### Neighborhood unions and regularity in graphs

O. Favaron, Y. Redouane
2001 Theoretical Computer Science
One way to generalize the concept of degree in a graph is to consider the neighborhood N (S) of an independent set S instead of a simple vertex. The minimum generalized degree of order t of G is then deÿned, for 16t6 (the independence number of G), by ut = min{|N (S)|: S ⊂ V; S is independent and |S| = t}. The graph G is said to be ut-regular if |N (S1)| = |N (S2)| for every pair S1; S2 of independent sets of t elements, totally ut-regular (resp. totally ut6s-regular where s is given 6 ) if it
more » ... s ut-regular for every t6 (resp. for every t6s), strongly ut-regular (resp. strongly ut6s-regular) if |N (S1)| = |N (S2)| for every pair S1; S2 of independent sets of G (resp. every pair of independent sets of order at most s). We determine the strongly ut62-regular graphs and give some properties of the totally ut62-regular and totally ut-regular graphs. Some of our results improve already known results.