Partial product of graphs and Vizing's conjecture

Abstract

Let G and H be two graphs with vertex sets V 1 = { u 1 , . . . , u n 1 } and V 2 = { v 1 , . . . , v n 2 } , respectively. If S ⊂ V 2 , then the partial Cartesian product of G and H with respect to S is the graph G □ S H = ( V , E ) , where V = V 1 × V 2 and two vertices ( u i , v j ) and ( u k , v l ) are adjacent in G □ S H if and only if either ( u i = u k and v j ∼ v l ) or ( u i ∼ u k and v j = v l ∈ S ). If A ⊂ V 1 and B ⊂ V 2 , then the restricted partial strong product of G and H with respect to A and B is the graph G A \boxtimes B H = ( V , E ) , where V = V 1 × V 2 and two vertices ( u i , v j ) and ( u k , v l ) are adjacent in G A \boxtimes B H if and only if either ( u i = u k and v j ∼ v l ) or ( u i ∼ u k and v j = v l ) or ( u i ∈ A , u k ∉ A , v j ∈ B , v l ∉ B$ , u i ∼ u k and v j ∼ v l ) or ( u i ∉ A , u k ∈ A , v j ∉ B , v l ∈ B , u i ∼ u k and v j ∼ v l ). In this article we obtain Vizing-like results for the domination number and the independence domination number of the partial Cartesian product of graphs. Moreover we study the domination number of the restricted partial strong product of graphs.