Let G = (V,E) be a graph. A set S ⊆ V is a total restrained dominating set if every vertex in V is adjacent to a vertex in S and every vertex of V −S is adjacent to a vertex in V −S. The total restrained domination number of G, denoted by γtr(G), is the minimum cardinality of a total restrained dominating set of G. A unicyclic graph is a connected graph that contains precisely one cycle. We sho...

A neighborhood total dominating set in a graph G is a dominating set S of G with the property that the subgraph induced by N(S), the open neighborhood of the set S, has no isolated vertex. The neighborhood total domination number γnt(G) is the minimum cardinality of a neighborhood total dominating set of G. Arumugam and Sivagnanam introduced and studied the concept of neighborhood total dominat...

A set S of vertices of a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. The independent domination number of G, denoted by i(G), is the minimum cardinality of an independent dominating set of G. In this paper, some new classes of graphs with equal domination and independent domination numbers are presented and exa...

Let β(G) and Γ(G) be the independence number and the upper domination number of a graphG, respectively. A graph G is called Γ-perfect if β(H) = Γ(H), for every induced subgraph H of G. The class of Γ-perfect graphs generalizes such well-known classes of graphs as strongly perfect graphs, absorbantly perfect graphs, and circular arc graphs. In this article, we present a characterization of Γ-per...

In this paper, we consider theMinimum Independent Dominating Set problem and develop exact exponential algorithms that break the trivial O(2|V |) bound. A simple O∗( √ 3 |V | ) time algorithm is developed to solve this problem on general graphs. For sparse graphs, e.g. graphs with degree bounded by 3 and 4, we show that a few new branching techniques can be applied to these graphs and the resul...

A k-dominating set is a set D k V such that every vertex i 2 V nD k has at least k i neighbours in D k. The k-domination number k (G) of G is the cardinality of a smallest k-dominating set of G. For k 1 = ::: = kn = 1, k-domination corresponds to the usual concept of domination. Our approach yields an improvement of an upper bound for the domination number found then the notion of k-dominating ...

Wir befassen uns in der Arbeit mit dem Entwurf von exakten Algorithmen für verschiedene NP-vollständige Optimierungsprobleme auf Graphen, wie beispielsweise Vertex Cover, Independent Set oder Dominating Set. Im Vordergrund der Arbeit stehen exakte Lösungsverfahren mit beweisbaren Laufzeitschranken. Wir verfolgen dabei den jüngst vorgeschlagenen Ansatz sogenannter “parametrisierter Algorithmen”....

We investigate polynomial-time approximability of the problems related to edge dominating sets of graphs. When edges are unit-weighted, the edge dominating set problem is polynomially equivalent to the minimum maximal matching problem, in either exact or approximate computation, and the former problem was recently found to be approximable within a factor of 2 even with arbitrary weights. It wil...

Let γ(G), i(G), γS(G) and iS(G) denote the domination number, the independent domination number, the strong domination number and the independent strong domination number of a graph G, respectively. A graph G is called γi-perfect (domination perfect) if γ(H) = i(H), for every induced subgraph H of G. The classes of γγS-perfect, γSiS-perfect, iiS-perfect and γiS-perfect graphs are defined analog...