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Domination and independence subdivision numbers of graphs

Teresa W. HaynesSandra M. HedetniemiStephen T. Hedetniemi — 2000

Discussiones Mathematicae Graph Theory

The domination subdivision number s d γ ( G ) of a graph is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the domination number. Arumugam showed that this number is at most three for any tree, and conjectured that the upper bound of three holds for any graph. Although we do not prove this interesting conjecture, we give an upper bound for the domination subdivision number for any graph G in terms of the minimum degrees of adjacent vertices...

Downhill Domination in Graphs

Teresa W. HaynesStephen T. HedetniemiJessie D. JamiesonWilliam B. Jamieson — 2014

Discussiones Mathematicae Graph Theory

A path π = (v1, v2, . . . , vk+1) in a graph G = (V,E) is a downhill path if for every i, 1 ≤ i ≤ k, deg(vi) ≥ deg(vi+1), where deg(vi) denotes the degree of vertex vi ∈ V. The downhill domination number equals the minimum cardinality of a set S ⊆ V having the property that every vertex v ∈ V lies on a downhill path originating from some vertex in S. We investigate downhill domination numbers of graphs and give upper bounds. In particular, we show that the downhill domination number of a graph is...

γ-graphs of graphs

Gerd H. FrickeSandra M. HedetniemiStephen T. HedetniemiKevin R. Hutson — 2011

Discussiones Mathematicae Graph Theory

A set S ⊆ V is a dominating set of a graph G = (V,E) if every vertex in V -S is adjacent to at least one vertex in S. The domination number γ(G) of G equals the minimum cardinality of a dominating set S in G; we say that such a set S is a γ-set. In this paper we consider the family of all γ-sets in a graph G and we define the γ-graph G(γ) = (V(γ), E(γ)) of G to be the graph whose vertices V(γ) correspond 1-to-1 with the γ-sets of G, and two γ-sets, say D₁ and D₂, are adjacent in E(γ) if there exists...

Domination Subdivision Numbers

A set S of vertices of a graph G = (V,E) is a dominating set if every vertex of V-S is adjacent to some vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set of G, and the domination subdivision number s d γ ( G ) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the domination number. Arumugam conjectured that 1 s d γ ( G ) 3 for any graph G. We give a counterexample to this conjecture. On the other hand, we show...

Offensive alliances in graphs

A set S is an offensive alliance if for every vertex v in its boundary N(S)- S it holds that the majority of vertices in v's closed neighbourhood are in S. The offensive alliance number is the minimum cardinality of an offensive alliance. In this paper we explore the bounds on the offensive alliance and the strong offensive alliance numbers (where a strict majority is required). In particular, we show that the offensive alliance number is at most 2/3 the order and the strong offensive alliance number...

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