Let / be an integer-valued function defined on the vertex set V(G) of a graph G. A subset D of V(G) is an /-dominating set if each vertex x outside D is adjacent to at least f(x) vertices in D. The minimum number of vertices in an /-dominating set is denned to be the /-domination number, denoted by 7/(G). In a similar way one can define the connected and total /-domination numbers 7 C| /(G) and...

A set S V is a dominating set of a graph G = (V;E) if each vertex in V is either in S or is adjacent to a vertex in S. A vertex is said to dominate itself and all its neighbors. The domination number (G) is the minimum cardinality of a dominating set of G. A set S V is an independent set of vertices if no two vertices in S are adjacent. The independence number, B0 (G), is the maximum cardinalit...

Optimality Theory (OT) and Harmonic Grammar (HG) differ because the former assumes a model of constraint interaction based on strict domination, while the latter assumes a weighted model of interaction. As Prince and Smolensky (1997) admit, “that strict domination governs grammatical constraint interaction is not currently explained”. Yet, Legendre et al. (2006, 911-912) make two suggestions. T...

We show that the diameter of a total domination vertex-critical graph is at most 5(γt −1)/3, and that the diameter of an independent domination vertex-critical graph is at most 2(i− 1). For all values of γt ≡ 2 (mod 3) there exists a total domination vertex-critical graph with the maximum possible diameter. For all values of i ≥ 2 there exists an independent domination vertex-critical graph wit...

This paper studies Upper Domination, i.e., the problem of computing the maximum cardinality of a minimal dominating set in a graph, with a focus on parameterised complexity. Our main results include W[1]-hardness for Upper Domination, contrasting FPT membership for the parameterised dual Co-Upper Domination. The study of structural properties also yields some insight into Upper Total Domination...

Humans may one day create superintelligence, artificially intelligent machines that surpass mankind’s intellect. Would these artificial intelligences choose to play games with us, and if so, which games? We believe this question is relevant for the ethics of general AI, the current widespread integration of AI systems into daily life, and for game AI research. We present a catalog of scenarios,...

Let $$\gamma (G)$$ and _{t}(G)$$ be the domination number total of a graph G, respectively, let _g(G)$$ _{tg}(G)$$ game respectively. Then, G is _g$$ -perfect (resp. _{tg}$$ -perfect) if every induced subgraph F satisfies _g(F)=\gamma (F)$$ _{tg}(F)=\gamma _t(F)$$ ). A recursive characterization graphs derived. The yields polynomial recognition algorithm for graphs. It proved that minimally -im...

Let γg(G) be the game domination number of a graph G. Rall conjectured that if G is traceable graph, then γg(G)≤12n(G). Our main result verifies conjecture over class line graphs. Moreover, in this paper we put forward δ(G)≥2, We show both conjectures hold true for claw-free cubic further prove upper bound γg(G)≤1120n(G) graphs minimum degree at least 2. Computer experiments supporting new and ...

Let G = (V , E) be a graph. A subset D ⊆ V is a dominating set if every vertex not in D is adjacent to a vertex in D. A dominating set D is called a total dominating set if every vertex in D is adjacent to a vertex in D. The domination (resp. total domination) number of G is the smallest cardinality of a dominating (resp. total dominating) set of G. The bondage (resp. total bondage) number of a...