H-forming sets in graphs

For graphs G and H , a set S ⊆ V (G) is an H -forming set of G if for every v∈V (G) − S, there exists a subset R ⊆ S, where |R|= |V (H)| − 1, such that the subgraph induced by R∪{v} contains H as a subgraph (not necessarily induced). The minimum cardinality of an H -forming set of G is the H -forming number {H}(G). The H -forming number of G is a generalization of the domination number (G) because (G)= {P2}(G). We show that (G)6 {P3}(G)6 t(G), where t(G) is the total domination number of G. For a nontrivial tree T , we show that {P3}(T )= t(T ). We also de3ne independent P3-forming sets, give complexity results for the independent P3-forming problem, and characterize the trees having an independent P3-forming set. c © 2002 Elsevier Science B.V. All rights reserved.