Mustapha Chellali BOUNDS ON THE 2-DOMINATION NUMBER IN CACTUS GRAPHS
نویسنده
چکیده
A 2-dominating set of a graph G is a set D of vertices of G such that every vertex not in S is dominated at least twice. The minimum cardinality of a 2-dominating set of G is the 2-domination number γ2(G). We show that if G is a nontrivial connected cactus graph with k(G) even cycles (k(G) ≥ 0), then γ2(G) ≥ γt(G) − k(G), and if G is a graph of order n with at most one cycle, then γ2(G) > (n+ l− s)/2 improving Fink and Jacobson’s lower bound for trees with l > s, where γt(G), l and s are the total domination number, the number of leaves and support vertices of G, respectively. We also show that if T is a tree of order n > 3, then γ2(T ) 6 β(T ) + s− 1, where β(T ) is the independence number of T .
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