A mixed dominating set is a collection of vertices and edges that dominates all graph. We study the complexity exact parameterized algorithms for \textsc{Mixed Dominating Set}, resolving some open questions. In particular, we settle problem's by treewidth pathwidth giving an algorithm running in time $O^*(5^{tw})$ (improving current best $O^*(6^{tw})$), as well lower bound showing our cannot be...

A subset X of edges in a graph G is called an edge dominating set of G if every edge not in X is adjacent to some edge in X. The edge domination number γ′(G) of G is the minimum cardinality taken over all edge dominating sets of G. Let m,n and k be positive integers with n − 1 ≤ m ≤ (n 2 ) , G(m,n) be the set of all non-isomorphic connected graphs of order n and size m, and G(m,n; k) = {G ∈ G(m...

A subset X of edges of a graph G is called an edge dominating set of G if every edge not in X is adjacent to some edge in X . The edge domination number γ′(G) of G is the minimum cardinality taken over all edge dominating sets of G. An edge Roman dominating function of a graph G is a function f : E(G) → {0, 1, 2} such that every edge e with f(e) = 0 is adjacent to some edge e′ with f(e′) = 2. T...

In this paper, we study both concepts of geodetic dominatingand edge geodetic dominating sets and derive some tight upper bounds onthe edge geodetic and the edge geodetic domination numbers. We also obtainattainable upper bounds on the maximum number of elements in a partitionof a vertex set of a connected graph into geodetic sets, edge geodetic sets,geodetic domin...

In this paper we describe several new clustering algorithms for nodes in a mobile ad hoc network. We propose to combine two known approaches into a single clustering algorithm which considers connectivity as a primary and lower ID as secondary criterion for selecting clusterheads. The goal is to minimize number of clusters, which leads toward dominating sets of smaller sizes (this is important ...

. Let G be a connected graph. A set S ⊆ V (G) is an outer-connected 2-resolving hop dominating of if and = or the subgraph ⟨V (G)\S⟩ induced by (G)\S connected. The domination number G, denoted γ^c2Rh(G) smallest cardinality G. This study aims to combine concept with sets graphs. main results generated in this include characterization join, corona, edge corona lexicographic product graphs, as w...