We establish sharp bounds on the sum and product of the independent domination numbers of a graph and its complement.

We study the function f G de ned for a graph G as the smallest integer k such that the join ofG with a stable set of size k is not jV G j choosable This function was introduced recently in order to describe extremal graphs for a list coloring version of a famous inequality due to Nordhaus and Gaddum Some bounds and some exact values for f G are determined

The minimum rank of a graph has been an interesting and well studied parameter 6 investigated by many researchers over the past decade or so. One of the many unresolved questions on 7 this topic is the so-called graph complement conjecture, which grew out of a workshop in 2006. This 8 conjecture asks for an upper bound on the sum of the minimum rank of a graph and the minimum rank 9 of its comp...

The First-Fit (or Grundy) chromatic number of G, written as χFF(G), is defined as the maximum number of classes in an ordered partition of V(G) into independent sets so that each vertex has a neighbor in each set earlier than its own. The well-known Nordhaus–Gaddum inequality states that the sum of the ordinary chromatic numbers of an n-vertex graph Contract grant sponsor: Hungarian National Sc...

A total dominating set of a graph G with no isolated vertices is subset S the vertex such that every adjacent to in S. The domination number minimum cardinality set. In this paper, we study middle graphs. Indeed, obtain tight bounds for terms order graph. We also compute some known families graphs explicitly. Moreover, Nordhaus-Gaddum-like relations are presented

Let G be a nontrivial connected and vertex-colored graph. A subset X of the vertex set is called rainbow if any two vertices in have distinct colors. The graph vertex-disconnected for x y G, there exists S such that when are nonadjacent, belong to different components $$G-S$$ ; whereas adjacent, $$S+x$$ or $$S+y$$ $$(G-xy)-S$$ . Such an x–y vertex-cut G. For vertex-disconnection number denoted ...