The linear-width of a graph G is de ned to be the smallest integer k such that the edges of G can be arranged in a linear ordering (e1; : : : ; er) in such a way that for every i = 1; : : : ; r 1, there are at most k vertices incident to edges that belong both to fe1; : : : ; eig and to fei+1; : : : ; erg. In this paper, we give a set of 57 graphs and prove that it is the set of the minimal for...

For a hereditary property P let kP(G) denote the number of forbidden subgraphs contained in G. A graph G is said to be weakly Psaturated, if G has the property P and there is a sequence of edges of G, say e1, e2, . . . , el, such that the chain of graphs G = G0 ⊂ G0+e1 ⊂ G1 + e2 ⊂ . . . ⊂ Gl−1 + el = Gl = Kn (Gi+1 = Gi + ei+1) has the following property: kP(Gi+1) > kP(Gi), 0 ≤ i ≤ l − 1. In thi...

Amylases are one of the most important enzymes in present-day biotechnology. The present study was concerned with the production and partial characterization of extracellular amylase from Bacillus amyloliquefaciens P-001. The effect of various fermentation conditions on amylase production through shake-flask culture was investigated. Enzyme production was induced by a variety of starchy substra...

Chordal graphs, also called triangulated graphs, are important in algorithmic graph theory. In this paper we generalise the definition of chordal graphs to the class of directed graphs. Several structural properties of chordal graphs that are crucial for algorithmic applications carry over to the directed setting, including notions like simplicial vertices, perfect elimination orderings, and ch...

AgraphG is a k-leaf power if there is a tree T such that the vertices ofG are the leaves of T and two vertices are adjacent in G if and only if their distance in T is at most k. In this situation T is called a k-leaf root of G. Motivated by the search for underlying phylogenetic trees, the notion of a k-leaf power was introduced and studied by Nishimura, Ragde and Thilikos and subsequently in v...

We introduced an analysis to identify structural characterization of two-dimensional regular and amorphous networks. The analysis was shown to be reliable to determine the global network rigidity and can also identify local floppy regions in the mixture of rigid and floppy regions. The eigenmode analysis explores the structural properties of various networks determined by eigenvalue spectra. It...

In this paper we investigate the structural properties of k-path separable graphs, that are the graphs that can be separated by a set of k shortest paths. We identify several graph families having such path separability, and we show that this property is closed under minor taking. In particular we establish a list of forbidden minors for 1-path separable graphs.

The b-chromatic number (G) of a graphG is defined as the largest number k for which the vertices of G can be colored with k colors satisfying the following property: for each i, 1 i k, there exists a vertex xi of color i such that for all j = i, 1 j k there exists a vertex yj of color j adjacent to xi . A graph G is b-perfect if each induced subgraph H of G has (H) = (H), where (H) is the chrom...

In this paper we investigate the ratio of the connected vertex cover number to the vertex cover number in graphs, called the Price of Connectivity (PoC). For general graphs, this ratio is strictly bounded by 2. We prove that for every (P5, C5, C4)-free graph the ratio equals 1. We prove also that for every (P5, C4)-free graph the ratio is bounded by 4/3 and that for every (P7, C6,∆1,∆2)-free gr...