We study combinatorial and algorithmic questions around minimal feedback vertex sets in tournament graphs. On the combinatorial side, we derive strong upper and lower bounds on the maximum number of minimal feedback vertex sets in an n-vertex tournament. We prove that every tournament on n vertices has at most 1.6740 minimal feedback vertex sets and that there is an infinite family of tournamen...

This paper describes an extremely fast polynomial time algorithm, the Near Optimal Vertex Cover Algorithm (NOVCA) that produces an optimal or near optimal vertex cover for any known undirected graph G (V, E). NOVCA is based on the idea of (i) including the vertex having maximum degree in the vertex cover and (ii) rendering the degree of a vertex to zero by including all its adjacent vertices. T...

This paper describes an extremely fast polynomial time algorithm, the NOVCA (Near Optimal Vertex Cover Algorithm) that produces an optimal or near optimal vertex cover for any known undirected graph G (V, E). NOVCA is based on the idea of (1) including the vertex having maximum degree in the vertex cover and (2) rendering the degree of a vertex to zero by including all its adjacent vertices. Th...

A graph with at least 2k vertices is said to be k-linked if for any ordered k-tuples (s1, . . . , sk) and (t1, . . . , tk) of 2k distinct vertices, there exist pairwise vertex-disjoint paths P1, . . . , Pk such that Pi connects si and ti for i = 1, . . . , k. For a given graph G, we consider the problem of finding a maximum induced subgraph of G that is not k-linked. This problem is a common ge...

Inspired by the study of community structure in connection networks, we introduce the graph polynomial Q (G;x, y), the bivariate generating function which counts the number of connected components in induced subgraphs. We give a recursive definition of Q (G; x, y) using vertex deletion, vertex contraction and deletion of a vertex together with its neighborhood and prove a universality property....

We give a bi-criteria approximation algorithm for the Minimum Nonuniform Partitioning problem, recently introduced by Krauthgamer, Naor, Schwartz and Talwar (2014). In this problem, we are given a graph G = (V,E) on n vertices and k numbers ρ1, . . . , ρk. The goal is to partition the graph into k disjoint sets P1, . . . , Pk satisfying |Pi| ≤ ρin so as to minimize the number of edges cut by th...

The PI polynomial of a molecular graph is defined to be the sum X(|E(G)|-N(e)) + |V(G)|(|V(G)|+1)/2 - |E(G)| over all edges of G, where N(e) is the number of edges parallel to e. In this paper, the PI polynomial of the phenylenic nanotubes and nanotori are computed. Several open questions are also included.

Generalizing a previous one-variable " interlace polynomial " , we consider a new interlace polynomial in two variables. The polynomial can be computed in two very different ways. The first is an expansion analogous to the state space expansion of the Tutte polynomial; the differences are that our expansion is over vertex rather than edge subsets, the rank of the subset appears positively rathe...

Let $G$ be a simple graph of order $n$ and size $m$.The edge covering of $G$ is a set of edges such that every vertex of $G$ is incident to at least one edge of the set. The edge cover polynomial of $G$ is the polynomial$E(G,x)=sum_{i=rho(G)}^{m} e(G,i) x^{i}$,where $e(G,i)$ is the number of edge coverings of $G$ of size $i$, and$rho(G)$ is the edge covering number of $G$. In this paper we stud...

Using a recently proposed gauge covariant diagonalization of $\pi a_1$-mixing we show that the low energy theorem $F^{\pi}=e f_\pi^2 F^{3\pi}$ current algebra, relating anomalous form factor $F_{\gamma \to \pi^+\pi^0\pi^-} = and neutral pion $F_{\pi^0 \gamma\gamma}=F^\pi$, is fulfilled in framework Nambu-Jona-Lasinio (NJL) model, solving long standing problem encountered extension including vec...