A cycle in a graph is dominating if every vertex lies at distance at most one from the cycle and a cycle is D-cycle if every edge is incident with a vertex of the cycle. In this paper, first we provide recursive formulae for finding a shortest dominating cycle in a Hahn graph; minor modifications can give formulae for finding a shortest D-cycle. Then, dominating cycles and D-cycles in a Halin g...

We first consider some problems related to the maximum number of dominating (or strong dominating) sets in a regular graph. Our techniques, centered around Shearer’s entropy lemma, extend to a reasonably broad class of graph parameters enumerating vertex colorings that satisfy conditions on the multiset of colors appearing in neighborhoods (either open or closed). Dominating sets and strong dom...

We give the first linear kernels for Dominating Set and Connected Dominating Set problems on graphs excluding a fixed graph H as a topological minor. In other words, we give polynomial time algorithms that, for a given H-topological-minor free graph G and a positive integer k, output an H-topological-minor free graph G′ on O(k) vertices such that G has a (connected) dominating set of size k if ...

A dominating set D ⊆ V is a split dominating set of a graph G = (V,E) if the induced subgraph of 〈V −D〉 is disconnected. The split domination number γs(G) is the minimum cardinality of a split dominating set of a graph G. In this article, we establish some results on split domination number of Pm⊕ Pn, Pm ⊕ Cn and Cm ⊕ Cn.

In a graph a vertex is said to dominate itself and all its neighbours. A doubly dominating set of a graph G is a subset of vertices that dominates every vertex of G at least twice. A doubly dominating set is exact if every vertex of G is dominated exactly twice. We prove that the existence of an exact doubly dominating set is an NP-complete problem. We show that if an exact double dominating se...

Vertex Cover (VC): A vertex cover in an undirected graph G = (V,E) is a subset of vertices V ′ ⊆ V such that every edge in G has at least one endpoint in V ′. The vertex cover problem (VC) is: given an undirected graph G and an integer k, does G have a vertex cover of size k? Dominating Set (DS): A dominating set in a graph G = (V,E) is a subset of vertices V ′ such that every vertex in the gra...

A double Roman dominating function on a graph $G$ with vertex set $V(G)$ is defined in cite{bhh} as a function$f:V(G)rightarrow{0,1,2,3}$ having the property that if $f(v)=0$, then the vertex $v$ must have at least twoneighbors assigned 2 under $f$ or one neighbor $w$ with $f(w)=3$, and if $f(v)=1$, then the vertex $v$ must haveat least one neighbor $u$ with $f(u)ge 2$. The weight of a double R...

Given a graph G = (V,E) together with a nonnegative integer requirement on vertices r : V → Z+, the annotated edge dominating set problem is to find a minimum set M ⊆ E such that, each edge in E −M is adjacent to some edge in M , and M contains at least r(v) edges incident on each vertex v ∈ V . The annotated edge dominating set problem is a natural extension of the classical edge dominating se...

A set D of vertices in a graph G is a dominating set if every vertex of G, which is not in D, has a neighbor in D. A set of vertices D in G is convex (respectively, isometric), if all vertices in all shortest paths (respectively, all vertices in one of the shortest paths) between any two vertices in D lie in D. The problem of finding a minimum convex dominating (respectively, isometric dominati...