In the article m dimensional lexicographic noncooperative games   ) ,..., ( 1 m   are defined for the players’ } ,..., 2 , 1 { n N  for which there exists a characteristic function T m v v v v ) ,..., , ( 2 1  . Some main features are proved of v function in a lexicographic case. A lexicographic cooperative game is called a couple   v N , , where v is a real vectorfunction on N subsets ...

In the domination game studied here, Dominator and Staller alternately choose a vertex of a graph G and take it into a set D. The number of vertices dominated by the set D must increase in each single turn and the game ends when D becomes a dominating set of G. Dominator aims to minimize whilst Staller aims to maximize the number of turns (or equivalently, the size of the dominating set D obtai...

The paired domination number γpr(G) of a graph G is the smallest cardinality of a dominating set S of G such that 〈S〉 has a perfect matching. The generalized prisms πG of G are the graphs obtained by joining the vertices of two disjoint copies of G by |V (G)| independent edges. We provide characterizations of the following three classes of graphs: γpr(πG) = 2γpr(G) for all πG; γpr(K2 G) = 2γpr(...

A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The paired-domination number of G, denoted by γpr(G), is the minimum cardinality of a paired-dominating set of G. In [1], the authors gave tight bounds for paired-dominating sets of generalized claw-free graphs. Yet, ...

A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The paired-domination number of G, denoted by γpr(G), is the minimum cardinality of a paired-dominating set of G. In [?], the authors gave tight bounds for paired-dominating sets of generalized claw-free graphs. Yet, ...

A paired dominating set of a graph G without isolated vertices is a dominating set of G whose induced subgraph has a perfect matching. The paired domination number γpr(G) of G is the minimum cardinality amongst all paired dominating sets of G. The graph G is paired domination edge-critical (γprEC) if for every e ∈ E(G), γpr(G+ e) < γpr(G). We investigate the diameter of γprEC graphs. To this ef...

A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The maximum cardinality of a minimal paired-dominating set of G is the upper paired-domination number of G, denoted by pr(G). We establish bounds on pr(G) for connected claw-free graphs G in terms of the number n of v...

A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and the subgraph induced by S contains a perfect matching. The minimum cardinality of a paired-dominating set of G is the paireddomination number of G, denoted by γpr (G). In this work, we present several upper bounds on the paired-domination number in terms of the maximum degre...

The dual notions of domination and packing in finite simple graphs were first extensively explored by Meir and Moon in [15]. Most of the lower bounds for the domination number of a nontrivial Cartesian product involve the 2-packing, or closed neighborhood packing, number of the factors. In addition, the domination number of any graph is at least as large as its 2-packing number, and the invaria...

In the domination game, introduced by Brešar, Klavžar, and Rall in 2010, Dominator and Staller alternately select a vertex of a graph G. A move is legal if the selected vertex v dominates at least one new vertex – that is, if we have a u ∈ N [v] for which no vertex from N [u] was chosen up to this point of the game. The game ends when no more legal moves can be made, and its length equals the n...