Domination game: Effect of edge- and vertex-removal

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Domination game: Effect of edge- and vertex-removal

The domination game is played on a graph G by two players, named Dominator and Staller. They alternatively select vertices of G such that each chosen vertex enlarges the set of vertices dominated before the move on it. Dominator’s goal is that the game is finished as soon as possible, while Staller wants the game to last as long as possible. It is assumed that both play optimally. Game 1 and Ga...

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Effect of edge-subdivision on vertex-domination in a graph

Let G be a graph with ∆(G) > 1. It can be shown that the domination number of the graph obtained from G by subdividing every edge exactly once is more than that of G. So, let ξ(G) be the least number of edges such that subdividing each of these edges exactly once results in a graph whose domination number is more than that of G. The parameter ξ(G) is called the subdivision number of G. This not...

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Domination parameters and edge-removal-critical graphs

For any of the basic domination parameters ir, , i, , or IR, we study graphs for which increases whenever an edge is removed ( -ER-critical graphs) and graphs for which decreases whenever an edge is removed ( −-ER-critical graphs). The latter case is only possible if = i or (perhaps) = ir. We give examples of classes of -ER-critical graphs for an upper domination parameter and characterize thes...

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Roman domination with respect to nondegenerate graph properties: vertex and edge removal

For a graph property P and a graph G, a subset S of the vertices of G is a P-set if the subgraph induced by S has the property P. A P-Roman dominating function on a graph G is a labeling f : V (G) → {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2 and the set of all vertices with label 1 or 2 is a P-set. The P-Roman domination number γPR(G) of G is the minimum of Σv∈V (...

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Double domination stable graphs upon edge removal

In a graph G = (V (G), E(G)), a vertex dominates itself and its neighbors. A subset S of V (G) is a double dominating set if every vertex of V (G) is dominated at least twice by the vertices of S. The double domination number of G is the minimum cardinality among all double dominating sets of G. We consider the effects of edge removal on the double domination number of a graph. We give a necess...

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ژورنال

عنوان ژورنال: Discrete Mathematics

سال: 2014

ISSN: 0012-365X

DOI: 10.1016/j.disc.2014.04.015