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...

It is proven that if G is a 3-connected claw-free graph which is also Z3-free (where Z3 is a triangle with a path of length 3 attached), P6-free (where P6 is a path with 6 vertices) or H1-free (where H1 consists of two disjoint triangles connected by an edge), then G is hamiltonian-connected. Also, examples will be described that determine a finite family of graphs L such that if a 3-connected ...

Consider a tree network that has been contaminated by a persistent and active virus: when infected, a network site will continuously attempt to spread the virus to all its neighbours. The decontamination problem is that of disinfecting the entire network using a team of mobile antiviral system agents, called cleaners, avoiding any recontamination of decontaminated areas. A cleaner is able to de...

A (1,≤ 2)-identifying code is a subset of the vertex set C of a graph such that each pair of vertices intersects C in a distinct way. This has useful applications in locating errors in multiprocessor networks and threat monitoring. At the time of writing, there is no simply-stated rule that will indicate if a graph is (1,≤ 2)-identifiable. As such, we discuss properties that must be satisfied b...

We design an O(n) algorithm to find a minimum weighted coloring of a (P5, P 5)free graph. Furthermore, the same technique can be used to solve the same problem for several classes of graphs, defined by forbidden induced subgraphs, such as (diamond, co-diamond)-free graphs.