We show that the DOMINATING SET problem parameterized by solution size is fixed-parameter tractable (FPT) in graphs that do not contain the claw (K1,3, the complete bipartite graph on four vertices where the two parts have one and three vertices, respectively) as an induced subgraph. We present an algorithm that uses 2 2)nO(1) time and polynomial space to decide whether a claw-free graph on n v...

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...

A subset D?V is an independent (or stable) dominating set of a graph G=(V,E) if D (i.e., there are no edges between vertices D) and dominates all the G (each vertex V?D has neighbour in D). In this paper, we study generalisation classical notion. Namely, instance our problem partition ?=(V1,…,Vk) V. Each Vi ? called obligation. An Independent Dominating with Obligations (IDO) (G,?) additional p...

A set S of vertices in a graph G is a total dominating set of G if every vertex is adjacent to a vertex in S. A fundamental problem in total domination theory in graphs is to determine which graphs have two disjoint total dominating sets. In this paper, we solve this problem and provide a constructive characterization of the graphs that have two disjoint total dominating sets.

We prove that line graphs and path graphs have bounded neighbourhood Helly. As a consequence, we obtain output-polynomial time algorithms for enumerating the set of minimal dominating sets of line graphs and path graphs. Therefore, there exists an output-polynomial time algorithm that enumerates the set of minimal edge-dominating sets of any graph.

A connected dominating set in a graph is a dominating set of vertices that induces a connected subgraph. We introduce and study the class of connected-domishold graphs, which are graphs that admit non-negative real weights associated to their vertices such that a set of vertices is a connected dominating set if and only if the sum of the corresponding weights exceeds a certain threshold. We sho...

In this paper, we are interested in the enumeration of minimal dominating sets in graphs. A polynomial delay algorithm with polynomial space in split graphs is presented. We then introduce a notion of maximal extension (a set of edges added to the graph) that keeps invariant the set of minimal dominating sets, and show that graphs with extensions as split graphs are exactly the ones having chor...