The least point covering and domination numbers of a graph

AbstractA set S ⊂ V of a graph G = (V,E) is a total point cover (t.p.c.) if S is a point cover containing all isolates of G, if any. The number αt(G) is the minimum cardinality of a t.p.c. A t.p.c. S is a least point cover (l.p.c.) if αt(〈S〉)⩽ αt(〈S1〉) for any t.p.c. S1, where 〈S〉 is the subgraph induced by S. The least point covering number α1(G) of G is the minimum cardinality of a l.p.c. A dominating set D of G is a least dominating set (l.d.s.) if γ(〈D〉) ⩽ γ(〈D1〉) for any dominating set D1 (γ denotes domination number). The least domination number γ1(G) of G is the minimum cardinality of a l.d.s. If γt is the total domination number, we prove among 0other things: (i) γ1 ⩽ γt, and (ii) for a tree, γ1 ⩽ α1.Conjectures. For any graph G of order p ⩾ 2, (1) γ1 ⩽ α1, (2) γ1 ⩽3p5, if G is connected