The size of a minimum critically m-neighbor-scattered graph

Abstract

It seems reasonable that for a connected representing graph of a spy network, the more edges it has, the more jeopardy the spy network is in. So, a spy network which has the minimum number of edges is the comparatively reliable network we want. As a special kind of graph, a critically m-neighbor-scattered graph is important and interesting in applications in communication networks. In this paper, we obtain some upper bounds and a lower bound for the size of a minimum critically m-neighbor-scattered graph with given order p and 4 - p ≤ m ≤ -1. Moreover, we construct a (1 + ε)-approximate graph for the minimum critically m-neighbor-scattered graph of order p for sufficiently small m and sufficiently large p. © Springer-Verlag Berlin Heidelberg 2007.