We first devise a branching algorithm that computes a minimum independent dominating set with running time O∗(1.3351n) = O∗(20.417n) and polynomial space. This improves upon the best state of the art algorithms for this problem. We then study approximation of the problem by moderately exponential time algorithms and show that it can be approximated within ratio 1 + ε, for any ε > 0, in a time s...

The local tree-width of a graph G = (V;E) is the function ltwG : N ! N that associates with every r 2 N the maximal tree-width of an r-neighborhood in G. Our main graph theoretic result is a decomposition theorem for graphs with excluded minors, which says that such graphs can be decomposed into trees of graphs of almost bounded local tree-width. As an application of this theorem, we show that ...

A Roman dominating function on a graph G = (V, E) is a function f : V → {0, 1, 2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of a Roman dominating function is the value f(V ) = ∑ u∈V f(u). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G. In this pape...

A Roman dominating function on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of a Roman dominating function is the value f(V (G)) =

In this paper we provide algorithms faster thanO(2) for several NP-complete dominationtype problems. More precisely, we provide: • an algorithm for CAPACITATED DOMINATING SET that solves it in O(1.89), • a branch-and-reducealgorithm solving LARGEST IRREDUNDANT SET inO(1.9657) time, • and a simple iterative-DFS algorithm for SMALLEST INCLUSION-MAXIMAL IRREDUNDANT SET that solves it in O(1.999956...

A Roman dominating function (RDF) on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex v for which f(v) = 0, is adjacent to at least one vertex u for which f(u) = 2. The weight of a Roman dominating function f is the value f(V (G)) = ∑ v∈V (G) f(v). The Roman domination number of G, denoted by γR(G), is the minimum weight of an RDF on G. For a given graph,...

A Roman dominating function on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of a Roman dominating function is the value f(V (G)) = ∑ u∈V (G) f(u). The Roman domination number of G, γR(G), is the minimum weight of a Roman dominating function on G. In this paper, we...