A graph is called well-covered if every maximal independent set of vertices of G is a maximum independent set; recall that S is independent if no two of its vertices are adjacent. In this paper we define the concept of well-covered by monochromatic paths graphs which is a variation of well-covered graphs. We consider some classical constructions of graphs: G-join of graphs and duplication of a ...

AIM To compare alcohol purchasing and consumption by ill drinkers in Edinburgh with wider alcohol sales in Scotland. DESIGN Cross-sectional. SETTING Two hospitals in Edinburgh in 2008/09. PARTICIPANTS A total of 377 patients with serious alcohol problems; two-thirds were in-patients with medical, surgical or psychiatric problems due to alcohol; one-third were out-patients. MEASUREMENTS ...

Let G be a graph. A set S of vertices in G dominates the graph if every vertex of G is either in S or a neighbor of a vertex in S. Finding a minimal cardinality set which dominates the graph is an NP-complete problem. The graph G is well-dominated if all its minimal dominating sets are of the same cardinality. The complexity status of recognizing well-dominated graphs is not known. We show that...

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) = s0 + s1x + ... + sαx α is the independence polynomial of G [12], where α = α(G) is the size of a maximum stable set. In this paper we prove that I(G;−1) satisfies |I(G;−1)| ≤ 2, where ν(G) equals the cyclomatic number of G, and the...

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...

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 circulant is a Cayley graph over a cyclic group. A well-covered graph is a graph in which all maximal stable sets are of the same size α = α(G), or in other words, they are all maximum. A CIS graph is a graph in which every maximal stable set and every maximal clique intersect. It is not difficult to show that a circulant G is a CIS graph if and only if G and its complement G are both well-co...

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...