcher, and Nowakowski conjectured that the independence sequence (i0(G), i1(G), . . . , iq(G)) was unimodal for any well-covered graph G with independence number q. Michael and Traves disproved this conjecture. Instead they posite...

Let G be a well covered graph, that is, all maximal independent sets of G have the same cardinality, and let ik denote the number of independent sets of cardinality k in G. We investigate the roots of the independence polynomial i(G, x)= ∑ ik xk . In particular, we show that if G is a well covered graph with independence number β, then all the roots of i(G, x) lie in in the disk |z| ≤β (this is...

A graph is said to be well-covered if all its maximal independent sets are of the same size. In 1999, Yamashita and Kameda introduced a subclass of well-covered graphs, called localizable graphs and defined as graphs having a partition of the vertex set into strong cliques, where a clique in a graph is strong if it intersects all maximal independent sets. Yamashita and Kameda observed that all ...

A graph G is well-covered if every maximal independent set of maximum. (k,ℓ)-partition a partition its vertex into k sets and ℓ cliques. (k,ℓ)-well-covered it admits (k,ℓ)-partition. The recognition graphs polynomial-time solvable for the cases (0,1), (0,2), (1,0), (1,1), (1,2), (2,0), hard, otherwise. In sandwich problem property Π, we are given pair G1=(V,E1) G2=(V,E2) with E1⊆E2, asked wheth...

A graph with n vertices is well-covered if every maximal independent set is a maximum independent set, and very well-covered if every maximal independent set has size n ⁄2. In this paper, we study these graphs from an algorithmic complexity point of view. We show that wellcovered graph recognition is co-NP-complete and that several other problems are NP-complete for well-covered graphs. A numbe...