Abstract Motivated by analogous questions in the setting of Steiner triple systems and Latin squares, Nenadov, Sudakov Wagner [Completion deficiency problems, Journal Combinatorial Theory Series B, 2020] recently introduced notion graph . Given a global spanning property $\mathcal P$ $G$ , $\text{def}(G)$ with respect to is smallest non-negative integer t such that join $G*K_t$ has In particula...

A graph is k-extendable if every independent set of size k is contained in a maximum independent set. This generalizes the concept of a B-graph (i.e. I-extendable graph) introduced by Berge and the concept of a well-covered graph (i.e. k-extendable for every integer k) introduced by Plummer. For various graph families we present some characterizations of well-covered and k-extendable graphs. We...

A CIS graph is a graph in which every maximal stable set and every maximal clique intersect. A graph is well-covered if all its maximal stable sets are of the same size, co-well-covered if its complement is well-covered, and vertex-transitive if, for every pair of vertices, there exists an automorphism of the graph mapping one to the other. We show that a vertex-transitive graph is CIS if and o...

A graph is well-covered if all its maximal independent sets are of the same size (M. D. Plummer, 1970). A well-covered graph (with at least two vertices) is 1-wellcovered if the deletion of every vertex leaves a graph which is well-covered as well (J. W. Staples, 1975). In this paper, we provide new characterizations of 1-well-covered graphs, which we further use to build 1-well-covered graphs ...

A graph is well-covered if all its maximal independent sets are of the same cardinality [25]. If G is a well-covered graph, has at least two vertices, and G − v is well-covered for every vertex v, then G is a 1-well-covered graph [26]. We call G a λ-quasi-regularizable graph if λ · |S| ≤ |N (S)| for every independent set S of G. The independence polynomial I(G;x) is the generating function of i...

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 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