‎an  oriented perfect path double cover (oppdc) of a‎ ‎graph $g$ is a collection of directed paths in the symmetric‎ ‎orientation $g_s$ of‎ ‎$g$ such that‎ ‎each arc‎ ‎of $g_s$ lies in exactly one of the paths and each‎ ‎vertex of $g$ appears just once as a beginning and just once as an‎ ‎end of a path‎. ‎maxov{'a} and ne{v{s}}et{v{r}}il (discrete‎ ‎math‎. ‎276 (2004) 287-294) conjectured that ...

A perfect transversal of a graph is a set of vertices that meets any maximal clique, or any maximal stable set of the graph in exactly one vertex. In this paper we give some properties of perfect transversals in graph products. One of these properties is the characterization of the existence of perfect transversals in some graph products via perfect transversals of the factors. As an applicatio...

We give a polynomial time algorithm that given a graph which admits a bisection cutting a fraction (1 − ε) of edges, finds a bisection cutting a (1 − g(ε)) fraction of edges where g(ε) → 0 as ε→ 0. One can take g(ε) = O( 3 √ ε log(1/ε)). Previously known algorithms for Max Bisection could only guarantee finding a bisection that cuts a fraction of edges bounded away from 1 (in fact less than 3/4...

A graph is perfect if, in all its induced subgraphs, the size of a largest clique is equal to the chromatic number. Examples of perfect graphs include bipartite graphs, line graphs of bipartite graphs and the complements of such graphs. These four classes of perfect graphs will be called basic. In 1960, Berge formulated two conjectures about perfect graphs, one stronger than the other. The weak...

The intersection graph $mathbb{Int}(A)$ of an $S$-act $A$ over a semigroup $S$ is an undirected simple graph whose vertices are non-trivial subacts of $A$, and two distinct vertices are adjacent if and only if they have a non-empty intersection. In this paper, we study some graph-theoretic properties of $mathbb{Int}(A)$ in connection to some algebraic properties of $A$. It is proved that the fi...

Let β(G), Γ(G) and IR(G) be the independence number, the upper domination number and the upper irredundance number, respectively. A graph G is called Γperfect if β(H) = Γ(H), for every induced subgraph H of G. A graph G is called IR-perfect if Γ(H) = IR(H), for every induced subgraph H of G. In this paper, we present a characterization of Γ-perfect graphs in terms of some family of forbidden in...

The purpose of this paper is to further the study of weakly injective and weakly projective modules as a generalization of injective and projective modules. For a locally q.f.d. module M , there exists a module K ∈ σ[M ] such that K ⊕N is weakly injective in σ[M ], for any N ∈ σ[M ]. Similarly, if M is projective and right perfect in σ[M ], then there exists a module K ∈ σ[M ] such that K ⊕ N i...

A Set S of vertices in a graph G(V,E) is called dominating set,if every vertex v V id either an element or adjacent to S. vertice Total set if S, total G with no isolated subset the such that In this paper we will find perfect Interval graph. We also out induced subgraph regarding which were splitted.