A family F of graphs is said to have the Erdős–Pósa property, if for every integer k there is an integer f(k,F) such that every graph G either contains k vertex disjoint subgraphs each isomorphic to a graph in F or a set C of at most f(k,F) vertices such that G−C has no subgraph isomorphic to a graph in F . The term Erdős–Pósa property arose because in [3] Erdős and Pósa proved that the family ...

This paper presents an O(1.2738 + kn)-time polynomialspace parameterized algorithm for Vertex Cover improving the previous O(1.286+kn)-time polynomial-space upper bound by Chen, Kanj, and Jia. The algorithm also improves the O(1.2745k+kn)-time exponentialspace upper bound for the problem by Chandran and Grandoni.

Motivated by certain conjectures regarding immanants of Jacobi-Trudi matrices, Stanley has recently defined and studied a symmetric function generalization XG of the chromatic polynomial of a graph G. Independently, Chung and Graham have defined and studied a directed graph invariant called the cover polynomial. The cover polynomial is closely related to the chromatic polynomial and to the rook...

We consider the concepts of a t-total vertex cover and a t-total edge cover (t ≥ 1), which generalize the notions of a vertex cover and an edge cover, respectively. A t-total vertex (respectively edge) cover of a connected graph G is a vertex (edge) cover S of G such that each connected component of the subgraph of G induced by S has least t vertices (edges). These definitions are motivated by ...

We introduce the concept of minimum edge cover for an induced subgraph in a graph. Let [Formula: see text] be unicyclic graph with unique odd cycle and its ideal. compute exact values all symbolic defects using describe one method to find quasi-polynomial associated ideal text]. classify class graphs when some power maximal annihilates any fixed Also those graphs, we Hilbert function module

In this paper we show that certain trigonometric polynomial shearlets which are special cases of directional de la Vall\'{e}e Poussin type wavelets able to detect singularities along boundary curves periodic characteristic functions. Motivated by recent results for discrete in two dimensions, provide lower and upper estimates the magnitude corresponding inner products. proof use localization pr...

Consider a hypergraph of rank r > 2 with m edges, independence number and edge cover number . We prove the inequality (r − 2)m+ r − 1 . One application of this inequality is a special case of a conjecture of Aharoni and the first author extending Ryser’s Conjecture to matroids. © 2007 Elsevier B.V. All rights reserved.

Bipartite graphs with equal edge domination number and maximum matching cardinality are characterized. These two parameters are used to develop bounds on the vertex cover and total vertex cover numbers of graphs and a resulting chain of vertex covering, edge domination, and matching parameters is explored. In addition, the total vertex cover number is compared to the total domination number of ...

We show that the expected time for a random walk on a (multi-)graph G to traverse all m edges of G, and return to its starting point, is at most 2m; if each edge must be traversed in both directions, the bound is 3m. Both bounds are tight and may be applied to graphs with arbitrary edge lengths, with implications for Brownian motion on a finite or infinite network of total edge-length m.

formalized in terms of a hypergraph G = (V,E) whose edges arrive one-by-one and the goal is to construct an edge cover F ⊆ E with the objective of minimizing the cardinality (or cost in the weighted case) of F . We further consider a parameterized relaxation of this problem, where given some 0 ≤ < 1, the goal is to construct an edge (1− )-cover, namely, a subset of edges incident to all but an ...