On locating and locating-total domination edge addition critical graphs

A set of vertices S of a graph G = (V, E) is a locating-dominating set, abbreviated (LDS), if for every pair of distinct vertices u and v in V - S, the neighborhoods of u and v in S are nonempty and different. A locating-total dominating set, abbreviated (LTDS), is a LDS whose induced subgraph has no isolated vertices. The locating-domination number, gamma(L)(G), of G is the minimum cardinality of a LDS of G, and the locating-total domination number, gamma(t)(L)(G), of G is the minimum cardinality of a LTDS of G. The addition of any missing edge e in a graph G, can increase, decrease or remain unchanged the locating (locating-total, respectively) domination number. A graph G is gamma(+)(L)-EA-critical (gamma(-)(L)-EA-critical, respectively) if gamma(L)(G) < gamma(L)(G+e) (gamma(L)(G+e) < gamma(L)(G), respectively) for every e is not an element of E. gamma(t+)(L)-EA-critical and gamma(t-)(L)-EA-critical graphs are defined similarly. In this paper, we give characterizations of pi(+)-EA-critical graphs and pi(-)-EA-critical trees where pi is an element of{gamma(L), gamma(t)(L)}.