Let G = (V,E) be a graph. A set S ⊆ V is a restrained dominating set if every vertex not in S is adjacent to a vertex in S and to a vertex in V − S. The restrained domination number of G, denoted by γr(G), is the smallest cardinality of a restrained dominating set of G. It is known that if T is a tree of order n, then γr(T ) ≥ d(n+2)/3e. In this note we provide a simple constructive characteriz...

Let G be a graph. A total dominating set of G is a set S of vertices of G such that every vertex is adjacent to at least one vertex in S. The total domatic number of a graph is the maximum number of total dominating sets which partition the vertex set of G. In this paper we would like to characterize the cubic graphs with total domatic number at least two.

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...

A triangle-free graph is maximal if the addition of any edge produces a triangle. A set S of vertices in a graph G is called an independent dominating set if S is both an independent and a dominating set of G. The independent domination number i(G) of G is the minimum cardinality of an independent dominating set of G. In this paper, we show that i(G) ≤ δ(G) ≤ n 2 for maximal triangle-free graph...

A set S ⊆ V is a global dominating set of a graph G = (V,E) if S is a dominating set of G and G, where G is the complement graph of G. The global domination number γg(G) equals the minimum cardinality of a global dominating set of G. The square graph G of a graph G is the graph with vertex set V and two vertices are adjacent in G if they are joined in G by a path of length one or two. In this p...

For an integer k ≥ 1 and a graph G = (V,E), a subset S of V is kindependent if every vertex in S has at most k − 1 neighbors in S. The k-independent number βk(G) is the maximum cardinality of a kindependent set of G. In this work, we study relations between βk(G), βj(G) and the domination number γ(G) in a graph G where 1 ≤ j < k. Also we give some characterizations of extremal graphs.

A total dominating set of a graph G is a set S of vertices of G such that every vertex is adjacent to a vertex in S. The total domination number of G, denoted by γt(G), is the minimum cardinality of a total dominating set. Let G be a connected spanning subgraph of Ks,s and letH be the complement of G relative to Ks,s; that is, Ks,s = G⊕H . The graph G is k-supercritical relative to Ks,s if γt(G...

In this paper, we study the minimality of the boundary of a Coxeter system. We show that for a Coxeter system (W,S) if there exist a maximal spherical subset T of S and an element s0 ∈ S such that m(s0, t) ≥ 3 for each t ∈ T and m(s0, t0) = ∞ for some t0 ∈ T , then every orbit Wα is dense in the boundary ∂Σ(W,S) of the Coxeter system (W,S), hence ∂Σ(W,S) is minimal, where m(s0, t) is the order ...

We investigate the parameterized complexity of Maximum Independent Set and Dominating Set restricted to certain geometric graphs. We show that Dominating Set is W[1]-hard for the intersection graphs of unit squares, unit disks, and line segments. For Maximum Independent Set, we show that the problem is W[1]-complete for unit segments, but fixed-parameter tractable if the segments are axis-paral...

A set of vertices of a graph G is a total dominating set if each vertex of G is adjacent to a vertex in the set. The total domination number of a graph γt (G) is the minimum size of a total dominating set. We provide a short proof of the result that γt (G) ≤ 2 3 n for connected graphs with n ≥ 3 and a short characterization of the extremal graphs.