A secure (total) dominating set of a graph G = (V, E) is a (total) dominating set X ⊆ V with the property that for each u ∈ V − X , there exists x ∈ X adjacent to u such that (X − {x}) ∪ {u} is (total) dominating. The smallest cardinality of a secure (total) dominating set is the secure (total) domination number γs(G) (γst(G)). We characterize graphs with equal total and secure total domination...

We generalise the family of (σ, ρ)-problems and locally checkable vertex partition problems to their distance versions, which naturally captures well-known problems such as distance-r dominating set and distance-r independent set. We show that these distance problems are XP parameterized by the structural parameter mim-width, and hence polynomial on graph classes where mim-width is bounded and ...

Let G = (V ,E) be a graph without isolated vertices. A set S ⊆ V is a paired-dominating set if every vertex in V − S is adjacent to a vertex in S and the subgraph induced by S contains a perfect matching. The paired-domination problem is to determine the paired-domination number, which is the minimum cardinality of a paired-dominating set. Motivated by a mistaken algorithm given by Chen, Kang a...

Given a graph G = (V,E), a perfect dominating set is a subset of vertices V ′ ⊆ V (G) such that each vertex v ∈ V (G) \ V ′ is dominated by exactly one vertex v ∈ V . An efficient dominating set is a perfect dominating set V ′ where V ′ is also an independent set. These problems are usually posed in terms of edges instead of vertices. Both problems, either for the vertex or edge variant, remain...

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, ...

Let G be a graph and S ⊆ V (G). For each vertex u ∈ S and for each v ∈ V (G) − S, we define d(u, v) = d(v, u) to be the length of a shortest path in 〈V (G)−(S−{u})〉 if such a path exists, and∞ otherwise. Let v ∈ V (G). We define wS(v) = ∑ u∈S 1 2d(u,v)−1 if v 6∈ S, and wS(v) = 2 if v ∈ S. If, for each v ∈ V (G), we have wS(v) ≥ 1, then S is an exponential dominating set. The smallest cardinalit...

We introduce the graph parameter boolean-width, related to the number of different unions of neighborhoods across a cut of a graph. For many graph problems this number is the runtime bottleneck when using a divide-and-conquer approach. Boolean-width is similar to rank-width, which is related to the number of GF [2]-sums (1+1=0) of neighborhoods instead of the Boolean-sums (1+1=1) used for boole...

A total dominating set of a graph G = (V, E) with no isolated vertex is a set S ⊆ V such that every vertex is adjacent to a vertex in S. A total dominating set S of a graph G is a locating-total dominating set if for every pair of distinct vertices u and v in V − S, N(u) ∩ S 6= N(v) ∩ S, and S is a differentiating-total dominating set if for every pair of distinct vertices u and v in V , N [u]∩...

The concept of inverse domination was introduced by Kulli V.R. and Sigarakanti S.C. [9] . Let D be a  set of G. A dominating set D1  VD is called an inverse dominating set of G with respect to D. The inverse domination number   (G) is the order of a smallest inverse dominating set. Motivated by this definition we define another parameter as follows. Let D be a maximum independent set in G. ...

The domination subdivision number sdγ(G) of a graph is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the domination number. Arumugam showed that this number is at most three for any tree, and conjectured that the upper bound of three holds for any graph. Although we do not prove this interesting conjecture, we give an upp...