For a graphical property P and a graph G, we say that a subset S of the vertices of G is a P-set if the subgraph induced by S has the property P. Then the P-domination number of G is the minimum cardinality of a dominating P-set and the P-independence number the maximum cardinality of a P-set. We show that several properties of domination, independent domination and acyclic domination hold for ...

For a graphical property P and a graph G, we say that a subset S of the vertices of G is a P-set if the subgraph induced by S has the property P. Then the P-domination number of G is the minimum cardinality of a dominating P-set and the P-independence number the maximum cardinality of a P-set. We show that several properties of domination, independent domination and acyclic domination hold for ...

The k-domination number γk(G) of a simple, undirected graph G is the order of a smallest subset D of the vertices of G such that each vertex of G is either in D or adjacent to at least k vertices in D. In 2010, the conjecture-generating computer program, Graffiti.pc, was queried for upperbounds on the 2-domination number. In this paper we prove new upper bounds on the 2-domination number of a g...

A subset W \(\subseteq\) V (G) of a graph G is an independent semitotal dominating set G, abbreviated ISTd-set if and every element exactly distance 2 from at least one other W. The domination number denoted by \(\gamma\)it2(G), the minimum cardinality G. In this paper, we study concept in graphs investigate conditions for on which ISTd-sets exist. Further, join any two are examined. Consequent...

An independent dominating set of a graph, also known as maximal set, is S pairwise non-adjacent vertices such that every vertex not in adjacent to some S. We prove for Δ=4 or Δ≥6, connected n-vertex graph maximum degree at most Δ has an size (1−Δ⌊Δ2/4⌋+Δ)(n−1)+1. In addition, we characterize all graphs having the equality and show other have (1−Δ⌊Δ2/4⌋+Δ)n.