Domination in graphs with minimum degree two

Abstract

The domination number y ( G ) of a graph G = (V E ) is the minimum cardinality of a subset of Vsuch that every vertex is either in the set or is adjacent to some vertex in the set. We show that if a connected graph G has minimum2degree two and is not one of seven exceptional graphs, then y ( g ) I ~l V l .We also characterize those connected graphs with y ( G ) = FJVI.

Pairs of Disjoint Dominating Sets and the Minimum Degree of Graphs

“…By the results of McCuaig and Shepherd [20], there would be infinitely many extremal graphs for this estimate. In fact, we believe that the edge-minimal extremal graphs for the bound in Conjecture 7 are the same as those described in [20] for the bound γ(G) ≤ …”

“…Among the most fundamental results on the domination number are upper bound for graphs which satisfy a minimum degree condition [1,2,4,[20][21][22][23].…”

“…For graphs G = (V, E) of minimum degree at least 2, Blank [4] and -independentlyMcCuaig and Shepherd [20] proved that…”

“…The first result is inspired by McCuaig and Shepherd's [20] work and their seven exceptional graphs H 1 , H 2 , . .…”

On Domination Number of 4-Regular Graphs

“…Figure 1. Graphs in family A Theorem 2 [2]. If G is a connected graph with δ(G) 2 and G / ∈ A , then γ(G) 2n/5.…”