A virus is a local configuration that, if present in a graph or a digraph, forbids these graphs or digraphs to have a specific property. The aim of this article is to sketch the evolution of the virus theory from its birth in 1991. Moreover some new results and open questions are given. The properties with its known viruses, that will be discussed in this work, are the following: hamiltonian, t...

Let D s (n; h, d = k) denote the number of h-strongly connected digraphs of order n and diameter equal to k. In this paper it is shown that: i) Ds(n; 11" d = 3) = 4(~) (3/4 + o(I))n for every fixed h ;::: 1; ii) Ds(n; h, d = 4) = 4(~) (2h 2 + 2-2 + o(I))n for every fixed h ;::: 2; iii) Ds(n; 11" d = k) = 4(~) ((2h+l I)2-kh+:3h-2 + o(I))n for every fixed h ;::: 1 and k ;::: 5. Similar asymptotic...

A digraph D is connected if the underlying undirected graph of D is connected. A subgraph H of an acyclic digraph D is convex if there is no directed path between vertices of H which contains an arc not in H. We find the minimum and maximum possible number of connected convex subgraphs in a connected acyclic digraph of order n. Connected convex subgraphs of connected acyclic digraphs are of int...

An orientation of a complete graph is a tournament, and an orientation of a complete n-partite graph is an n-partite tournament. For each n 2:: 4, there exist examples of strongly connected n-partite tournament without any strongly connected subtournaments of order p 2:: 4. If D is a digraph, then let d+ (x) be the out degree and d(x) the indegree of the vertex x in D. The minimum (maximum) out...

it has been proved that sphericity testing for digraphs is an np-complete problem. here, we investigate sphericity of 3-connected single source digraphs. we provide a new combinatorial characterization of sphericity and give a linear time algorithm for sphericity testing. our algorithm tests whether a 3-connected single source digraph with $n$ vertices is spherical in $o(n)$ time.

The eccentricity e(v) of a vertex v in a strongly connected digraph G is the maximum distance from v. The eccentricity sequence of a digraph is the list of eccentricities of its vertices given in nondecreasing order. A sequence of positive integers is a digraphical eccentric sequence if it is the eccentricity sequence of some digraph. A set of positive integers S is a digraphical eccentric set ...

Let D be a strongly connected digraph. The average distance σ̄(v) of vertex v is the arithmetic mean distances from to all other vertices D. remoteness ρ(D) and proximity π(D) are maximum minimum D, respectively. We obtain sharp upper lower bounds on as function order n describe extreme digraphs for bounds. also such strong tournaments. show that tournament T, we have π(T)=ρ(T) if only T regular...

This lecture covers the proof of the Bessy-Thomassé Theorem, formerly known as the Gallai Conjecture. Also, we discuss the cyclic stable set polytope, and show that it is totally dual integral (TDI) (see lecture 5 for more on TDI systems of inequalities). In this section we provide a brief recap of some definitions we saw in the previous lecture. Also we answer a question that remained unanswer...

We propose a novel method for Web page grouping based only on hyperlink information. Because of the explosive growth of the World Wide Web, page grouping is expected to provide a general grasp of the Web for effective information search and netsurfing. The Web can be regarded as a gigantic digraph where pages are vertices and links are arcs. Our grouping method is a generalization of decomposit...

A flippable edge in an acyclic digraph is an edge whose reorientation leaves the graph acyclic. We characterize the spanning trees T of an undirected graph G such that there exists an acyclic orientation of G whose set of flippable edges is T . In particular for every edge e ∈ E(G) we give a linear algorithm returning an acyclic orientation and a spanning tree T containing e such that T is the ...