Aslam presents an algorithm he claims will count the number of perfect matchings in any incomplete bipartite graph with an algorithm in the function-computing version of NC, which is itself a subset of FP. Counting perfect matchings is known to be #P-complete; therefore if Aslam's algorithm is correct, then NP=P. However, we show that Aslam's algorithm does not correctly count the number of per...

An edge of a graph is called critical, if deleting it the stability number of the graph increases, and a nonedge is called co-critical, if adding it to the graph the size of the maximum clique increases. We prove in this paper, that the minimal imperfect graphs containing certain configurations of two critical edges and one co-critical nonedge are exactly the odd holes or antiholes. Then we ded...

Let γ(G), i(G), γS(G) and iS(G) denote the domination number, the independent domination number, the strong domination number and the independent strong domination number of a graph G, respectively. A graph G is called γi-perfect (domination perfect) if γ(H) = i(H), for every induced subgraph H of G. The classes of γγS-perfect, γSiS-perfect, iiS-perfect and γiS-perfect graphs are defined analog...

Our graphs are “Michigan” except that they have vertices and edges rather than points and lines. If G is a graph, then y1 = y1 (G) denotes the number of its vertices, ar = (Y(G) denotes the size of its largest stable (independent) set of vertices and o = o(G) denotes the size of its largest clique. The graphs that we are interested in have the following three properties: (i) n =0X0+1, (ii) ever...

The class of even-hole-free graphs is structurally quite similar to the class of perfect graphs, which was the key initial motivation for their study. The techniques developed in the study of even-hole-free graphs were generalized to other complex hereditary graph classes, and in the case of perfect graphs led to the famous resolution of the Strong Perfect Graph Conjecture and their polynomial ...

A b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbor in all other color classes, and the b-chromatic number of a graph G is the largest integer k such that G admits a b-coloring with k colors. A graph is b-perfect if the b-chromatic number is equal to the chromatic number for every induced subgraph of G. We prove that a graph is ...

A graph G with a perfect matching is Pfaffian if it admits an orientation D such that every central cycle C (i.e. C is of even size and G − V (C) has a perfect matching) has an odd number of edges oriented in either direction of the cycle. It is known that the number of perfect matchings of a Pfaffian graph can be computed in polynomial time. In this paper, we show that every embedding of a Pfa...

In early 70s Berge conjectured that any bridgeless cubic graph contains five perfect matchings such that each edge belongs to at least one of them. In 1972 Fulkerson conjectured that, in fact, we can find six perfect matchings containing each edge exactly twice. By introducing the concept of an r-graph (a remarkable generalization of one of bridgeless cubic graph) Seymour in 1979 conjectured th...

We consider the question of the computational complexity of coloring perfect graphs with some precolored vertices. It is well known that a perfect graph can be colored optimally in polynomial time. Our results give a sharp border between the polynomial and NP-complete instances, when precolored vertices occur. The key result on the polynomially solvable cases includes a good characterization th...

A graph admitting a perfect matching has the Perfect-Matching-Hamiltonian property (for short PMH-property) if each of its matchings can be extended to Hamiltonian cycle. In this paper we establish some sufficient conditions for $G$ in order guarantee that line $L(G)$ PMH-property. particular, prove happens when is (i) with maximum degree at most $3$, (ii) complete graph, or (iii) an arbitraril...