Elusiveness of Finding Degrees

Abstract

We address the complexity of finding a vertex with specific (or maximum) (out)degree in undirected graphs, directed graphs and in tournaments in a model where we count only the probes to the adjacency matrix of the graph. Improving upon some earlier bounds, using adversary arguments, we show that the following problems require ${n \atopwithdelims ()2}$ probes to the adjacency matrix of an n node graph:

• determining whether a given directed graph has a vertex of outdegree k (for a non-negative integer $k \le (n+1)/2$);

• determining whether an undirected graph has a degree 0 or 1 vertex;

• finding the maximum (out)degree in a directed or an undirected graph, and

• finding all vertices with the maximum outdegree in a tournament.

A property of a simple graph is elusive, if any algorithm to determine the property requires all the relevant probes to the adjacency matrix of the graph in the worst case. So the above results imply that determining whether a directed graph has a vertex of (out)degree k (for a non-negative integer $k \le (n+1)/2$) or an undirected graph has a vertex of degree 0 or 1 vertex are elusive properties.

In contrast, we show that one can find a maximum outdegree in a tournament using at most ${n \atopwithdelims ()2} -1$ probes. By substantially improving a known lower bound, we show that, for this problem ${n \atopwithdelims ()2} -2$ probes are necessary if n is odd, and ${n \atopwithdelims ()2} -n/2 -2$ probes are necessary if n is even. For determining the existence of a vertex with degree $k > 1$ in an undirected graph, we give a lower bound of $.42n^2$ improving on the earlier lower bound of $.25n^2$.