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Nearly perfect sets in the n-fold products of graphs

Perl M.

Opuscula Mathematica

2007

Vol. 27, no. 1

83-88

The study of nearly perfect sets in graphs was initiated in [2], Let S ⊆ V(G). We say that S is a nearly perfect set (or is nearly perfect) in G if every vertex in V(G) - S is adjacent to at most one vertex in S. A nearly perfect set S in G is called 1-maximal if for every vertex u ∈ V(G) - S, S ∪ {u} is not nearly perfect in G. We denote the minimum cardinality of a 1-maximal nearly perfect set in G by np(G). We will call the 1-maximal nearly perfect set of the cardinality np(G) an np(G) - set. In this paper, we evaluate the parameter np(G) for some n-fold products of graphs. To this effect, we determine 1-maximal nearly perfect sets in the n-fold Cartesian product of graphs and in the n-fold strong product of graphs.

A note on Vizing's generalized conjecture

Blidia M., Chellali M.

Opuscula Mathematica

2007

Vol. 27, no. 2

181-185

In this note we give a generalized version of Vizing's conjecture concerning the distance domination number for the cartesian product of two graphs.

Nearly perfect sets in products of graphs

Kwaśnik M., Perl M.

Opuscula Mathematica

2004

Vol. 24, no. 2

177-180

The study of nearly perfect sets in graphs was initiated in [2]. Let S ⊆ V(G). We say that S is a nearly perfect set (or is nearly perfect) in G if every vertex in V(G) - S is adjacent to at most one vertex in S. A nearly perfect set S in G is called maximal if for every vertex u ∈ V(G) - S, S ∪ {u} is not nearly perfect in G. The minimum cardinality of a maximal nearly perfect set is denoted by np(G). It is our purpose in this paper to determine maximal nearly perfect sets in two well-known products of two graphs, i.e. in the Cartesian product and in the strong product. Lastly, we give upper bounds of np(G1 x G2) and np(G1 ⊗ G2), for some special graphs G1,G2.