The neighbourhood polynomial G , is generating function for the number of faces of each cardinality in the neighbourhood complex of a graph. In other word $N(G,x)=sum_{Uin N(G)} x^{|U|}$, where N(G) is neighbourhood complex of a graph, whose vertices are the vertices of the graph and faces are subsets of vertices that have a common neighbour. In this paper we compute this polynomial for some na...

the neighbourhood polynomial g , is generating function for the number of faces of each cardinality in the neighbourhood complex of a graph. in other word $n(g,x)=sum_{uin n(g)} x^{|u|}$, where n(g) is neighbourhood complex of a graph, whose vertices are the vertices of the graph and faces are subsets of vertices that have a common neighbour. in this paper we compute this polynomial for some na...

This paper describes an incremental neighbourhood tabu search heuristic for the generalized vehicle routing problem with time windows. The purpose of this work is to offer a general tool that can successfully be applied to a large variety of specific problems. The algorithm builds upon a previously developed tabu search heuristic by replacing its neighbourhood structure. The new neighbourhood i...

We examine neighbourhood polynomials, which are generating functions for the number of faces of each cardinality in the neighbourhood complex of a graph. We provide explicit polynomials for hypercubes, for graphs not containing a four-cycle and for the graphs resulting from joins and Cartesian products. We also show that the closure of the roots are dense in the complex plane except possibly in...

An exponential neighbourhood for the traveling salesman problem (TSP) is a set of tours, which grows exponentially in the input size. An exponential neighbourhood is polynomial time searchable if we can nd the best among the exponential number of tours in polynomial time. Deineko and Woeginger asked if there exists polynomial time searchable neighbourhoods of size at least bnc!, for some > 1 2 ...

A magnet is a pair u, v of adjacent vertices such that the proper neighbours of u are completely linked to the proper neighbours of v. It has been shown that one can reduce the graph by removing one of the vertices of a magnet together with all edges linking the other one to its proper neighbours without changing the stability number. We characterize a class of graphs such that by repeated use ...

A universal labeling of a graph G is a labeling of the edge set in G such that in every orientation l of G for every two adjacent vertices v and u, the sum of incoming edges of v and u in the oriented graph are different from each other. The universal labeling number of a graph G is the minimum number k such that G has universal labeling from {1, 2, . . . , k} denoted it by −→ χu(G). We have 2∆...

Maire introduced the class of those graphs without long holes, for which every induced subgraph possesses a vertex whose neighbourhood has no P4. Those graphs are called slightly triangulated graphs and are proven to be perfect in [14]. In this paper, we introduce a wider class of graphs, obtained by relaxing the neighbourhood condition. We also provide polynomial algorithms to recognize and to...

We prove that line graphs and path graphs have bounded neighbourhood Helly. As a consequence, we obtain output-polynomial time algorithms for enumerating the set of minimal dominating sets of line graphs and path graphs. Therefore, there exists an output-polynomial time algorithm that enumerates the set of minimal edge-dominating sets of any graph.

We improve an estimate obtained in [Br] for the average number of limit cycles of a planar polynomial vector field situated in a neighbourhood of the origin provided that the field in a larger neighbourhood is close enough to a linear center. The result follows from a new distributional inequality for the number of zeros of a family of univariate holomorphic functions depending holomorphically ...