Connected domination game

Abstract

In this paper we introduce a domination game based on the notion of connected domination. Let G = (V, E) be a connected graph of order at least 2. We define a connected domination game on G as follows: The game is played by two players, Dominator and Staller. The players alternate taking turns choosing a vertex of G (Dominator starts). A move of a player by choosing a vertex v is legal, if (1) the vertex v dominates at least one additional vertex that was not dominated by the set of previously chosen vertices and (2) the set of all chosen vertices induces a connected subgraph of G. The game ends when none of the players has a legal move (i.e., G is dominated). The aim of Dominator is to finish as soon as possible, Staller has an opposite aim. Let D be the set of played vertices obtained at the end of the connected domination game (D is a connected dominating set of G). The connected game domination number of G, denoted γcg(G), is the minimum cardinality of D, when both players played optimally on G. We provide an upper bound on γcg(G) in terms of the connected domination number. We also give a tight upper bound on this parameter for the class of 2-trees. Next, we investigate the Cartesian product of a complete graph and a tree, and we give exact values of the connected game domination number for such a product, when the tree is a path or a star. We also consider some variants of the game, in particular, a Staller-start game.

Hydrogen Production from Catalytic Steam Reforming of Bio‐Oils: A Critical Review

“…In order to solve the carbon deposition problem, methods for increasing the coking resistance by enhancing the carbon gasification are being developed. Mainly, the deposited carbon can be further gasified with steam to keep the activity of the catalysts [41][42][43]:…”

Effect of predomination and vertex removal on the game total domination number of a graph

“…The connected domination game was introduced very recently in [1] as a game played on a graph G by two players, Dominator and Staller, with the following rules. Dominator's goal is to finish the game with the smallest possible number of moves, while Staller aims to prolong the game.…”

“…In the first paper on the connected domination game [1], the connected game domination number of trees and 2-paths is determined, along with an upper bound on 2-trees. The game is studied on the Cartesian product graphs.…”

“…The number of moves in a game on G when Staller can pass at most k times isγ k cg (G). It was proved in [1] that γ cg (G) ≤γ k cg (G) ≤ γ cg (G) + k. Another important variation of the connected domination game presented in [1] is the connected domination game with Chooser which turns out to be an extremely useful tool for proving some bounds for the connected game domination number. The rules of the game are essentially the same as in the usual game, except that there is another player, Chooser, who can make zero, one or more moves after any move of Dominator or Staller.…”

“…Note that Chooser has no specific goal, he can help either Dominator or Staller. For later use we also state the Chooser Lemma from [1]. Lemma 1.1 (Chooser Lemma) Consider the connected domination game with Chooser on a graph G. Suppose that in the game Chooser picks k vertices, and that both Dominator and Staller play optimally.…”