Abstract.— The paper considers the concept of centrality in an undirected graph. A system of axioms and an index for centrality satisfying the axioms are presented. The index is based on the degrees of the vertices in a given undirected graph, and it will enlarge the class of comparable graphs with respect to a centrality measure.

The domination number a(G) of a graph G is the size of a minimum dominating set, i.e., a set of points with the property that every other point is adjacent to a point of the set. In general a(G) can be made to increase or decrease by the removal of points from G. Our main objective is the study of this phenomenon.For example we show that if T is a tree with at least three points then a(T -u) > a(T) if and only if u is in every minimum dominating set of 7'. Removal of a set of lines from a graph G cannot decrease the domination number. We obtain some upper bounds on the size of a minimum set of lines which when removed from G increases the domination number.

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